reports of the midwest category seminar v
TRANSCRIPT
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich
195
M. Andr6, M. Barr, M. Bunge, A. Frei, J. W. Gray, P. A. Grillet, P. Leroux, F. E. J. Linton, J. MacDonald, P. Palmquist, P. B. Shay, F. Ulmer
Reports of the Midwest Category Seminar V Edited by J. W. Gray, University of Illinois at Urbana-Champaign and Forschungsinstitut for Mathematik, ETH Z0rich and S. Mac Lane, University of Chicago
Springer-Verlag Berlin. Heidelberg New York 19 71
AMS Subject Classifications (1970): 18 A xx, 18 C 15, 18 D 10, 18 E xx , 18 H 05
I S B N 3-540-05442-1 Spr inge r -Ver l ag Ber l in • H e i d e l b e r g • N e w York I S B N 0-387-05442-1 Spr inge r -Ver l ag N e w Y o r k • H e i d e l b e r g - Ber l in
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TABLE OF CONTENTS
Michel Andr@, Hopf and Eilenberg-MacLane Algebras . . . . . . . . I
P.Brian Shay, Discoherently Associative Bifunctors on Groups 29
P.A.Grillet, Directed Colimits and Sheaves in Some Non-Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . 36
Marta C.Bunge, Bifibration Induced Adjoint Pairs ........ 70
P.H.Palmquist, The Double Category of Adjoint Squares ...... 123
Pierre Leroux, Structure et S@mantique Abstraites: Extension des Categories de ~orphismes d'une Paire de Foncteurs Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . 154
A.Frei and J.L.MacDonald, Limit-Colimit Commutation in Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . 199
Michael Barr, Non-Abelian Full Embedding; Announcement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
F.E.J.Linton, The Multilinear Yoneda Lemmas . . . . . . . . . . . 209
Friedrich Ulmer, Locally ~-Presentable and Locally ~-Generated Categories . . . . . . . . . . . . . . . . . . . ...... 230
John W.Gray, The Meeting of the Midwest Category Seminar in Zurich, August 24-30, 1970 . . . . . . . . . . . . . . . . . 248
HOPF AND EILENBERG-MACLANE ALGEBRAS
Michel Andr6
Received June, 1970
The purpose of this note is to give a short proof of Cartan's
structure theorem on Eilenberg-MaeLane algebras, in using a structure
theorem on Hopf algebras with divided powers. The proof appearing
here follows Cartan's ideas with two improvements. On the one hand,
we do not use very mueh the multiplicative structures in the induc-
tive proof; on the other hand we can introduce the Eilenberg-MacLane
simplicial sets into the homological machinery more or less in any
form.
In characteristic 0, according to Milnor-Moore, a connected co-
commutative Hopf algebra is the enveloping algebra of a graded Lie
algebra. Dually a connected commutative Hopf algebra is the enveloping
ooalgebra of a graded Lie coalgebra. In characteristic p, that result
does not hold in general, but it does hold if the Hopf algebra has
divided powers (the comultiplication being a homomorphism of algebras
with divided powers).
Now let us consider a field
K(~,n) and its singular homology
K, an Eilenberg-MacLane space
H(~,n,K) . Actually H(~,n,K) is
a Hopf algebra with divided powers ; consequently H(~,n,K) is the
enveloping eoalgebra of a graded Lie coalgebra. Since the Hopf algebra
is oooommutative, the Lie coalgebra is abelian, in other words it is
a graded vector space. It remains to compute this graded vector space
depending on w, n and K .
The ground field K is fixed. Its characteristic is p ~ 0,2
For the case of characteristic 0 or 2, see the end of this note.
I. HOPF ALGEBRAS WITH DIVIDED POWERS
A Hopf algebra with divided powers is both an algebra with di-
vided powers and a Hopf algebra, the eomdltiplieation being a homo-
morphism of algebras with divided powers. For more details see [3]
for algebras with divided powers, [4] for Hopf algebras and [2] for
Hopf algebras with divided powers. The notion of a graded Lie coalge-
bra is dual to the notion of a graded Lie algebra. To a graded Lie
coalgebra L there corresponds an enveloping coalgebra U(L) which
is actually a Hopf algebra with divided powers.
Theorem i. Let H be a connected Hopf algebra with divided powers.
Then there is one and only one graded Lie coalgebra L (up to an
isomorphism) which appears in an isomorphism H ~ U(L) of Hopf alge-
bras with divided powers.
For the proof see [2] . That result can be rephrased in the
following way.
Theorem 2. Let H be the category of connected Hopf algebras with
divided powers and ~ the category of positively graded Lie coalge-
bras. Then the categories H and ~ are equivalent through U .
Actually we only need the abelian case of that result.
Theorem 3. Let C be a connected eocommutative Hopf algebra with
divided powers. Then there is one and only one graded vector space V
(up to an isomorphism) which appears in an isomorphism C ~ U(V) of
Hopf algebras with divided powers.
Theorem 4. Let C be the category of connected cocommutative Hopf
algebras with divided powers and ~ the category of positively
graded vector spaces. Then the categories ! and ~ are equivalent
through U .
In the abelian case there is an explicit description of the
Hopf algebras with divided powers U(V) On the one hand we define
E (x,2q-l) ~ U(V) P
where the graded vector space V has exactly one generator x ,
appearing in degree 2q-i . We have
Ep(x,2q-l) : K.I + K.x
where x belongs to U2q_I(V) ; the multiplication maps x ® x onto
0 and the comultiplication maps x onto x ® i + i ® x . On the
other hand we define
P (y,2q) ~ U(V) P
where the graded vector space V has exactly one generator y
appearing in degree 2q . We have
Pp(y,2q) = ~ K.y k kZ0
where Yk belongs to U2kq(V) ; the multiplication maps Yi ® Yj
onto (i,j)yi+ ~ J and the eomultiplication maps Yk onto E Yi ® Y j; i+j=k
the k-th divided power of Ym is equal to (m,m-l)(2m,m-l) ....
(m(k-l),m-l)Ymk
Proposition 5. Let V be a positively graded vector space with the
generators x i in degree 2qi-i (i ~ I) and the generators yj in
degree 2qj (j e J). Then there is a natural isomorphism of Hopf alge-
bras with divided powers U(V) = [ ® Ep(Xi,2qi-l) ]® [ ® Pp(yj,2qj)]. iEl j~J
Of the Hopf algebra with divided powers P (y,2q) we shall use P
later essentially the algebra structure. Let us define the following
graded algebra
where z k
onto zi+ j
Lemma 6.
The element
Let
Qp(z,2q) : ~ K.z k
0Nk<p
appears in degree 2kq ; the multiplication maps
if i+j < p and onto 0 otherwise.
There is a natural isomorphism of graded algebras
Pp(y,2q) ~ ® Qp (Zk,2pkq). k~0
z k corresponds to the k+l-st divided power of
Let us study some functors of the category
be the homomorphism mapping
i:ZZ÷ ZZ /pZZ
z i ® zj
y •
A of abelian groups.
i onto I mod p and let
ditions :
i) the functor F is additive
F(~) + F(~') ~ F(~ + ~')
2) the functor F is union preserving
lim F(~.) ~ F(Uw.) ÷ l l
3) the homomorphism F(i) is a monomorphism
4) the homomorphism F(Jn) is an epimorphism for any n_>l
Lemma 7. Let F be a functor from the category of abelian groups to
the category of vector spaces over the field K of characteristic
p > 0 . Let us suppose that the functor satisfies the following con-
rated abelian groups, the following result can be proved.
be the homomorphism mapping i onto i mod pn and i mod p onto
n-i pn p mod . By means of the structure theorem of the finitely gene-
j :Zg + ~ /pZ~ ,Zg /pnzz n
5) the dimension of the vector space F(~/pn~) is finite and
independent of n ~ i.
Then the functor F is completely determined by the two vector spaces
F(ZZ) and F(TZ/pZZ) .
There is a result of the same type for graded vector spaces.
Theorem 4 allows us to write it in the following way.
Proposition 8. Let F be a functor from the category ~ of abelian
groups to the category ~ of connected cocommutative Hopf algebras
with divided powers over the field K of characteristic p > 0 Let
us suppose that the functor satisfies the following conditions :
i) the functor F is sum preserving
F(~) ® F(~') ~ F(~ + ~')
2) the functor F is union preserving
lim F(w.) ~ F(Uw i) + i
3) the homomorphism F(i) is a monomorphism
4) the homomorphism F(Jn) is an epimorphism for any n ~ 1
5) in each degree the dimension of the graded vector space
F(~ /pn~ ) is finite and independent of n k 1 .
Then the functor F is completely determined by the two graded vec-
tor spaces
F(?Z) and F(TZ,/p?Z) .
II. CONSTRUCTIONS
We do not use the word construction in the usual sense. Here a
construction consists of
I a differential graded algebra A .
II a differential graded module T over A .
III a bigraduation of T with
T : Z T. .. n i+j:n 1'3
Some properties are requested.
i) If a is an element of A of positive degree, then a p
is equal to 0
2) The vector space Ho[A ] is 1-dimensional.
3) The vector space H0[T ] is 1-dimensional and the vector
space Hn[T ] is 0-dimensional for n > 0
4) The vector space AiTj, k is contained in Ti+j, k •
5) The differential d of T has the following form
d -- d O + d I + d 2 + .....
d. mapping Tj into i ,k Ti+j-l,k-i
Actually T has a second differential d : d O mapping Ti, j into
• The corresponding homology HIT] is bigraded and has the Ti-l,j
structure of a H[A]-module with
Hi[A]H j,k[T] C Hi+j,k IT]
Then we request the following property•
6) There is a graded vector space N and there is an iso-
morphism
HIT] ~ H[A] ® N
of H [A] -modules mapping Hi,j IT] onto Hi[A] ® Nj
Since dod I + dld o is equal to 0 , the homomorphism H[dl] is
well defined. Since d0d 2 + dld I + d2d 0 is equal to 0 , it is a
differential 2. This differential ~ maps ~i,j IT] into Hi,j_I[T]
and is H [A] -linear. Since H 0[A] is 1-dimensional, the corresponding
differential of H[A] ® N must have the form
!d ® d
where d is a certain differential of N Up to an isomorphism, the
complex N is determined by the isomorphism
N
The differential algebra A is called the initial algebra of the
construction and the new complex N is called the final complex of
the construction. We shall see later that H[N] is given by H[A] if
this graded algebra is a Hopf algebra with divided powers.
A homomorphism of constructions consists of a homomorphism
~:A ÷ A' of differential graded algebras and of a homomorphism
~:T ÷ T' of differential graded modules, with the property
[Ti,j]c z T, k~0 i+k,j-k
The homomorphism of constructions (~,~) gives a homomorphism of
complexes
~:N + N'
The homomorphism ~ is called the initial homomorphism and the homo-
morphism v is called the final homomorphism.
It is clear how to define the tensor product (finite or infinite)
of construction. The resulting initial algebra (final complex) is the
tensor product of the given initial algebras (final complexes).
Let ~ be the category of positively graded vector spaces and
let R be the unique funotor of V into V with the following p -- _
properties:
i) the functor R is direct sum preserving P
2) if V
2q-l, then R (V) P
3) if V
2q, then R (V) P
is 1-dimensional with one generator in odd degree
has one generator in degree 2q .
is 1-dimensional with one generator in even degree
has one generator in odd degrees 2pkq + i and one
generator in even degrees 2pk+lq + 2, k Z 0
Lemma 9. There is a construction with the initial algebra E (x,2q-l) P
@ifferential 0), with the final complex Pp(y,2q) (differential 0)
and with d o and d I equal to 0 .
Give to the tensor product
the differential
Ep(X,2q-l) ® Pp(y,2q)
d with
dy k = xYk_ I dxy k = 0
Lemma i0. There is a construction with the initial algebra Qp(z,2q)
(differential 0), with the final complex
Ep(x,2q+l) ® Pp(y,2pq+2)
(differential 0) and with d O and d I equal to
Give to the tensor product
0 .
Qp(z,2q) ® E (x,2q+l) ® P (y,2pq+2) P P
the differential d with
dyj : Zp_iXyj_l
dziY j = 0
dzixY j = Zi+lY j
dZp_lXY j = 0
if i ~ 0
if i ~ p-i
Proposition ii. For each positively graded vector space V there is
a natural construction C (V) with the initial algebra U(V) (dif- P
ferential 0), with the final complex U(R V) (differential 0) and P
with d o and d I equal to 0 .
Use tensor products of constructions and the lemmas 9, 6 and i0.
Proposition 12. If the initial homomorphism e of a homomorphism of
constructions gives an isomorphism H[~] , then the final homomorphism
v of the homomorphism of constructions gives an isomorphism H[v]
This result is due to J. Moore and proved in ~3 : spectral se-
quence arguments.
Proposition 13. Let (A,T) be a construction and let # be a homo-
morphism of graded algebras of U(V) into H[A] . Then there exists
a homomorphism of constructions
(~,T) : C (V) ÷ (A,T) P
such that H[~] is equal to ~ .
In the proof, using tensor products of constructions, we can re-
place the construction C (V) by the constructions of lemmas 9 and P
i0. Then using the property 1 of the construction (A,T) we construct
a homomorphism ~ of differential graded algebras with H[~] equal
to ~ . Finally using the property 3 of the construction (A,T) we
construct a homomorphism ~ of differential graded modules.
Theorem 14. Let A and N be the initial algebra and the final
complex of a construction. If the graded algebra H[A] is a coeommu-
tative Hopf algebra with divided powers, then, up to an isomorphism,
there is a unique graded vector space V appearing in an isomorphism
U(V) [ H[A] of Hopf algebras with divided powers. Further there is
an isomorphism
10
U(R V) ~ H[N] P
o f g r a d e d v e c t o r s p a c e s .
F o r t h e p r o o f a p p l y t h e o r e m 3, p r o p o s i t i o n 13 and p r o p o s i t i o n
12.
The p r e c e d i n g t h e o r e m i s n a t u r a l i n t h e f o l l o w i n g s e n s e .
Proposition 15. Let ~:A ÷ A' and v:N ÷ N' be the initial homo-
morphism and the final homomorphism of a homomorphism of constructions.
If the homomorphism H[a] is a homomorphism of cocommutative Hopf
algebras with divided powers, then there is a homomorphism ~:V ÷ V'
of graded vector spaces appearing in a commutative diagramm of homo-
morphisms of Hopf algebras with divided powers
U(V) . U(V') U(~)
.HEA']
Further there is a commutative diagram of homomorphisms of graded
vector spaces
U(R V) ~ U(R V') P U(R ~) P
P
H[N] Hb] , H[N']
Theorem 4 gives the homomorphism ~ . Then U(R ~) corresponds P
to H[v] if proposition 13 generalizes in the following way. There
is a commutative diagram of homomorphisms of constructions
C (V) ' C (V')
(A,T) ~ (A',T')
11
For the proof use the exact sequences
0 ÷ W ÷ V ÷ W' ÷ 0 V : W + W'
O + W' ÷ V' + W" + 0 V' : W' + W"
and the following homomorphisms of constructions due to proposition
13.
C (W) ÷ (A,T) ÷ C (W') P P
Cp(W") ÷ (A',T')
Theorem 16. Let a and v be the initial homomorphism and~the fi-
nal homomorphism of a homomorphism of constructions. If the homomor-
phism H[a] is a homomorphism of cocommutative Hopf algebras with
divided powers and if the homomorphism H[~] is a monomorphism (res-
pectively an epimorphism) of graded vector spaces, then the homomor-
phism H[~] is a monomorphism (respectively an epimorphism) of
graded vector spaces.
Apply the preceding proposition and the exactness property of
R P
III. SIMPLICIAL THEORY
If E is a simplicial set, then C(E,K) is the usual complex
giving the singular homology of E with coefficients in K
H[C(E,K)] : H(E,K) .
According to Eilenberg-Zilber, if E and F are simplicial sets,
there is a natural homomorphism of complexes
C(E,K) ® C(F,K) ÷ C(E x F,K)
with nice properties (associativity, commutativity, divisibility,
12
isomorphism) quite enough for proving the two following lemmas.
Lemma 17. If F is a simplicial abelian group, then C(F,K) is a
differential algebra in a natural way ; the p-th power of any of its
elements of positive degree is equal to 0 . If F acts on the simpli-
cial set E , then C(E,K) is a differential module over the differ-
ential algebra C(F,K) in a natural way.
Lemma 18. If F is a simplicial abelian group, then H(F,K) is a
cocommutative Hopf algebra with divided powers in a natural way.
Let us denote by A n the simplicial set characterized by the
following equality for any simplicial set E
Hom(A ,E) = E n n
Further let us consider the following simplicial set E for any sim-
plioial set E . An element of Em is a set {a0, .... ,a m } of maps
of simplieial sets
a m am_ I a 0 A. P A. ~ ..... ~. ~ E
im im- I ~ 0
with any integers io,...,i m . The following equalities define the
face and degeneracy maps
i . . . . ,am} em {aO'''''am} = {aO' "'aiai+l
0_< i < m
m
~m {aO'''''am} = {a0''''am-1}
i { "'''am } = {aO, ai,l d .. am } am ~0' "'' 'ai+l'" ' 0 < i ~ m .
Lemma 19. For a simplicial set E there is a natural isomorphism
H(E,K) ~ H(E,K)
IEI
For the proof we use a bisimplicial set IEI . An element of
m,n is a set {a 0,...,an+ 1 } of maps of simplicial sets
13
an+l ~ eO n ~n-i A- , A. ~ A. , .... A. * E m i n in_ I l 0
with any integers i0, .... ,i n . The following equalities define the
face maps of both types
,i {~0' } e m "'''~n+l
n {~O'''''~n+l}
= {~O,...,~n,~n+l e~}
: {aO' 3 3 ± " ~n+l}
the map emi being the usual i-th injection of Am_ I into Am Then
we consider the corresponding bicomplex C(IEI,K) ; in degree (m,n)
the vector space has one generator for each element of IEIm,n The
isomorphism of the lemma is a consequence of the following isomor-
phisms
M [C(LEI,K)] : o
me 0
n~ 0
The generalization of this lemma and of its proof is the spec-
tral sequence of a fibre map. Let us consider the case where the
fibre map is a principal fibration. A principal fibrationconsists of
a simplicial group F acting on a simplicial set E and of a sur-
jeetion of simplicial sets
¢ : E ÷ B
with the following properties
I) fe = f'e if and only if
2) ¢(e) = ¢(e') if and only if
f : ft
e' = fe .
14
Let us notice the two following properties of principal fibrations. A
simplicial map C ÷ B and a principal fibration E ÷ B of group F
give a principal fibration E ×B C ÷ C of group F . A principal fi-
bration E ÷ A n of group F is isomorphic to the trivial principal
fibration F × A + A n n
To a principal fibration #:E ÷ B of group F there corres-
ponds a bisimplicial set I¢I • An element of I#Im,n is a set
{eO' .... '~n+l ;8} of maps of simplicial sets
A m ] ~n+l
o~ n
A. ~ A. i n
, E
an-i ~0 .... A. ~ B
in_ I 10
with any integers i0, .... ,i n and with the equality
¢6 = ~0 ...... an-lanan+l "
The following equalities define the face maps of both types
,i {~ , "~} = {~O i i m 0 .... 'an+l' ' .... '~n'an+l em;Sem}
c"J {~ " : {~0 ~jej+l' '~n+l '6} n 0' .... '~n+l '~} ' .... ' ..... "
Then we consider the corresponding bicomplex C(I¢I,K) . We denote
by H' , H" , H the homology for the first, second, total differen-
tials.
Lemma 20. Let ¢:E ÷ B be a principal fibration of group F
w0(B) = 0 = Wl(B) Then
H'[C(I¢I,K)] --" H(F,K) ® C(B,K)
with
15
One of the spectral sequences of the bicomplex is degenerated
H " ~ ( I + I K)] : o n
H 8~(1¢1,K)3 ~ c(m,K)
and gives the first isomorphism
n~ 0
H [ C ( I ¢ I , K ) ] -'- H(E,K)
For computing H'[C( el,K)] we use the following isomorphisms
Cm,n(l~ ,K) ~ Z Cm(E XB Ai ,K) A. ÷...A. ÷B n z n z 0
Hm,n[C( e l , K ) ] ~ Z Hm(E x B A i ,K) A. +...A. +B n 1 10 n
H(E x B A i ,K) ~ H(F ×&i ,K) Z H(F,K) n n
even in a natural way since ~0(B) = 0 = ~I(B) . Thus we get the iso-
morphism
H'Ec ( I¢ I ,~< ) ] ~' H(r,~<) ® C(m,~<)
If we go a step further we get the spectral sequence of a principal
fibration
H (F,K) ® H (B,K) Z H (F,K) ® H (B,K) ~ H (E,K) m n m n
We do not use it here.
Proposition 21. Let ¢:E ÷ B be a principal fibration of group F
with the following properties
i) the group F is abelian
2) the vector space H(E,K) is 1-dimensional
3) the homotopy groups ~0(B) and ~I(B) are equal to 0 .
16
Then there exists a natural construction such that H(F,K) is
the homology of the initial algebra and H(B,K) the homology of the
final complex.
We use the complex A : C(F,K) and the bicomplex T : C(I¢I,K).
By lemma 17, the complex A is a differential algebra. We have the
isomorphism
Cm,n(l¢l,K) ~ Z Cm(E ×B Ai ,K) A. ÷...A. ÷B n i n i 0
But F acts on E ×B Ai , consequently C(E ×B Ai ,K) is a n n
C(F,K)-module according to lemma 17. Thus C(I¢I,K) is a C(F,K)-mo-
dule and we have
d'(at) = da.t + a.d't
d"(at) : a.d"t
consequently
d(at) = da.t + a.dt
Now T is a differential module over the differential algebra A .
We have to verify the six properties of the definition of a
construction. Property 1 : see first part of lemma 17. Property 2 :
use ~0(E) = 0 and ~I(B) = 0 Property 3 : see first part of lemma
20. Property 4 : clear out of the definitions. Property 5 : use
d = d' ~ d" . Property 6 : see second part of lemma 20. Thus we have
a construction with the final complex
N = c(~,K)
Lemma 19 gives the isomorphism
H[N] ~ H(B,~<) .
17
Theorem 22. Let %:E ÷ B be a principal fibration of group F with
the following properties
i) the group F is abelian
2) the vector space H(E,K) is 1-dimensional
3) the homotopy groups T0(B) and ~I(B) are equal to 0
Then H(F,K) is a cocommutative Hopf algebra with divided powers and,
up to an isomorphism, there is a unique graded vector space V
appearing in an isomorphism U(V) ~ H(F,K) of Hopf algebras with
divided powers. Further there is an isomorphism
U(RpV) ~ H(B,K)
of graded vector spaces.
Apply lemma 18, proposition 21, theorem 14.
If ~:E ÷ B is a principal fibration of group F and ~':E' ÷ B'
a principal fibration of group F', then a homomorphism of principal
fibra~ions of ~ into ~' consists of a homomorphism e:F ÷ F', of a
map B:E ~ E' and a map y:B ÷ B' with the following equalities
B(fe) : ~(f)B(e) ~' 0 ~ : ¥ o ~
Theorem 23. Let %:E ÷ B and }':E' ÷ B' be principal fibrations
of groups F and F' with the following properties
I) the groups F and F' are abelian
2) the vector spaces H(E,K) and H(E',K) are 1-dimensional
3) the homotopy groups n0(B) , ~I(B), ~0(B'), ~I(B') are equal
to 0 .
Let (~,B,y) be a homomorphism of the principal fibration % into
the principal fibration }' . If
H(~,K) : H(F,K) ÷ H(F',K)
18
is a monomorphism (respectively an epimorphism) of vector spaces,
then
H(y,K) : H(B,K) ÷ H(B',K)
is a monomorphism (respectively an epimorphism) of vector spaces.
Apply lemma 18 and theorem 16.
IV. EILENBERG-MACLANE ALGEBRAS
We denote by H(~,n,K) the homology of the Eilenberg-MacLane
space K(~,n) . Actually there is a simplicial abelian group K(~,n) .
Consequently
H(w,n,K) = H(K(~,n),K)
is a connected cocommutative Hopf algebra with divided powers. We
shall determine it. We shall use the following functor L from the P
category of abelian groups to the category of graded vector spaces
over K . In all degrees L (7) is equal to 0 except in degree i P
where we get ~ ® ~ K and in degree 2 where we get p~®~ ; the
abelian group ~ consists of the elements x of ~ with px equal to 0 P
If B is any simplicial abelian group with n0(B) = 0 , there
is a natural short exact sequence of simplicial abelian groups
0 ÷ F ÷ E + B ÷ 0
with HIE] = ~(E) = 0 . If B is a K(~,n) , then F is a K(~,n-l).
Consequently for n > i there exists a natural principal fibration
with the following properties
i) the group F is abelian
2) the vector space H(E,K) is 1-dimensional
3) the homotopy groups n0(B) and ~I(B) are equal to 0
19
4) the Hopf algebra with divided powers H(F,K) is isomorphic
to H(~,n-I,K)
5) the graded vector space H(B,K) is isomorphic to H(~,n,K)
We can apply theorem 22 and theorem 23 and prove by induction on n
the following results.
Proposition 24. If the graded vector space H(~,I,K) is finite di-
mensional in each degree, then the graded vector space H(~,n,K) is
finite dimensional in each degree. Further there is a unique graded
vector space V , finite dimensional in each degree, appearing in two
isomorphisms
H(w,I,K) ~ U(V)
of graded vector spaces.
H(~,n,K) ~ U(R n-I V) P
Proposition 25. If the homomorphism H(e,I,K) is a monomorphism
(respectively an epimorphism), then the homomorphism H(e,n,K)
a monomorphism (respectively an epimorphism).
Now we can prove Cartan's theorem.
is
Theorem 26. For n ~ i, the two following functors from the category
of abelian groups to the category of Hopf algebras with divided
powers
H(.,n,K) U o R n-I o L P P
are isomorphic.
Let us use proposition 8. The functor
the five conditions. The functor H( ,n,K)
U o R n-I 0 L fulfills P P
fulfills the first two
conditions. Further the functor H( ,n,K) fulfills the last three
conditions if the functor H( ,I,K) does, according to the preceding
propositions. The graded vector spaces H(~ ,n,K) and H(~ /p~ ,n,K)
20
are isomorphic to the graded vector spaces U(Rn-IL 2) and P P
U(Rn-IL ~/p~) for any n ~ i , if it holds for n = i , according P P
to proposition 24. Thus Cartan's theorem is proved by proposition 8
if the following holds :
i) dim Hi(~ ,I,K) = i (i = 0,i) , = 0 (i > i)
2) dim Hi(~ /pn~ ,I,K) = i
3) H(~ ,I,K) ÷ H(~ /p~ ,I,K) is a monomorphism
4) H(~ + ~/pZ ,I,K) ÷ H(~ /pn~ ,I,K) is an epimorphism.
The third condition is obvious since the homomorphism HI(i,I,K) is
nothing but the homomorphism i ® ~ K (use ~i ). The fourth condition
is satisfied if the homomorphism H2(~ /p~ ,I,K) + H2(~ /pn~ ,I,K)
is an epimorphism. Thus it remains to prove
i) dim Hi(Z,I,K) = i (i = 0,i)
2) dim Hi(z/pn~,K) = i
3) the canonical homomorphism
: 0 (i > i)
H2(~ /p~,l,K) + H2(% /pnz,l,K)
is not equal to 0.
That is easily proved by the homology theory of groups.
V. CHARACTERISTIC 0
In characteristic 0 we can use the same proof with the follow-
ing modifications. At first we can forget lemma 6, proposition 15,
theorem 16, theorem 23, proposition 25 and we can forget divided
powers in theorems 1,2,3,4, in proposition 5, in theorem 14, in lemma
18, in theorem 22, in theorem 26 and pthpowers in lemma 17. Further
we have to modify some definitions. We define
P0(Y,2q) ~ U(V)
21
where the graded vector space V has exactly one generator y
appearing in degree 2q . We have
k P0(Y,2q) = Z K.y
k~0
k where y
onto yi+j
i i belongs to U2kq(V) ; the multiplication maps y ® y
and the comultiplication maps yk onto Z (i,j)yi ® yi. i+j=k
In thedefinition of a construction we forget the first condition. In
the definition of the functor R 0 we still have the first two con-
ditions and we modify the third one in the following way: if V is
1-dimensional with one generator in even degree 2q , then R0(V)
has one generator in degree 2q + i . In all degrees L0(~) is equal
to 0 except in degree i where we get ~ ®~ K Finally we have to
modify some propositions. Lemma 7 must be read in the following way:
let F be a functor from the category of abelian groups to the
category of vector spaces over the field K of characteristic 0 .
Let us suppose that the functor is additive and union preserving.
Then the functor F is completely determined by the vector space
F(~ ) Proposition 8 must be read in the following way: let F be
a functor from the category of abelian groups to the category of
connected cocommutative Hopf algebras over the field K of charac-
teristic 0 . Let us suppose that the canonical homomorphisms
F(~) ® F(~') ÷ F(~+~') lim F(~.) ÷ F(U~.) ÷ i i
are isomorphisms and that F(~ ) is finite dimensional in each de-
gree. Then the functor F is completely determined by the graded
vector space F(~ ) . Lemma i0 must be read in the following way:
there is a construction with the initial algebra P0(Y,2q) (differen-
tial 0), with the final complex E0(x,2q+l) (differential 0) and
with d O and d I equal to 0 . For the proof give to the tensor
product
the differential
22
Pg(y,2q) ~ Eg(x,2q+l)
d with
dy n = 0 n n+l and dxy = y
with the definitions of L 0 and of
theorem in eharaeteristie 0 .
R 0 given above, we get Cartan's
Theorem. For n ~ i , the two following functors from the category
of abe!Jan groups to the category of Hopf algebras
are isomorphic.
H(.,n,K) n-i o LO U o RO
We use the same proof as for theorem 26. It suffices to know
that H(Z,I,K) is 1-dimensional in degree 0 and I and 0-dimensional
in higher degrees.
VI. CHARACTERISTIC 2
Now the ground field K has characteristic 2. We have to modi-
fy the definitions in the following way. A Hopf algebra with divided
powers is both an algebra with divided powers and a Hopf algebra, the
comultiplication being a homomorphism of algebras with divided powers.
Now the divided powers are defined for any homogeneous element of
positive degree, not only for any homogeneous element of positive and
even degree. The definition of the functor U can be modified in a
consequent way. The definition of U(L) in characteristic 2 corres-
ponds to the definition of U(L) in characteristic p ~ 2 with the
further condition that L is equal to 0 in all odd degrees. See
the modified proposition 5 below.
23
We define E2(x,n) and P2(Y,n) for any n > 0
Q2(z,n) . The definition of the graded algebra E2(x,n)
following
and we forget
is the
E2(x,n) = K.I + K.x
with x appearing in degree n . The definition of the Hopf algebra
with divided powers P2(Y,n) is the following
P2(Y,n) : ~ K.y k k~0
with Yk appearing in degree kn , the multiplication maps Yi ® Yj
onto (i,j)yi+ j and the comultiplication maps Yk onto
Z Yi ® yj ; the k-divided power of Ym is equal to i+j=k
(m,m-l)(2m,m-l) .... (m(k-l),m-l)Ymk •
The functor R 2 is defined by the following properties
i) the functor R 2 is direct sum preserving
2) if V is 1-dimensional with one generator in degree n
then R2(V) has one generator in degrees 2kn + i, k ~ 0
Finally the functor R o L has to be replaced by a functor M 2 P P
from the category of abelian groups to the category of graded vector
spaces over K . In all degrees M2(~) is equal to 0 except in de-
gree 2 where we get ~ ® zK and in degree 2 k + i (k = 1,2 .... )
where we get 2 ~ ® ~ K .
With those definitions the following results remain unmodified:
theorems i, 2, 3, 4 (structure of Hopf algebras with divided powers),
lemma 7, proposition 8 (structure of functors of abelian groups)
propositions ii, 12, 13, 15, theorems 14, 16 (initial algebras, final
complexes of constructions) lemmas 17, 19, 20, proposition 21 (sim-
plicial theory).
24
We must rewrite proposition 5, lemmas 6, 9, i0. That is of no
importance for the final result.
Proposition 5. Let V be a positively graded vector space with the
generators x. in degree n. (i E I). Then there is a natural iso- 1 1
morphism of Hopf algebras with divided powers
U(V) ~ ~ P2(xi,ni) . iel
Lemma 6. There is a natural isomorphism of graded algebras
P2(Y,n) { ® E2(Zk,2kn) kz0
the element z k corresponds to the k+l-st divided power of y .
Lemma 9/10. There is a construction with the initial algebra E2(x,n)
(differential 0), with the final complex P2(Y,n+l) (differential 0)
and with d o and d I equal to 0
It is important to notice that lemma 18 does not hold in general.
For example if ~ is the free group with one generator, the homology
H(~,K) is too trivial for being a Hopf algebra with divided powers.
We must modify lemma 18 and consequently theorems 22, 23 in the
following way.
Lemma 18. If F is a simplicial abelian group and if HI(F,K) is
equal to 0 , then H(F,K) is a cocommutative Hopf algebra with
divided powers in a natural way.
Theorem 22. Let #:E ÷ B be a principal fibration of group F with
the following properties:
i) the group F is abelian
2) the vector space H(E,K)
3) the homotopy groups
is 1-dimensional
T0(B), ~I(B) and ~2 (B) are equal to 0.
25
Then H(F,K) is a cocommutative Hopf algebra with divided powers and,
up to an isomorphism, there is a unique graded vector space V
appearing in an isomorphism U(V) ~ H(F,K) of Hopf algebras with
divided powers. Further there is an isomorphism
of graded vector spaces.
U(R2V) { H(B,K)
Theorem 23. Let ~:E ÷ B and ~':E' ÷ B' be principal fibrations
of groups F and F' with the following properties
I) the groups F and F' are abelian
2) the vector spaces H(E,K) and H(E',K) are 1-dimensional
3) the homotopy groups n0(B) , ~I(B), ~2(B), ~0(B'), ~I(B'),
(B') are equal to 0 ~2
Let (~,6,y) be a homomorphism of the principal fibration
the principal fibration ~' If
into
H(e,K) : H(F,K) ÷ H(F',K)
is a monomorphism (respectively an epimorphism) of vector spaces, then
H(y,K) : H(B,K) ÷ H(B,,K)
is a monomorphism (respectively an epimorphism) of vector spaces.
Let us consider now Eilenberg-MacLane algebras. We have to
modify propositions 24,25.
Proposition 24. If the graded vector space H(~,2,K) is finite
dimensional in each degree, then the graded vector space H(~,n,K)
with n ~ 2 is finite dimensional in each degree. Further there is
a unique graded vector space V , finite dimensional in each degree,
appearing in two isomorphisms
26
H(~,2,K) ~ U(V)
of graded vector spaces.
H(~,n,K) ~ U(R~-2V)
Proposition 25. If the homomorphism H(~,2,K) is a monomorphism
(respectively an epimorphism), then the homomorphism H(~,n,K) with
n k 2 is a monomorphism (respectively an epimorphism).
Now we have Cartan's theorem in characteristic 2.
Theorem 26. For n a 2, the two following functors from the category
of abelian groups to the category of Hopf algebras with divided
powers
H(.,n,K) n-2 o M 2 U o R 2
are isomorphic.
Proof. As in the case of characteristic p ~ 2 , we use an inductive
proof. Everything remains the same except the beginning of the in-
duction: here we start with n = 2 We have to prove the following
assertions.
i) the vector spaces H(~ ,2,K) and U(M2(~ )) are isomorphic
in all degrees.
2) the vector spaces H(~ /2n~ ,2,K) and U(M2(~ /2nz)) are
isomorphic in all degrees.
3) the homomorphism H(~ ,2,K) ÷ H(~ /2 ~ ,2,K) is a monomor-
phism.
4) the homomorphism H(Z + Z/22~,2,K) ÷ H(~. /2nz,2,K) is an
epimorphism.
Actually we can use
ing way.
H(~,I,K) for the proofs and that in the follow-
27
Let A be the functor exterior algebra for the category of
graded vector spaces. This functor A generalizes E 2 as the functor
U generalizes P2 . Then we can use lemma 9/10 completely and genera-
lize theorems 22, 23 in the following way.
Theorem 22' Let #:E ÷ B be an principal fibration of group F
with the following properties
i) the group F is abelian
2) the vector space H(E,K) is 1-dimensional.
3) the homotopy groups ~o(B) and ~I(B) are equal to 0
4) the graded algebra H(F,K) is an exterior algebra A(V) .
Then the graded vector space W with Wi+ I equal to V i appears in
an isomorphism
U(W) ~ H(B,K)
of graded vector spaces.
Theorem 23' Let %:E + B and #':E' ÷ B' be principal fibrations
of groups F and F' with the following properties
I) the groups F and F' are abelian
2) the vector spaces H(E,K) and H(E',K) are 1-dimensional
3) the homotopy groups n0(B) , ~I(B), ~0(B'), ~I(B') are equal
to 0
4) the graded algebras H(F,K) and H(F',K) are exterior
algebras A(V) and A(V') .
Let (~,6,7) be a homomorphism of the principal fibration ~ into
the principal fibration ~' such that H(~,K) is equal to A(~)
for a certain homorphism ~:V ÷ V' . If H(~,K) is a monomorphism
(respectively an epimorphism), then H(~,K) is a monomorphism (res-
pectively an epimorphism).
28
Consequently propositions 24, 26 hold in a weaker form for the
step from H(~,I,K) to H(~,2,K) It remains to prove the following
assertions.
i) the algebra H(~ ,I,K9 is an exterior algebra A(V) with
V i equal to 0 in all degrees except in degree i where we get K .
2) the algebra H(~ /2n~ ,I,K) is an exterior algebra A(W n)
with W~ equal to 0 in all degrees except in degree z
2 k (k = 0,1,2, .... ) where we get K .
3) the homomorphism
H(~ ,I,K) = A(V) ÷ H(~ /2~ ,I,K) = A(W I)
is a monomorphism due to a homomorphism V ÷ W I
4) the homomorphism
H(~ + ~/2~,I,K) = A(V+W I) ÷ H(~ /2n~,l,K) = A(W n)
is an epimorphism due to a homomorphism V + W I ÷ W n .
That is easily proved by the homology theory of groups.
[1-1
[2]
[33
[4]
Andr6 Mo Limites et fibr6s. Comptes Rendus Acad6mie Sciences.
Paris 260, 756-759 (1965)
Andr6 M. On the structure of Hopf algebras with divided powers.
To appear Journal of Algebra (1971)
Cartan H. S6minaire Ecole Normale Sup6rieure 1954-1955.
Benjamin (1967)
Milnor J. , Moore J. On the structure of Hopf algebras.
Annals of Mathematics 81, 211-265 (1965).
Tu!ane University. March 1970
DISCOHERENTLY ASSOCIATIVE BIFUNCTORS ON GROUPS
P. Brian Shay
Received July 27, 1970
i. Introduction. Let G be a category with one object whose endo-
morphisms, also denoted G, form a group isomorphic to
(a,b: bka-(k+l)bak+lb-ka-lb-la = l, for all k ~ 0). We construct
below a discoherently associative bifunctor on G. This example
is universal for groups in the following sense: Suppose M is a
group with a discoherently associative bifunctor. Then M has a
subgroup isomorphic to a non-abelian quotient of G. In particu-
lar, all associative bifunctors on finite groups are coherently
associative.
It has not been known previously whether or not any category
could be furnished with a discoherently associative bifunctor°
We are very much indebted to A. Heller, who suggested the problem
to us, for his continuous helpful advice, and to G. M. Bergman,
who has provided a proof that G is not abelian.
2. Terminology. A bifunctor, Q, on a category, C, is a func-
tor, ~ : C~ C ~ C, where C ~ C is, of course, the category
whose objects and morphisms are pairs of objects and morphisms of
C with composition defined in the obvious way. We consider only
covariant functors, and use the notation O : (A,B) ~ A ~ B , O:
((f,g): (A,B) --~-- (C,D)) --~-- (fOg: A®B--~--C~D).
30
: C~ C ~ C is associative if there is a natural isomorphism,
, b e t w e e n t h e f u n c t o r s ~ o ( ~ ~ I d ) and G o ( I d ~ ~ ) :
C ~ C ~ C ~ C. Such an i s o m o r p h i s m , n o t n e c e s s a r i l y u n i q u e ~ i s
said to be an association for ~. An association,~ , for ~,
is said to be coherent if the following pentagonal diagram com-
mutes for each quadruple of objects A, B, C, D of C.
( ( ( A ~ B ) Q C ) ~ D ) ~ A ~ B ' O ' D ~ ( ( A ~ B ) ~ ( C ~ D))
~A,B,c~D
( ( A ~ ( B ~ C ) ) Q D )
~ A , B Q C , D ~
(A® ((B® C) QD))
An a s s o c i a t i v e b i f u n c t o r i s c o h e r e n t l y a s s o c i a t i v e i f a n y o f i t s
a s s o c i a t i o n s i s c o h e r e n t . I t s h o u l d be n o t e d t h a t an a s s o c i a t i v e
b i f u n c t o r may h a v e a s s o c i a t i o n s w h i c h a r e c o h e r e n t and t h o s e w h i c h
a r e n o t . An e x a m p l e i s p r o v i d e d by t h e t e n s o r p r o d u c t i n t h e
c a t e g o r y o f a b e l i a n g r o u p s . The a s s i g n m e n t s
( a ~ b ) ~ c - - ~ a ~ ( b ~ c ) and ( a ~ b ) ~ c - - ~ - a ~ ( b ~ o ) a r e
b o t h a s s o c i a t i o n s , one c o h e r e n t , t h e o t h e r n o t . I f none o f t h e
a s s o c i a t i o n s f o r a b i f u n c t o r a r e c o h e r e n t , t h e b i f u n c t o r i s s a i d
t o be d i s c o h e r e n t l y a s s o c i a t i v e . One m i g h t r e f e r t o t h e b a s i c
p a p e r o f Mac L a n e , [ 1 ] , f o r a more s a t i s f a c t o r y , b u t more d i f f i -
c u l t t o s t a t e , d e f i n i t i o n o f c o h e r e n c e w h i c h i s e q u i v a l e n t t o t h e
one we use, as he proves.
~A,B,C (~ D
(A~ (B~ (C ~D)))
f A~ ~B,C,D
31
3. Associative Bifunctors on Groups. If we regard a group as a
category with one object whose endomorphisms are all invertible,
the concepts above may be applied to groups and are, of course,
greatly simplified. A bifunctor, ~, on a group, H, is a group
homomorphism from H X H to H. ~ is associative with association
a ~ H if (f~g) ~h = a-l(f ~ (g~h))a, for all f,g,h ~ H. Then
is coherently associative if there exists an association a ~ H
such that a 2 = (e ~a)a(a~e), where e ~ H is the identity elemenh
4. The Example. In particular, let us define a homomorphism
: G~G ~ G = (a,b: b-k(a-lba)b k = a-k(a-lba)ak, for all
k ~ O) as follows: age = b; b~e = a-lba; e(~a = e; e~b = e.
One must check to see that relations are preserved. This is easy
in the form they are written here. Also 'a' is an association
for (~. It suffices to check that the two homomorphisms,
x ~ (x~ e) ~e and x ~ a-l(x~e)a, agree on generators.
We now show Shat if c is a coherent association for ~ , then
G is abelian. The conditions on c are that (i) ((x~e) ~e)
= c-l(x(~ e)c = a-l(x~ e)a, and (ii) c 2 = (e~c).c-(c~ e), i.e.,
c = c ~e. From (i), with x = a, we get a-lba = c-lbc. From (i)
and (ii), with x = c, we get c = a-lca. Therefore, a-lcb = ca-lb
= bca -1 = ba-lc. Hence, a-lcb ~e = ba-lc ~e. Computing,
b-lc(a-lba) = (a-lba)b-lc. But a-lba = c-lbc, so the left hand
side is c, and e = a-lbab -1 by cancellation. We are forced, then,
32
to demonstrate that G is not abelian, and will do so in a later
paragraph.
It might be of interest that there are a countable number of
associations for (G, ~) ).
tion.
5. The Universal Property.
For all k ~ O, a(ba-l) k is an associa-
Let H be any group, ~ an associa-
tive bifunctor on H, ~ an association for ~ . Let ~ Q e : ~ ,
e ~ = ~ , eQ~ = ~ . If ~ commutes with ~ and ~, ~-2~%~
is a coherent association for Q . This is proven by the follow-
ing sequence of lemmas, whose proofs may be reconstructed with no
great difficulty. Frequent use is made of the fact that, since
is a group homomorphism, H ~)e commutes with e ~H.
Lemma i. For all ~ e H, ~ commutes with (e ~ )~ e.
Lemma 2. e ~D ~ - ~t = ~ ~ e.
Lemma 5. ~-i(~ ~ e)~ = ~ -i(~ ~ e ) ~ E H.
Lemma 4. ~ (e ~ ~)~-i = ~ (e ~ ~ )~-i~ ~ H.
l~roposition i. l~-2~ is an association -- i.e., for all
~ , ~ ,~ ~ H
(~-2~)((7 ~ ) ~ ~)(~-2~)-i = Y ~ (~ ~ ~ )
Proposition 2. ~ - 2 ~ is coherent -- i.e.,
(e ~ (flI~,-2r"~'~IZ))('VL.-21I~'I~)(~I~,-2C~'~) ~ e) : ('l~,-2~p'l~) 2.
z f ~l commutes w i t h ~1 ~ e and e Q ~ f o r any a s s o c i a t i o n ~ . , ®
is coherent. If ~ is discoherently associative, it must have an
33
association, ~ , which cannot, we have decided, commute both with
= ~ ~ e and ~ = e ~ . Gp(~ ,~)C H is isomorphic to a
non-abelian quotient of G under the mapping a ~ , b ~ if
and ~ do not commute. GP(~,~)C H is isomorphic to a non-
abelian quotient of G under the mapping a ~-i b ~-i if ~
and ~ do not commute. It need only be shown that relations are
preserved. It is easy to show, e.g., that the kth relation and
the k = i relation in ~ and ~ imply the (k + l)st relation. It
suffices to establish the relation ~-2~2 = ~-i~-i~.
But ~-2~2 = ~-i<~-i~ >~ = ~-l<~-l<q @ e>~>~
= ' ~ - l ( ( " r ' [ (~) e) @ e)"r~ = ~ - l (q:~ (~ e)"~. = (q:~ ( ~ e) (~) e
= ~[-l~ ~ e = ~ -l(~-lq~)~.
6. An Interesting Corollar2. That a discoherently associative
bifunctor can be assigned to a group has the consequence that dis-
coherently associative bifunctors often are at least as common as
coherently associative bifunctors in the sense of the following
theorem: Let C be a category with an associative bifunctor, ~.
In addition, let there be an object of C, A, such that either
Homc(A~A,A) or HOmc(A,A ~A) is non-empty. Then C may be imbed-
ded in a category C' with a discoherently associative bifunctor
' such that ~ 'IC~ C = ~ "
Proof: C' may be taken to be C~ G, where G is the group of the
example above. ~ is extended in the obvious way. Relatively
straightforward study of the appropriate diagrams will give the
34
theorem. [Connected categories, categories with initial or ter-
minal objects of course satisfy the condition required of the
object A.
7. G is not abelian. We give a proof which was communicated by
G. M. Bergman. Let H = (bk,k ~ Z U (-~): bib i = bibj+l, i~ j).
b~b ~ b~b~ ±i, One easily deduces (i) j i = i j+~ ' ~ '~ =
i< min(j,j+~) and more easily (2) b~b-~i i = i, ~ = Zl. Claim:
A normal form for H is given by the words not containing 2-1etter
subwords appearing on the left-hand sides of (i) and (2). The
reduction process defined by (i) and (2) clearly terminates. It
suffices, then, to show that in words formed by the overlap of two
words on the left-hand sides of (i) and/or (2) the same reduced
word is obtained, no matter which of the two indicated reductions
is carried out first, e.g., for bib i b i this is obvious; for
b~b~.b~ , with j~ min(k,k+~) and i ~ min(j,j+ ~), the indicated
• bkb i j+~ ' reductions give bjbk+~b and both of which further
to b~ b~+~ b~+~+ { . The other two cases are as straight- reduce
forward. H is clearly non-abelian: e.g., the reduced form of
bob I is bobl, but the reduced form of blb 0 is b0b 2.
But G and H are isomorphic under the maps induced by:
a-~-~-b_~; a-ibai~-~mbi , i ~ Z. Verification: Since
35
----(a-lba)(anb -n) = ----(anb-n)(a-lba), n ~ O, we must have
blb~,b~n = b~b~nb 1. But the right-hand side is the reduced
form of the left-hand side. In the other direction, since
bjb i = bibj+l, ±~ j, we must have (a-JbaJ)a = a(a-J-lba j+l) for
all j G Z (clear) and (a-JbaJ)(a-iba i) = (a-ibai)(a-J-lba j+l) for
i, j E Z, j = m + i, m ~ O. The latter quickly reduces to
a-mbamb = ba-m-lbam+l, m ~ 0° But b-l(a-mbam)b =
b-l(b-m+la-lbabm-1)b = b-ma-lbab m = a-m-lbam+l.
It is clear from the relations of G that no finite quotient
can be non-abelian. It can also be shown that every metabelian
quotient of G is abelian.
1.
REFERENCES
S. Mac Lane, Natural associativity and commutativity, Rice University Studies, 49, (1963), 28-46.
City University of New York
DIRECTED COLINITS AND SHEAVES IN SOME NON-ABELIAN CATEGORIES
Pierre Antoine Grillet
Received Dec.14, 1970
We extend, first Grothendieck's classical result characteri-
zing C 3 abelian categories, then some of Heller and Rowe's and Gray's
results about sheaves, to non-abelian cases of some generality.
The categories we consider are finitely complete categories
in which every morphism f has a regular decomposition, i.e. f = mp
for some monomorphism m and regular epimorphism p ; it is furthermore
assumed that if fg' = f'g is a pullback and if f is a regular epi-
morphism then so is f' For the sake of having a terminology we call
such categories regular. It is known that any category which is tri-
p,ease over the category of sets (for instance, every finitary or infi-
nitary variety of universal algebras) is a regular category; so is any
abelian category.
The result of Grothendieck we mentioned is extended to a
characterization of C 3 regular categories, i.e. cocomplete regular ca-
tegories in which directed colimits preserve finite limits (hence also
monomorphisms). In such a category, directed colimits behave very nice-
ly; for instance Gray's condition 32 holds. Assume furthermore that
the category is complete and 31 also holds (then we call it a C 4 re-
gular category), and that it is C~ in the sense that products respect
regular epimorphisms; then any product of directed colimits can be des-
cribed as a directed colimit of products.
We use this to obtain additional information about the cate-
gory ~(X,C) of sheaves over X in S If ~ is a C 4 regular category
then ~(X,C) is coreflexive in the category of presheaves; no further
assumption is required as to C having a generator or even being well-
powered; in fact, Heller and Howe's recursive construction of the asso-
ciated sheaf terminates in at most two steps (which answers a question
37
of Gray's). If furthermore ~ is C ~I (as defined above), then ~(X,C)
is a C 3 regular category; and the colimits and finite limits in ~(X,C)
can be safely computed on the stalks (this means that ~(X,C) is co-
tripleable under the category of presheaves and extends the similar re-
sult of Van Osdol, concerning the case when C ~s finitarily triplea-
ble over the category of sets, e.g. is a finitary variety).
The proofs of all these results use the properties of rela-
tions in regular categories.
We are much indebted to Professors Mac Lane, Michael Barr
and D.H. Van Osdol for a number of remarks concerning the manuscript.
We are also much indebted to Professor Van Osdol for references, and
also for suggesting that lemma 3.1 below might be true and yield an
answer to Gray's question.
i. Regular categories.
i. This section is preliminary in nature. First we compare
regular categories with some important previous types of categories
with decompositions. Then we list quite a few elementary results which
which will be used extensively in the next section and to a lesser ex-
tent in the last one.
2. We called a category ~ regular when it satisfies:
(I) C is finitely complete;
(II) every morphism f of C has a regular decomposition f = mp
for some monomorphism m and regular epimorphism p ) ;
(III) if fg' = f'g is a pullback and f is a regular epimorphism,
then f' is also a regular epimorphism.
Some laxity can be used in what is meant by a regular epi-
morphism. Consider the four following definitions: p is a regular ~p~-
morphism in case
38
a) p is the coequalizer of its kernel pair (= pair (x,y) such
that px = py is a pullback)(definition used in [2]);
b) p is a strict epimorphism (definition used in [I0]);
c) p is a coequalizer (cf.[l)]);
d) if f and p have same domain and pu = pv implies fu = fv ,
then f = tp for some unique t (cf.[8]).
Any of these can be used in axioms (II) and (III), and all four con-
cepts of regular categories obtained thus coincide. Furthermore, in a
regular category all four classes of epimorphisms coincide, and coinci-
de with the class of e~tremal epimorphisms [9] although the latter can-
not be used in the definition. When regular epimorphi~ms are defined
by a), axiom (II) can be replaced by the existence of a coequalizer
for each kernel pair (this is shown in [2]).
In a regular category, the classes of all monomorphisms and
of all regular epimorphisms form a bicategory structure in the sense
of [12] (as shown in [i], see also [ii]), and will yield a bicategory
structure in the sense of [14] if we are also provided with a selection
of one monomorphism from each class of equivalent monomorphisms, and
similarly for regular epimorphisms. In particular, composing two regu-
lar epimorphisms yields a regular epimorphism, and any two regular de-
compositions of a morphism are equivalent.
3- Here are some examples of regular categories: i) all abe-
lian categories (then any epimorphism is regular); 2) all (finitary or
infinitary) varieties (= equationally definable classes) of universal
algebras. More examples can be obtained from transfer theorems. If
(T,c,~) is a triple on a regular category C and T preserves regular
epimorphisms (e.g. if ~ is the category of sets), then the category
of T-algebras is regular. The last section gives another transfer theo-
rem for the category of sheaves. Finally, if C is a regular category,
then so is the functor category [~,C], where ~ is any small category;
39
then the monomorphisms of [~,~] coincide with the pointwise monomor-
phisms, and similarly for the regular epimorphisms.
Much light is thrown upon regular categories by Barr's non-
abelian full embedding theorem ([2],[3]), which among other things sta-
tes that a small regular category ~ can be fully embedded into a func-
tot category iS,Sets] in such a way that finite limits and regular
decompositions are preserved (hence also reflected). The elementary
properties which follow can of course be deduced from the axioms, but
this theorem provides alternate proofs of most of them.
4. In what follows, we consider subobjects as classes of
equivalent monomorphisms. Every morphism f in a regular category
yields a subobject Im f of its codomain (which is indeed an image as
defined e.g. in [18]), namely the class of all m in the regular decom-
positions f = mp of f ; the subobject defined by any monomorphism m
is then Im m • Clearly f is a regular epimorphism if and only if
Im f = i (where 1 is used to denote the greatest subobjeet of a given
object~ we also use I to denote identity morphisms).
Since G is finitely complete, it has inverse images~ we de-
note the inverse image of a subobject ~ under f by fsx, (the notation
f-l_ is more logical but would create confusion when we start dealing
with relations). In addition ~ also has direct images, which can be
defined by: fs Im m = Im fm • It is easy to show that: fsl = i ;
fsf x = f x ; fsf x = x when f is a monomorphism, fs I = Im f ; fs s-- s-- s-- --
fs fs Z = [ when f is a regular epimorphism; in general fsfS~ =
y a Im f ( a , /\ will be used to denote intersections of subobjects).
It is well-known that inverse images preserve intersections of subob-
jects~ in a regular category, direct images preserve [existing] unions.
(Unions of subobjects are also least upper bounds for the ordering on
subobjects and will be denoted by v , \/ ).
5. In a regular category a , a relation ~ :A ~B is de-
40
fined, as usual, as a subobject of A xB (the proper name is additive
relation, when G is abelian). In particular, morphisms a : D >A and
b : D )B determine (a,b) :D ~A xB , and thus a relation
= Im (a,b) : A ~B . The composition of ¢ : A ~ B and ~ :B ~ C
can be defined in two ways: first by Puppe's formula [~ = ps(qSe A rS~),
where p,q,r are the projections from A xBx C to A x C , A xB ,
Bx C ; next, by pullbacks, i.e. if ~ = Im(a,b) , ~ = Im (b',c), then
~ : Im (ax, cy) , where bx = b'y is a pullback. Because C is regula~
the second composition does not depend on the choice of (a,b) , (b',c)
and coincides with the first. It is clear from the second definition
that the composition of relations ls associative. The inverse ~-i of
a relation e is defined in the obvious way.
All this (and the remainder of this paragraph) extends well-
known results and concepts of the abelian case ([15],[19],[16]) and has
been considered by many authors in non-abelian cases; hence we shall a-
gain skip the proofs. It would be more convenient at the beginning to
define relations as pairs A > • < B of morphisms, but since we
shall need intersections and unions and direct and inverse images of
relations, the definition we gave is ultimately more convenient. Al-
though in a regular category all axioms of partially ordered categories
as set forth in [15] need not be satisfied nevertheless a number of ba-
sic properties of relations do hold.
First, the composition of relations is order-preserving and
-i ¢ ¢ = ~ always holds. One can identify each morphism f with the
relation Im(IA, f) , where A is the domain of f , and verify that if
f and g are morphisms and satisfy f S g (as relations), then f = g.
One can define the image of a = Im(a,b) by : Im ¢ = Im b ; when
\/ ~. exists, Im \/¢. : \/ Im ¢. . The image of a subobject under a iel l iel I iel i
relation can be defined by: a s Im m = Im am ; if f is a morphism,
= fs then fs has the same meaning as before and (f-l) s ; in general
= Im(a,b) implies ~ = ba -I and therefore ~ = b a s • The relation s s
41
a is a morphism if and only if a-la ~ c and a -Is ¢ (where e de-
notes any equality -- another name for the identity morphism, or diago-
nal); in fact, if and only if a-±a ~ e and ~Sl = I (where a s =
(-i) ). S
The direct image of a relation ~ : A---~ A under a morphism
can be defined either by f~f-i : B___~ B or as ~s a , whe- f: A > B
re f = f × f : A x A > B x B ; the reader may check that ~s ~ = f~f-i
always holds. Similarly, f-laf = ~sa whenever deflned. The first defi-
nition is perhaps more natural, but from the second we inherit all pro-
perties of direct or inverse images of subobjects. It is relevant to
note that if f is a regular epimorphism then so is ~; more generally,
any finite product of regular epimorphisms of C is again a regular
epimorphism (this can be deduced from (III) by noting that f ×g =
(fx i)(I × g) and inserting f ×i , I ×g into suitable pullbacks).
6. In a regular category, the congruence ker f induced by
a morphism f is the relation f-±f ; equivalently, it is Im(x,y) ,
where fx = fy is a pullback. For instance, f is a monomorphism if and
only if kerf = e
The inverse image of a congruence under a morphism is al-
ways a congruence; more precisely, ~S(ker g) = ker gf • The similar
property for direct images is of course false. However, if
kerf < ker g and f ls a speclal epimorphism, then ~ (ker g) is a -- S
congruence; in fact, ~s(ker g) = kert , for the assumptions imply
g = tf for some t • This factorization property will be used fairly
often. In the above, t is a monomorphism if and only if kerf = ker g.
It is immediate that every congruence ~ is reflexive
( a ~ ¢ ), symmetric ( a -I= a ) and transitive ( ~ ~ a ; in fact,
~a = a ). The converse is a condition which has been used by Lawvere
to characterize varieries [13] and it shall therefore be denoted by
(L). It holds in abelian categories and infinitary varieties as well.
42
All the other properties we need for the manipulation of con-
gruences have been given in the previous paragraphs 4,5.
7. We conclude with a list of properties which are more tech-
nical and therefore perhaps not so well-known to the reader.
Lemma I.i. If fg' = f'g is a pullback, then Im fg' :
Im gf' = Im f a Im g •
Proof. If f = mp , g = nq are regular decompositions, and
mn' : nm' , m'q' : qn" pn" = ' p'q" : 'p" , n'p , q , then, up to isomor-
phisms, g' = n"q" and f' = n"p" , and these are regular decomposi-
tions by (III); hence fg' (mn')(p'q") is also a regular decomposi-
tion, whence the result.
Lemma 1.2. Im(u,v) ~ ker f if and only if fu = fv
Lemma 1.3. Let D e [~,G] be a diagram in C with colimlt
(ci)ie @ : D > C (C eC). If (ai)ie $ : D--~ A (A e C) is a cocompatible
family and induces a : C---~ A , then Im a = \/ Im a In particular,
\/In c. = i • ie ~Z i
Proof. If Im m is a subobject of C and Im c i ! Im m for
all i, then every c i factors through m, inducing a cocompatible fa-
mily (di)ie @ with md i = c i and a morphism u with d i = uc i , for
all i • Then mu = 1 C , whence Im m = i • Therefore I = i~/Im C i
Then a 1 = ~/a Im c = \/ Im a. s ieas • ie~ i
Lemma 1.4. Let D be as above; assume furthermore that the
coproduct ~ D(i) , with injections m. (i e $) exists. Let
c : Y D(i) )C be such that
gular epimorphism, and ker c
all Im(mi,mjD(f)) with f : i
cm. = c for all i. Then c is a re- I i
is the smallest congruence containing
> j e ~
Proof. Applying 1.3 twice, we get Csl = \/ c Im m. = ie~ s i
= \/ Im c. = 1 so that c is a regular epimorphism. Also, (c i) ie @ l ' ie~
is a cocompatible family, hence by 1.5 must contain all Im(mi,mjD(f)).
43
If conversely ker d contains all these, then (dmi)ie $ is a cocompa-
tible family, hence factors through (ci)ie ~ = (cmi)ie ~ , so that d
factors through c and ker c S ker d •
Lemma 1.5. Let f,g : A ) B and m be a monomorphism of
codomain A. Then me Equ(f,g) if and only if Im(m,m) = g-lf A ¢ ;
s in particular, Equ(f,g) = ~A(g-lf) (where A A : A ) A xA is the
diagonal).
Proof. Let fx = gy be a pullback, so that g-lf = Im(x,y).
It suffices to verify that m is an equalizer of f and g if and only
if (x,y)k = gA m is a pullback for some k.
2. Directed colimits in regular categories.
i. The main result of this section is:
Theorem 2.1. Let ~ be a cocomplete regular category. Then
directed colimits in ~ preserve finite limits [and monomorphisms] if
and only if the following conditions hold in ~ :
(C~) Inverse images preserve directed unions of subobjects;
(C~) A directed union of congruences is a congruence;
(C~') For every functor • : I ~ C , i ! ~ X i , (i,j) ! ~ xij
( i ~ j ) , (where I is a directed preordered set), such that each
morphism z . is a monomorphism, there exists an object Ae C and a ij
[not necessarily cocompatible] family of monomorphisms X[ > A
When all this holds, we say that ~ is a C 3 regular cate-
gory •
This statement calls for a few remarks. First, (C}) implies
i~e/l \/ (z ) whenever the familiar C 3 condition: ~A( ~i) = l~I -- a~i
(Yl)iel iS directed (Grothendieck's condition A.B.5 [6]); since C
has regular decompositions, it is equivalent to the conjunction of
A.B.5 and the condition that inverse images under regular epimorphisms
44
preserve directed unions of subobjects. A consequence of (C~) (though
apparently not of A.B.5 alone, in general), which we shall need and
use constantly, is that the composition of relations preserves direc-
ted unions (i.e. (ie~/l ~i)(~/j ~j) = (i,j)~e/l×j ~i~j when (~i)iel ,
(~j)jeJ are directed); this is immediate from Puppe's formula°
The other two conditions are very mild. When (C~) holds,
it is clear from the above that Lawvere's condition (L) implies (C~).
As for (C~'), it holds in any category where the coproduct injections
are monomorphisms. It follows for instance that in the abelian case
there is no need for (C~) , (C~') in theorem 5.1. The resulting theorem
is not quite as good as the classical result as stated for instance in
[18]; thls apparently is due to the fact that in our general proof we
have to manipulate relations, which is not necessary in the abelian
case as monomorphisms can then be characterized in terms of kernels.
5. The easy part of the proof is that (C~), (C~), (C~')
hold in any cocomplete regular category ~ where directed colimits
preserve monomorphisms and finite limits. That (Cj) holds is clear
since directed colimits in C also preserve pullbacks, and we can al-
ways describe ~/I Im m i , when (Im mi)ie I is directed, by organi-
zing the family of the domains of the m. into a functor I >~ in l
the obvious way, so that the morphism m induced to the colimit is a
monomorphism by the hypothesis and satisfies Im m = \/ Im m. by 1.3. iel l
The verification of (C~) is similar. If (~i)iei is a
directed family of congruences on A e ~ , preorder I in the obvious
way and write ~i = Im(xi'Yi) = ker Pi ' where Pi : A > B. is a re- l
gular epimorphism and PiXi =piy i ls a pullback. Then (xi,Y i) :
K. > A ×A is a monomorphism and there is an obvious functor I ~ i
with i 0 > K. • Another functor I ) @ with i ! > B. is obtained i i
by noting that i ~ j implies ker Pi = ~i ~ aj : ker pj , hence
pj = bijPi for some unique bij as Pi is a regular epimorphism.
Taking the colimits we obtain morphisms p , x , y such that px = py
45
is again a pullback and Im(x,y) = iYl Im(xi'Yi) (by 1.3). It follows
that i~/l ai = ker p is a congruence.
To verify (C~') it suffices to prove that, if a functor
I > ~ , i I > A. (i j) ~ > a.. ( i < j ), (where I is direc- i ' ' lJ --
ted p~reordered), is monic, i.e. all morphisms a.. are monomorphisms, iJ
then the morphisms A. > collm A. are also monomorphisms. This is i i
well-known in the abelian case and holds, more generally, in any cate-
gory with finite intersections and directed colimits, where directed
colimits preserve monomorphisms, i.e. a pointwise monomorphism
~ ~ , where Z , ~ : I > C , induces a monomorphism to the colimits.
To show this, let C denote such a category, and I be a
preordered set, and I : I ~ C be a monic functor, with i! ~ X i ,
(i,j) ! ~ x.. ( i < j ) The proof that X. > colim ~ is also a ij - i
monomorphism is immediate in the case when I is in fact a directed
m-semilattice. In that case one has for each i ~ I a functor
: I > C defined by: Yj = XiAj ' Yjk = XiAj,iAk ( j S k ) ~ then
(XiAj,j)j~ I is a monomorphism from ~ to Z and induces to the co-
limits a monomorphism which is just X. > colim Z • i
If I is any directed preordered set, then we can come back
to the case of a directed a-semilattice as follows. First we find the
semilattice. For each kc I , let S k be the set of all finite inter-
sections of subobjects Im Xik of X k with i ~ k • We note that S k
is an A-semilattice. If k ~ ~ ~ I , then Zk~ is a monomorphism, hen-
ce (xk~) s preserves intersections and therefore induces a semilatti-
ce homomorphism Sk~ : S k ~ S~ which is clearly injective. Take
S = colim S k ; this again is an A-semilattice and it comes with injec- k~l
tire homomorphisms s k : S k > S with S = k~l Sk(Sk) . A mapping
i b > T of I into S can be defined by: T = s ( Im x i ) = si(1) ; i i
it is order-preserving and we see that ~ = [ T ~ i ~ I ] is cofinal
in S, so that S is directed.
46
For each s e S
morphism Ysk: Ys
yt:Yt > X£ ,
t ~ m ~ we see that
, select k~ I with s e Sk(S k) , and a mono-
> X k such that s = Sk( Im Ysk) . If s,t £ S ,
t = s ( Im yt~), then there exists me I with k ~ m,
s = Sm(Im XkmYsk) , t = Sm(Im X~mYt~) ; since
s m is injective, s ~ t implies Im ~kmYsk ~ Im x mYt~ and there
exists a unique Yst : Y > Yt such that s XkmYsk X~mYteYst ; since
I is directed Yst does not depend on the choice of m. This cons-
truction yields a monic functor ~: S -> C .
It is clear that Y:- --- X for each i e I and we can ex- I i
pand this to obtain a functor ~}' : I ~ C , i i :~ Y-- which is iso- I '
m o r p h i c t o I The f i r s t p a r t o f t h e p r o o f shows t h a t , f o r e a c h i ,
YT ~ colim ~ is a monomorphism, and using the isomorphisms
colim ]~ - colim ~' =colim ]} it follows that X > colim I is also i
a monomorphism, q.e.d.
3. We now start proving the converse. Thus, we let ~ be a
cocomplete regular category in which (C~) , (C~) , (C~') hold. Note
that, for each directed preordered set I , the category [I,C] of all
functors I > C is regular, and its monomorphisms coincide with the
pointwise monomorphisms while finite limits are pointwise too. Unlike
the abelian case, it does not suffice to prove that directed colimits
preserve monomorphisms; however, it would suffice to show that they
preserve finite limits. The unusual length of the proof comes from the
fact that this is more complex, and also, unfortunately, requires that
preservation of monomorphisms be shown first anyway. The proof will be
divided into four parts: preliminary results, then preservation of mo-
nomorphisms, finite products, equalizers. We shall use the following
convention, that if Z denotes a functor I > C , then % : i ~ > X i ,
(i,j) I ~ xij ( i _< j) and we call X = colim ~ and x i :X.l ~ X ;
and similarly for ~ ,~ .
Lemma 2.2. Let a i :X i > A ( i e I ) be a cocompatible
47
family for l , inducing
Proof. Clearly
-i a :X > A . Then a : i~e/i aix i
(aixi-l)ie I is a directed family of rela-
tions. It follows from (Cj) that
\/ x-i -I -i \/ a,x-i)( \/ a.x-l) -I < kel axk k XkXk a = iel i i jel J J --
: \/ (axk)(axk)-i S e ; kel
also,
\/ a.x-l)si = \/ (x i aS I = \/ Im x . = 1 iel i I iel s iel i '
by 1.3. This shows that b = \/ a x -I is a morphism. But it is clear iel i l
that, for every i, by i < ax whence bx. : ax. and b = a • -- • ' i i
Note that if in the above each a. is a monomorphism, then i
kera = ( \/ a.xTl)-l( \/ aj~j i) < k~/i -i -I < e iel i i jel -- Xkak akXk --
and therefore a is also a monomorphism.
Lemma !.3. If Z : I > ~ is monic, then every x i is a
monomorphism.
Proof. Let C = LI x. be the coproduct, with injectlcns ic7 i
m i :X i > C It follows from (C~') that each m i is a monomorphism.
Also, by 1.4, there exists a regular epimorphism c : C ) X such
that x i = cm i for all i , and ker c is the smallest congruence on
C which contains all Im(mi,mjxij) with i ~ j •
Let ~ be the set of all finite subsets of the preorder re-
lation [ (i,j) e I × I ; i S J ] For each F e ~ , the subdiagram
of ~ consisting of all X. but only those xij i ' . . with (i,j) e F ,
has a colimit in ~; by i.$ again, it follows that there exists a smal-
lest congruence eF on C which contains all Im(mi,mjxij) With
(i,j) e F . It follows from (C~) that F~/~ ~F is a congruence; it is
clear that \/~F = ker c Fe~
48
We want to prove that cm. is a monomorphism; for this we 1
shall show that ~s : e for all F. Specifically, pick i e I , F e ~. i
There ezists te I with i _< t and j <_ t , k <_ t whenever (j,k)eF.
Let C t = II X. and f : C t > X t be induced by all xjt , j < t ; j_<t J
' = ~ Xj n • C t : > C CtL] C~ , g : X t > XtU C~ be the let C t j t '
' > XtU ' . We see that g is injections, and h : fLJl : C : CtlJ C t C t
a monomorphism (since m t is a monomorphism) and that gf = hn ,
hmj : gxjt whenever j _< t . If (j,k) e F , then hmj : hmkX.jk ; then
it follows from I.Z that ~F -< ker h • Therefore
~s ~s mi ~F -< mi ker h = ker hm i : ker gxit : ¢
By (C~) , this implies
ker x : ~s ker c : \/ ~s ~F < ¢ i i Fe~ i -- '
so that x is a monomorphism. i
The next lemma is of interest in itself.
Proposition E.G. If C is a C 3 regular category and I is a
directed preordered set, then for any Z : I ~ C : ker x. = \/ ker x.. l j~i lj
for every i e I
Proof. By (C~),
i e I . Put ~i : ker Pi '
phism. If i A j , then by
~. = \/ ker z . is a congruence for every l j~i iJ
where Pi : Xi > Y'l is a regular epimor-
ker pjxij = ~.( \/ ker Xjk) : \/ lj k~j k~i
ker Xik = ker Pi '
whence pjxij : yijpi for some unique Yij ' where Yij is in fact a
monomorphism; in this way we obtain a monic functor B : I > G • Using
the same factorization property, it is easy to show that ~ and ~ have
"the same" colimit, i.e. that (yipi)iel is another colimit of Z •
It follows that ker x i = ker yip i ; since Yi is a mono-
morphism, by 2.3, this shows that ker x i = ker Pi ' q.e.d.
49
It is clear that the construction of directed colimits in
that proof is very much like the construction of directed colimits in,
say, the category of sets.
4. That directed colimits in C preserve monomorphisms will
presently follow from 2.~ and
Lemma 2.5. Let a i : A i > A be a family of morphisms such
that (Im ai)ie I is directed. If i~e/i Im a i = I , then ie~/i Im ai : i .
Proof. Recall that ai = aix a i : (a i x IA)(IA x ai). If I
p : A xA > A , Pi : Ai xA ) A i are the "first" projections, then
p(a i × i A) = alP i is a pullback, and it follows from (C~) that
i~/l Im a i : i implies i~e/l Im(a i xl A) : i . Then, similarly,
~/ Im(l A ~a ) = i for every i e I Therefore jeI i J
(i,j) \/elxIlm(aixa')J = iel\/(j~l \/ (aiXlA)slm(iA.xaz J ) =
= \/ Im(a i x l A) = i ; ieI
the result follows if we observe that (Im(a i xai))ie I is cofinal in
(Im(a i x aj))(i ,j)elxl
Now let ~I : (mi)ie I : Z ~ } be a monomorphism of functors
I ~ C (so that each m. : X. > Y. is a monomorphism), and i i i
m : X ~ Y be induced by ~ ; By 1.3,2.5, i~e~ Im ~i = I Then it
follows from 2.4 and (C}) that
ker m = ~/ (ker m a lm ~ ) = ~/ (~i) ~s ker m iel i iel s i '
xi~S ker m = ml-S ker Yi = j~i\/ (m~ ker Yij =
~/ ~s ker m \/ -s : . . -- x..¢ : \/ ker x.. = ker z i j>i ~J J j>i ~J j_>i ~J '
ker m : ieI~/ (Zi)s ker zi --< e '
which proves that m is a monomorphism.
5. We now have shown that directed colimits in C preserve
50
monomorphisms, and turn to equalizers.
Let g ~> Z ~_ ~ be an equalizer diagram of functors
I > ~ , with I directed preordered. We want to show that the dia-
E m> X f{ Y induced at the colimits is also an equalizer dia- gram g
gram (in G ). By the above, we know that m is a monomorphism; also,
fm : gm • Now a description of Equ(f,g) is given by 1.5, and with
this in mind we begin to evaluate g-lf me • For each i,
x i fx ^ x_Ix- = i i i
-i -i a x[iz i : : gi Yi Yifi
( k/ -: j>i_ gi YijYijfi ) ^( \/k_>_i xilkXik)
: \/ ( -i -i A ~[l ) t>i xitgt ftxit tzit
(by (Cj) and 2.4)
(by (C})
~/ ~s (g~ift me) = \/ ~s In( m t) t>i it t>i it mt'
(by 1.51.
Therefore
(g-I -s -i f a e) alm xi = (Xi)s xi (g f h e) =
: \/ (~i) ~s im(m t mt ) : \/ (~t)s( ~ ) ~s im(m t mt ) t>i s it ' t>i it s it '
--< t>i \/ (~t)s Im(mt'mt) = t>i~/ Im(xtmt' xtmt) -- < Im(m,m) ,
since Im xtm t S Im m . Then it follows from (Cj) and 2.5 that
g-lfa e S Im(m,m). On the other hand, fm = gm implies Im m S Equ(f,g)
and Im(m,m) S g-lf a c . Therefore Im(m,m) = g-lf a e and since m is
a monomorphism it follows from 1.5 that m e Equ(f,g), q.e.d.
6. Finally, we show that directed colimits in ~ preserve
finite products. It is enough to show that colim( Z × A )m ~olim Z) ×A
naturally, for every A e ~ and functor Z : I > ~ ; indeed if Z , ~
are functors I > ~ , then we may consider Z ×4 , which is a point-
51
wise product, as a subfunctor of ~ : I x I > C , (i,j) , ) X i xYj ;
since the diagonal is a cofinal subset of I xl , we have a natural
isomorphism colim( ~ x ~) ~ colim ~ ; and then we also have natural
isomorphisms
colim ~ = colim(X i xY.) ~ colim (X. × colim ~) ~ colim I x colim i,j J i •
Thus take I : I > C
Y. = X. x A for all i ). Clearly i 1
for b, hence induces a morphism
I
and A eC • Put ~ : I x A (so that
(x i × iA)ie I is a cocompatible family
t : Y > X x A , which is natural in
and A ; we want to show that t is an isomorphism.
If p : X ×A > X is the projection, then, by 1.3 and (C~),
Im t = ts( \/ Im yi) = \/ Im ty : ~/ Im(x[ ×I A) = iel iel i iel
: \/ pS Im x. : pS( \/ Im x.) :pSl = i ,
iel l iel l
since p(x i x 1 A) = xiP i (where Pi : X.IX A > X. is the projection) 1
is a pullback. This shows that t is a regular epimorphism.
It now suf~ces to show that t is also a monomorphism. We
begin by proving that ker(x i ×i A) = ~/ ker(x.. × 1 A) • Indeed the j~i ~J
functor - ×A preserves pullbacks and regular decompositions; for each
f : B ~ B' ' are the pro- , p (f xl A) : fp is a pullback (where p,p'
jections B xA > B , B' ×A > B' ), and it follows from (Cj)
that our functor also preserves directed unions of subobjeets. Then
ker (x ×i A) = \/ ker (x × I A) follows from 2.4. Then it follows
from a.4, a.5 and (C~) that
ker t : ~/ (ker t a lm ~i ) : ~/ (~i) s ~s i ker t : iel iel
: \/ (Y.) ker ty : \/ (yi) ( \/ ker(x ×IA)) : iel i s i iel s j_>i ij
: \/ (~i) ( \/ ker yij) : ~/ (~i) ker Yi < ¢ " iel s j_>i iel s --
This shows that t is a monomorphlsm, and in fact completes the proof
52
of the whole theorem.
7. In the course of the proof, we have obtained additional
results, such as 2.4, showing how nicely directed colimits behave in a
C 3 regular category. We supplement this by two more results.
Proposition 2.6. In a C 3 regular category, Gray's condition
~2 holds.
Proof. We have to show that, if Z : I ) ~ , (with I direc-
ted preordered ) ,and xif = xig for some i e I and f,g : A > X i
then j_>i\/ Equ(xij f'xijg) = 1 (cf. [18],[4]). If /4 A : A ) A xA is
the diagonal, then, by 1.5, 2.4, (C~):
\/ Equ(xijf ' \/ s -i -I f) : j>i xij g) : j~i ~A ( g xijxij
s -I -I = A A ( g x i xif) = Equ(xif,xig) : i •
Proposition 2.?. Let ~ be a C 3 regular category, Z : I >
a monic functor,( with I directed preordered),and f,g : X ~ A . Then
Equ(f g : \/ (xi) s Equ(fxi,gx i) ' i¢I
Proof. Each x i is a monomorphism; hence (Xi)sEqu(fxi,gx i)
= Equ(f,g) Aim x i , as readily verified. Then the result follows from
l.J and (C~)
8. We shall conclude this section by showing that additional
hypotheses on ~ allow to write any product of directed colimits as a
directed colimit of products.
Namely, we assume that ~ is a C 4 regular category in the
sense that it is a complete and cocomplete C 3 regular category in which
Gray's condition 51 (cf. [18] , [4])( = A.B.6 [6]) holds:
51 : if ((~i)ielx)X¢ A is a non-empty family of non-empty, direc-
ted families of subobjects of Ae ~ , then
/k(\/ _x i) = V( /k z~ x) XcA i¢I k ~ET XeA
53
I X and the X coordinate of T is denoted by ~X . where T = A
We already know that any finite product of regular epimor-
phisms of ~ is a regular epimorphism, and call ~ a C* regular catego- I
ry if a__ny product of regular epimorphisms of £ is always a regular
epimorphism, too. This slightly abuses the language but is equivalent
to what is usually called C~ in the abelian case.
Note that a finitary variety satisfies all these conditions,
and so does any C4, C~ abelian category.
Theorem 2.8. Let ~ be a C 4 regular category and (]6~)keA
be a non-empty family of functors Z X : I X > ~ from directed preor-
dered sets I X If all Z X • are monic, or if ~ is also C* there is i ,
a natural isomorphism
colim Z x ~ colim ~ XTX , XeA veT XeA
whose inverse is induced by all x'T = X~A XTX (If ~ is not C~, then
in the second case the x's only induce a monomorphism). T
The notation is as before, except that we may always assume
that the I X a r e p a i r w i s e d i s j o i n t a n d i t i s l e g i t i m a t e t o w r i t e X. 1
etc. instead of X~l It should also be noted that T =X~ I X is a pre-
ordered directed set under the coordinate-wise preorder, and we have a
: ~ X~k and x = ~ xax,T k ( a S T). functor Z : T > G with X T keA aT XeA
9. We begin the proof with the following generalization of
~.5.
Lemma 2.9. Let (Ax)xe A be a non-empty family of objects
of ~ and, for each X, (fi)ielx be a non-empty family of morphisms
of codomain A x such that (Im fi)ielx is directed and ~/Im f = i • i61X l
Assume that all fi are monomorphisms or that ~ is also C*I • Then
\/ Im f = i where T is as before and f = ~ fTX TeT T ' T keA
Proof. Conslder first the case when all f. are monomor- 1
54
phisms. For each i e I v e T put:
A i = ( X A~) ×X i ; gi = ( X keAk H XeA\H ±Ax
hT H = ( ~ fTX) ×IX : X = ~ X X > A , keA\~ TU ¢ keA ~H
morphisms are monomorphisms and that f = gT hT,~
It is easy to verify that in fact f T
: /\ Im all the gtx ( X e A ). Hence Im fT XeA gTX
f. :X. > A ; i 1
) x fi : A. > A = ~ A k ; i keA
Note that all 0he new
for all T,~ •
is an intersection of
On the other hand,
if Pi : Ai > Xl , p~ : A > A~ are projections, then P~gi : fiPi
is a pullback, so that Im gi = P~ Im fi ' and, by (Cj) , ~/Im gi= I . iel
for each ~ . Since G is C# , then
~/T Im f = ~/ ( /~ Im gTA TET XeA
= PX(\/ Im 1 ~eA iel X gi) :
This takes care of the case when all f are monomorphisms. l
In the general case, we also aSSURe that ~ is C* i ; then products pre-
serve regular decompositions, and it is clear that by considering a re-
gular decomposition of each fi we can reduce this case to the pre-
vious case.
Armed with this lemma, we can now prove theorem Z.8. With
' = X ~X ( ~T ) the notation there, we want to show that the x T keA
induce an isomorphism X ~ h A~ XX . First it is clear that they induce
a morphism t : X > ~ X i such that x' = tx for all T If every XA T T
~k is monic or if ~ is C* then the lemma applies to the families I'
(z.) i iel X and yields \/ Im x' : i from which it is easily deduced
TET T '
that t is a regular epimorphism.
To complete the proof, we have to show that t is also a mo-
nomorphism. This can be done, as follows, without using C I
If all ~X are monic, then all z i , hence also all x' T '
are monomorphisms. When Q is C 3 , it follows that t is a monomorphism.
In the general case, we show that ker x' : \/ ker ~ =
ker x T • For each k and each i e I X with i _> <X , pick monomorphisms
55
mTX : K~X > XTX x XTX , mTx, i : KTX,i > XTX x XTX such that ker x X
= : \/ lm and : Im mTk , ker x k,i Im m X,i . By 2.~, Im mTk i~k mTk, i
there exists an obvious functor ~k : [ i e I X ~ i ~ TX ] > ~ ,
i ~ ) K X,i , with colimit KTk ; clearly ~ is monic. Therefore it
follows from the above that there exists an isomorphism
colim ~ K k,~ X ~ X ~eZ XeA XeA KTX '
where Z = X~^ [ i e I X ; i ~ ~X ] ~ note that E : [ o e T ; ~ k < ]
Applying ~ mTX , we deduce from 1.3 and this: Xei
\/ (Im Z = Im k o>~ XeA m~k, °X) keA mTX
(this does not use C~ since these products are products of monomor-
phisms). However, products preserve pullbacks, hence also congruences,
and therefore the last relation reads
Hence
of ~ 6
\/ ker x ker x' O~T OT T
ker x T' : ker xT ' by ~.~.- Then we use the reasoning at the end
to show that t is a monomorphism, which completes the proof.
3. Sheaves in regular categories.
i. Throughout, ~ denotes a C~ regular category, X a fixed
topological space, ~(X,~) and ~(X,~) the categories of presheaves
and sheaves, respectively, on X with values in ~ • Note that ~(X,~),
being a functor category, is also a regular category (with "pointwise"
regular decompositions), clearly in fact a C~ regular category. The
hypotheses on ~ will be relaxed whenever possible, but C will remain
complete and cocomplete throughout, and in that case regularity is a
rather mild condition since it follows from remarks in ~ 1.5 that it is
equivalent to our axiom (III) (= pullbacks carry regular epimorphisms).
2. We start by recalling He]ler and Rowe's construction [7]
56
of the associated sheaf. Unlike Heller and Rowe we find it more conve-
nient to define an open covering C of an open subset U of X as a fa-
mily (Ui)ie I of open subsets of U (with U = ie [jl Ul ) which is not
necessarily indexed by U. This implies minor modifications in the
construction, which we indicate along with the definitions we shall use
afterwards. The reader is referred in [7] for details.
First, if C = (Ui)ie I and ~ : (V) are open coverings j jeJ
of U then ~ refines C (written as: C < ~) in case each V. is , -- J
contained in some U i In general, C *~ = (U i ~ Vj)(i,j)el× J refines
C and @ . Under S , the set G(U) of all open coverings of U is
then a directed preordered set.
If P e~(X,G), we have for each open set
C = (Ui)ie I e [(U) a canonical diagram
P(u) u~ P(c) --~f p(c.c) g~
U ~ X and each
where P(C) : iel~ P(Ui) , P(C.C) = j,~el P(Uj N U k) , and
P ) f P ~ ((Puj,ujnUk)ke I) u : Uc = (Pu, u i iel ' : fC = jel
P ~ ((PUk,UjNUk) i ) are induced by the restriction maps. [For g : gc = k I je
any presheaf, fu = gu ; P is a monopresheaf if u is always a mono-
morphism, a sheaf if u is always an equalizer of f and g.] Let
u* : Ec(U) ) P(C) C
have u C UcCc (U)
A functor ~(U) > G
ted as follows. Let C : (Ui)ie I
Then there exists a mapping 9 : J
j e J • This yields maps
be the equalizer of f and g. Since
for some unique morphism Oc(U).
fu = gu, we
such that C I > Ec(U) is construe-
= (Vj)je J e G(U) satisfy C S
> I such that Vj ~ Ugj for all
P'(~) = i~I
P" 4) = j~i ke I
(Pui,Vp)peg-i i) : P(C) > P(~)
IPuj P q;q c 9-~k" n k,V nv )pe jl: P(c c) > P(~.~)
57
P P p. P It is easy to verify that fP P'(~) = P"(~)fc ' g~P'(~) = (~)gc '
and P'(~) u~ does not depend on the choice of ~ . This induces on the
equalizers a morphism Ec~(U) : Ec(U) > E@(U) unique such that
u~ Ec~(U) = P'(9)u~ and independant on the choice of 9. Let E(U) =
colim Ec(U) with maps pc(U) : Ec(U) ) E(U) . One sees that c(U) :
: Pc(U) cc(U) : P(U) > E(U) does not depend on C.
One makes E into a presheaf as follows. If V ~ U and
C = (Ui)ie I e ~(U) , then C nV = (UinV)ie I e {(V) and the morphisms
h' : i~I~ PUi,Uin v : P(C) > P(C nv) ,
h" = ~ PU j,kel j nUk,Uj nuknv : P(c.C) > P(CAV*CNV)
induce on equalizers a morphism E c U,V : Ec(U) > EC~v(V) (unique such
that Ucnv* ECU, V = h' u c* ). The restriction map EU, V : E(U) > E(V) is
induced by all E C (C e ~(U)) U,V
One sees that c : P ) E is now a morphism of presheaves
and furthermore every morphism from P to a sheaf factors uniquely
through c. Heller and Rowe's result is that, in an exact category
with products, exact directed colimits and a projective generator, this
construction, when iterated a sufficient number of times (by ordinal
induction), will eventually terminate at the associated sheaf of P [7].
In his review of [?] (MR 26 ~ 1887), Gray conjectured that in most ca-
tegories two steps should actually be enough. We shall see that this is
the case in a C% regular category.
3. The result has two steps and so does the proof. The first
Lemma 3.1. If ~ is a C% regular category, then for every
presheaf P, E is a monepresheaf.
Proof. Let C = (Ui)ie I e {(U) . By theorem 2.8, there is a
monomorphism [note that we do not assume that ~ is C~ ]
step is:
58
t : co l im .~ (U i ) met L I ETi
> ~ E(U.) i I l '
where T : ~ [(U i induced by all iel
write ~i = (Vj)je J (where the sets
let C T = (Vj jeJ e ~(U) , where J =
diagram
PT' = iel~ PTi " For each T ,
Jl are pairwise disjoint) and
i~l J'l Consider the commutative
X ie I ET i
ECT U)
le~l u~i u i ) > X P(Ti)
C~
> P(C T )
P i~l fmi
> ~ P(mi * Ti) iel
em > P(C~ *c T)
where the last vertical map is the projection ~ P(VjNV k) > j,keJ
~ P(VjAV k) (since U J. x J. ~ J x J ). A similar diagram iel j,keJ i iel z z
exists with the f's replaced by the g's • Since products preserve
equalizers, there exists a morphism u : EoT(U) > iel~ E i(U i) such
that u*cm = (iel ~ u*mi) u~ ; uCT* is a monomorphism, hence so is u T
' PC~ EC 7 = Now we prove that pmum = u~ PCT (where : > E(U)
colim Ec(U) Consider the (three-dLmentional) diagram fig. I below,
where the n's are projections from products, 9 : J~ > J is the in- I
clusion and h = ~ ((Pv U ^" ) ) We see that areas ~) and iel ~, .HV. jeJ i j l J
commute So do areas ~ , (~ , (~ by definition of the various E
maps. Finally, u~i~iu T ~ m ~l = niu by definition of u . Since u*. is
a monomorphism, it follows that area G commutes. Hence pTi~iu =
E EU, Ui pc T Taking products over I yields PSuT = Uc PC~
We now observe that the coverings of the form C T ( T e T )
form a coflnal subset of ~(U) (a remark we shall use again). If in-
deed ~ = (Wk)ke K e ~(U) , then let mi = (U i 0Wk)ke K ; we see that
e T and C = C*~ refines T
We now take the directed col~mit (over T ). This sends the
59
E(u)
PC ~ T
E c (U)
U ~ C~
EU, U i
P CT) = i~I P(Ti)
T P c T n u i ~ E (U) ~ ~6~ . E
"~ ~ie~l T i i :" ~- "
~, U ~a~ ui ) ECT f lU i
u c v n u i
P (C,r RU i )
1 )
Fig. i
(U) i i
U'b. TI
P Ti)
commutative square below left to a commutative square below right.
E(U)
PC~
E e (U) T
E u c
U T
E u C > X E(U.) E(U) > X E(U.)
iel I p~I iel It 1
> i~I E'i(Ui) E(U) > collm-l~ I E i(U i)
where we know that t is a monomorphism (beginning of the proof) and
u is induced by all uT, hence is also a monomorphism.
follows that u~ is a monomorphism, q.e.d. It
The second step of the proof is:
Lemma 3.2. Let C be a C 4 regular category. If P is a mono-
presheaf, then E is a sheaf.
Proof. Since ~ is complete, cocomplete and satisfies 31,
32 , some of Gray's results (the ones which do not depend on G being
well-powered as well) still apply, and one of these is that for every
monopresheaf P there exists a monomorphism m : P > F , where F is
a sheaf (see [18],[4]). If c : P > E is as defined in ~2, then m
60
induces a morphism n : E > F such that m = nc ; n can be obtained
from the morphisms no(U) : Ec(U) > F(U) induced on equalizers by m,
by going to the colimit. We note that each no(U) is a monomorphism
(since m is a monomorphism), hence so is n. It follows that each
Ec$(U) is also a monomorphism, so that each functor ~(U) > C ,
C ' ) E (U) is monic. C
We now start the proof as for the previous lemma, keeping
the same notation. This time, however, it follows from 9.8 that t is
an isomorphism(since each functor ~(U) > C , C > EC(U) is monic).
Consider the diagram in fig. 2 below,
EU,Uj E U UjnU k E(U) > E(Uj) J'
gc (u) T
u,< ] P(C T
P r "CT
> E . (u. ~1% T j 0
I U* j r J
~P(Tj)
®
E~ ,~o~ © hj
> E ujnu k)
PTjnUk~ ~) ~ j * T k
7E ~u (u'nuk)- ~ : . ~ ; , ~ , (u nu k) I ® ~U*T j *Tk U~Jn Uk P' (X)
> P(~j[~U k) >P(~j*~k)
Fig.2
= ~ PVp, VpAU k where: j,k e I , hj pe~j~ , X : Jj xJ k > J.j is the projec-
tion. Area O obviously commutes; areas (~,Q commute since the diagram
in fig.l is commutative; areas Q , Q ,Q commute by definition of the
various E maps. Taking products, we obtain the commutative diagram be-
low, where Q-®, ( 9 - O have bee merged, = )< u*. T1 ' = ~I U* " = k~e and the last horizontal map is i ~j.Tk ' P~ j, I PTj.Tk
easily seen to be fP (indUced by restriction maps). CT
Consider the diagram in fig.3 and the similar diagram with
g's instead of f's. Since all functors ~(U) ----> C under considera-
(,1
E(U)
PC T
E C
u ~- Cq-
P( fl '
C, r
P (Cn.~.r)
E E uc > E(C) fc
T U
U) ~ ~ ie~ ETi(Ui)
') P(C~) CT
Fig.3
> E(C.C)
P ('r j . ' r k )
~" p(cT.C ~) ]
" is a monomorphism; so is ~ ; therefore tion are monic, PT
Equ( E , E , P -- P - P P fop T , gCp T ) = Equ( FCTU , gcTU ) However, Equ( fCT ' g¢) =
Im u* = Im U u • Since ~ is also a monomorphism, it follows that C T T
E , E , P -- P -- Equ( f~PT ' gCp ~ ) : Equ( fcTU , gcTU ) = Im UT
Taking the colimits over T, we obtain from fig.3 a commuta-
tive square
E u c E(U) > E(C)
E(U) ~ colim iel~ E i ( U i )
where t i s a isomorphism, and the other vertical map is the identity
since the C T form a cofinal subset of {(U) Then it follows from
1.3, 2.7 that \/ Im = i and ~eT PCT
E E , Equ(~,gc ) : k/ (pT)s Equ ( E , TeT fcPT ' gcPT ) =
TYT ' = Im E = Im u~ : Im PTUT T~T UcPc~
62
E Since we already know that u
C
that E is a sheaf.
is a monomorphism (by 3.1), this proves
4.Putting the two lemmas together, we obtain:
Theorem 3.3. If C is a C 4 regular category, then for any
topological space X, ~(X,C) is coreflective in ~(X,C), and Heller
and Rowe's construction yields the coreflection in at most two steps.
In particular, the same is true if ~ is any C 4 abelian ca-
tegory (and it is a new result in that case, too, as far as we know).
The following result was shown incidentally (proof of 3.?):
Proposition 3.4. Let P be a monopresheaf and P ~ P be
its associated sheaf. If F is a sheaf and P ~ F is a monomorphism,
then (when G is C 4 regular) P > F is also a monomorphlsm.
We tried to arrange the proof of the theorem so that we
could look back and try if we could get rid of the regularity in the
proof. Except for the manipulation of subobjects at the end of the
proof, we have used only: completeness, existence of directed colimits,
preservation of monomorphlsms by directed colimits, ~i ' 52 (through
the result of Gray we quoted) and the conclusions of ~.8 in the cases
when C[ is not assumed. We do not knew whether the manipulation of
subobjects at the end of the proof can be bypassed, but it can clear-
ly be replaced by an g2-1ike conditlon.[In all fairness, we must add
that regularity occurs so often that it is not clear to us why one
should wish to do without it in that theorem].
5. We now use these results for a more detailed study of
$(X,C) when G is C 4 regular.
First, g(X,C), being coreflective in ~(X,C), is also com-
plete and cocomplete. We show that it is regular. Throughout
Cp : P ) P denotes the coreflection of P e P = P(X,C) in $ = ~(X,C).
First the kernel pair of f e ~ is the same in ~ and in
63
~, and it follows that f is a monomorphlsm of 3 if and only if it is
a monomorphism of ~ • Next, the regular epimorphisms of 3 are given by
the following:
Lemma 3.5. Let f e 3 have the regular decomposition f = mp
in @. Then f is a regular epimorphism of ~ if and only if ~ is an
isomorphism.
Proof. First assume that N is an isomorphism. Then a,be 5,
af = bf implies am = bm , ar~ = ~ = bN = b~ and a = b Now let
fx = fy be a pullback, and gc 3 be such that gx = gy • Since
px = py is also a pullback (in @) and p is a regular epimorphism,we
have g = tp for some t. Then also g = ~ and since N is an isomor-
phism g also factors through f, uniquely since f is an epimorphism
as shown above. Therefore f is a regular epimorphism.
For the converse, let M be the domain of m. Then m = ~CM,
= cMp • In particular c M is a monomorphism, and, by the first part
of the proof, ~ is a regular epimorphism (of 3) If ~ is an isomor-
phism, then f must also be a regular epimorphism.
Since @ has regular decompositions (it is a functor catego-
ry and ~ is regular), this argument also shows that any morphism of
3 has a regular decomposition in 3.
To show that ~ is regular, it now suffices to verify our
pullback axiom (III). We first prove the following: if
m M ~ O
r'[ lq
N ---5--~n K
is a pullback in @ where G , K are sheaves and m , n monomorphisms,
and if ~ is an isomorphism, then ~ is an isomorphism.
Since M is a monopresheaf, then Heller and Rowe's construc-
tion gives M , hence also ~, in one step, from the diagram
6 4
G u C o ( u ) ~ o(c)
/ I~(U) \ ~c(U) |m , m(U)// ~ ~ u~ ]
/ /cM(U) ~.c(U) > M(c)
G fc
M fc
> O (C-C)
'I ! !
M(c~c)
where C = (Ui)ie I e {(U) , m' = i~l m(Ui)
and the similar diagram with g's instead of
induced on equalizers.
We have a similar diagram for N
F
m" = ~eI m(UjNUk) j,
f's, where ~c(U)
n K for which we use
instead of E. Putting both diagramstogether yields
G G u C G(U) ~ O(C) - fc
. m,/' I M(U) ' ' ~-TCM(C) fC
" M
uc ] q '
q(U) K ?'
u C /~K(U) ,K(C)
N u C
r C
r (U)
: G(C.C)
K fc
m/ M(C*C)
I"
r t! , K(C*C)
/ N(C*C)
where q' = X q(U i) etc., and where r C is induced on equalizers iel
(considering the similar commutative diagram with g's instead of f's).
Since mr = qn is a pullback, m(Ui)r(U i) = q(Ui)n(U i) is
a pullback For all 1, whence m'r' = q'n' m"r" = q"r" , are pullbacks
65
and induce on equalizers a pullback ~c(U) r e = q(U) ~c(U) • It follows
that q(U) s Im ~c(U) = Im ~c(U)
If now we assume that ~ is an isomorphism, then, by 1.3,
~u) Im ~c(U) = ~(U) ( ~/( Im pc) = ~(U) I = 1 ; then it follows C s C U) s
from (C~) that ~/ Im ~c(U) = I and the similar calculation cc¢(u)
shows that ~(U) s I = i , so that ~(U) (already a monomorphism by 3.4)
must be an isomorphism.
Thus we have proved that, in our pullback mr = qn , if
is an isomorphism then so is ~. Let now fg' = gf' be any pullback
in ~. If f = mp is a regular decomposition of f in ~, then we can
find pullbacks mh = gn , pk = hq such that g' = k , f' = nq ~ since
is regular, f' = nq is a regular decomposition of f' in ~ • Then
we note that mh = gn is a pullback which satisfies the requirements
for what we just proved; then it follows from 3.5 that, if f is a re-
gular epimorphism of 5, then ~ is an isomorphism, hence so is ~,
and therefore f' is also a regular epimorphism of 5. This completes
the proof of:
Proposition 3.6. If G is a C 4 regular category, then for
any topological space X, ~(X,C) is a regular category.
6. We can prove much more under the further assumption that
C is C~ .[There ought to be a way to make do without that condition
(it is not needed in the abelian case; see [18],[4])]. Then we can pro-
ve that the stalks behave as they should. The basic result is:
Lemma 3.7.1f C is a C 4 , C~ regular category, then the
stalk functor reflects isomorphisms.
[We are indepted to Van Osdol for the remark that this means
that ~ is cotripleable under ~ .]
Proof. The stalk functor S :~ ~ C X sends each sheaf F
Into (Fx)x¢ X • Ideutlfylug X with the discrete category on X , we
66
if f e $ and
isomorphism.
Put
~ ( u ) : X F X , x e U
tion maps; they come with monomorphisms m : F
and n :G > ~ , n(U) = (Gu, x)xe U such that
~(U) = ~ f : ~(U) > ~(U))(see, e.g., [18] xeU x
see that G x : iX,8] is a regular category with pointwise limits, co-
limits and regular decompositions. In particular, we have to prove that
fx is an isomorphism for every x¢ X , then f is an
f : F ~ G and let F , G be the sheaves defined by:
~(U) = ~ G with the projections as restric- XE U x '
F , m(U) = (Fu, x)x~ U
nf = ~m (where
for the details).
We shall prove that, when fi is a monomorphism, then
nf = ~m is a pullback. First we note that, by 2.8,
F(U) = X collm F(V) ~ colim X F(Tx) xeU xeV~U TeT xeU
where this time T is the set of all mappings which assign to each
xeU an open set
by all ~ F x~U T x , x
t i o n off ~ ( U ) .
F o r e a c h ~ e T
a commutative diagram
• x with x e ~x=U , and the isomorphism is induced
: ~ F(~x) > ~(U) . We have a similar descrip- xeU
we have a covering C = (TX)xe U e ~(U) and
F(U)
f(U)
G(u)
F
u c
G u C
F(C)
~ G ( e )
F fc ) F(C.C)
f.
fG c ~ c(c~c)
where f' = ~ f(~x) , f" = ~ f(Tyn~z) There is a similar dia- xeU y,zeU
gram with gC's instead of fc's . If f is a monomorphism, then so is
f" and since F,G are sheaves it follows from 1.5 that
Im u F Equ(f"f , gcj = Equ( f' , gcf') :
67
,-i, o,-i o y,s O)-1~O) = = AS ( f ~gc ~ f f') = As ((gc ~C
= f,s /ks ((gO)-ifOc) = f,s Im ucO •
Since f(U) , f' , u F , u o are monomorphisms, it follows that
O P u e f ( U ) = f ' u c i s a p u l l b a c k .
Going to the colimit over T we obtain a pullback
F(u) ---~Y(u)
f(U) I [ O(U) ~ ~ ( u )
which is easily seen to be but n(U) f(U) = ~(U)m(U) . It follows that
nf = ~m is a pullback (when f is a monomorphism).
The lemma follows at once: indeed, if S(f) is an isomor-
phism, then so is ~ ; then nf = ~m shows that f is a monomorphlsm;
therefore nf = ~m is a pullback and f must also be an isomorphism.
The obvious application of the lemma is as follows. Let
(ai)lel be a cocompatible family for some diagram for instance of
sheaves; assume that, for every xeX , (a~)le I is a colimit of the corresponding diagram of stalks at x. If a is the morphism of sheaves
induced by (al)lel from the colimit, then since S preserves colimlts
a x is an isomorphism for every x ; therefore a is an isomorphism, and
(ai)lel is a colimit of the diagram. This is what is meant by: I! coli-
mits in ~ can be safely computed on the stalks". The same applies to
finite limits, and to regular decompositions, since these are also
preserved by S.
It is then clear that directed colimits in ~ will preserve
finite limits (hence also monomorphlsms); for this holds in G. We have
just finished proving:
68
Theorem 3.?. Let C be a C 4 C* regular category. For any ' i
topological space X, ~(X,C) is a regular category, and all colimits,
finite limits and regular decompositions in ~(X,C) can be safely com-
puted on the stalks.
We also have shown incidentally that the coreflection
P(X,G) ~ ~(X,q) (which commutes with the stalk functors (see [18])
preserves colimits, finite limits and regular decompositions.
References
[I] M. Barr. Coequalizers and free triples. Math. Z. 116 (1970) 30?-322.
[2] M. Barr. Non-abelian full embedding, I. (to appear)
[3] M. Barr. Non-abelian full embedding, II. (to appear)
[4] J.W. Gray. Sheaves with values in a category. Topology 3 (1965)
1-18 [also, Notes, Columbia University (1962)].
[5] P.A. Grillet. Morphismes sp@ciaux et d@compositions, C.R. Acad. Sci.
Paris 266 (1968) [email protected] 397-398; Quelques propri@tSs des cate-
gories non-ab$1iennes, ibid. 550-552; La suite exacte d'homolo-
gie dans une cat~gorie non-ab$1ienne, ibid. 604-606.
[6] A. Grothendieck. Sur quelques points d'Alg~bre homologique. Tohuku
Math. J. 9 (1957) I19-2~I.
[?] A. Heller and K.A. Rowe. On the category of sheaves. Amer. J. Math.
84 (196~) 205-216.
[8] P. Hilton. Categorles non-abellennes. Notes, Universit$ de ontreal
(1964).
[9] J.R. Isbell. Subobjects, adequacy, completeness and categories of
algebras. Rozprawy Mat. 36 (1964) 32pp.
[10] J.R. Isbell. Structure of categories. Bull. Amer. Math. Soc. 72
(1966) 6~9-655.
[11] G.M. Kelly. Monomorphisms, epimorphisms and pullbacks. J. Austral.
Math. Soc. 9 (1969) 124-142.
[12] J. Kennison. Full reflective subcategories and generalized cove-
rings. IIi. J. Math. 12 (1968) 353-365.
[13] F.V. Lawvere. Functorial semantics of algebraic theories. (Doct.
69
Diss.) Columbia University (1963).
[i4] S. MacLane. Groups, categories and duality. Proc. Nat Acad. Sci.
USA 34 (1948) 263-267.
[15] S. MacLane. An algebra of additive relations. Proc. Nat. Acad. Sci.
~SA 47 (1961) I043-I051.
[16] S. MacLane. Homology. Springer, New York, 1963.
[17] E.G. Manes. A triple miscellany: some aspects of the theory of al-
gebras over a triple (Doc. Diss.). Wesleyan University, 1967 .
[18] B. Mitchell. Theory of categories. Academic Press, 1965.
[19] D. Puppe. Korrespondenzen in Abelschen Kategorien. Math. Ann. 148
(1962) 1-30.
[20] D.H. VanOsdol. Sheaves of algebras (to appear).
KANSAS STATE UNIVERSITY
BIFIBRATION INDUCED ADJOINT PAIRS
I Marta C.Bunge
Received November 4, 1970 and December 21, 1970
It i s well known (cf. Kan [63, L a w v e r e [73) tha t a f u n c t o r
Se t s f : Se t s B ~Sets A induced by c o m p o s i t i o n wi th f :A ~__B p o s -
s e s s e s both a lef t and a r i g h t ad jo in t p r o v i d e d tha t the c a t e g o r y A
be s m a l l . M o r e o v e r , t h e r e a r e l i m i t f o r m u l a s fo r c o m p u t i n g t h e i r
v a l u e s . An a l t e r n a t i v e d e s c r i p t i o n of the le f t ad jo in t ~ f has been
provided by Tierney ill]; his idea is to view functors as fibrations
of some sort and make use of well known constructions of the theory
of fibred categories (cf. Gray [2]). More precisely, let Qf:(f, _B)--~B__
be the 0-fibration (or opfibration) which best approximates f via a
functor @f: A ) (f,B) ; this functor is the right adjoint to the canonical
projection Pf:(f, B__) )A. Tierney's prescription for obtaining the
left Kan extension of a given functor F.A )Sets reads as follows:
(i) associate with F its corresponding 0-fibration QF over A ; (ii)
pull back QF along Pf; (iii) compose with Qf; (iv) make discrete in the
best possible way the fibres in the composite. The discrete 0-fibra-
tion over B so obtained has ~f(F):~ ) Sets as its functor counterpart.
1 R e s e a r c h p a r t i a l l y s u p p o r t e d by the N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a
unde r G r a n t No A7255.
71
I show that the above construction can be made part of a general
s c h e m a invo lv ing a b i f i b r a t i o n and a pa i r of ad jo in t f u n c t o r s a s s o c i a t e d
with it. I proceed to describe the categories and functors involved.
If C is any ca t ego ry , denote by (Cat, C) 0 the fu l l s u b c a t e g o r y of (Cat, C)
whose ob j ec t s a r e the sp l i t n o r m a l O - f i b r a t i o n s over C. C o n s i d e r a
p a i r (A, B) of c a t e g o r i e s and a (1, O ) - b i f i b r a t i o n (P, Q) over th i s p a i r
0 (Cat, A) 0 ~ (Cat, B) 0 by (cf. G r a y [2 ] ) . Define a f unc to r NIp, Q:
m i m i c k i n g the c o n s t r u c t i o n of Y'f above . Tha t is , on ob jec t s , the va l ue
0 of Mp, Q at a given QI._EI *A is obtained by first pulling back Q1
along P and then composing with Q. There results a o-fibration over
B with arbitrarily large fibres. If we were to insist that this construction
should preserve smallness of the fibres we would need the assumption
that Q itself has small fibres. If the bifibrationin question is (Pf, Qf),
arising from an f in the manner indicated before, this requirement is
surely met if A is small. A functor in the opposite direction is cons-
tructed by making use of the rest of the bifibration structure represented
by the functor (P,~): A__ -" ((Cat, B__)0)°P , whose rule at an object .A as-
signs the 0-fibration over B obtained by restricting Q to the fibre of P
above A. Given QI: El ~B_.__ z, a 0-fibration, denote by H Q1 the functor
((Cat,__B)0°P ~ Cat. The value of a functor N;, (-, Q1 ):
Q:(Cat, B) 0 >(Cat,A) 0
is defined to be the left vertical arrow in the pull-back diagram below:
pull-back ~ E(HQ1)
A (P' Q) ((Cat, B)o)OP
72
T h i s f u n c t o r t a k e s f i b r a t i o n s w i t h s m a l l f i b r e s o v e r B i n t o s i m i l a r o n e s
over A if and only if the functor H Q1 has small categories as values. If
B i s s m a l l , t h i s i s t h e c a s e . In s o m e i n s t a n c e s of t h e a d j o i n t n e s s , t h e
f u n c t o r j u s t d e s c r i b e d c a n b e o b t a i n e d w i t h o u t t h e r e q u i r e m e n t t h a t B
s h o u l d b e s m a l l . 0
For example, Npf, Qf c a n b e d e s c r i b e d m o r e s i m p l y
a s t h e f u n c t o r i n d u c e d b y p u l l i n g - b a c k a l o n g f ( w h e n f i b r a t i o n s a r e d i s c r e t e ) .
H o w e v e r , o t h e r e x a m p l e s a r e a v a i l a b l e t o s h o w t h a t t h e r e s t r i c t i o n i s ,
in general, necessary.
0 0 The schema refered to above says that h/[p,Q is left adjoint to Np, Q.
By imposing certain restrictions on this basic situation it is possible to
recover ~f -¢ Sets f, but also the dual situation Sets f -~ ~f, as well as
other familiar examples. In order to do so, one must restrict the fibres
in a suitable way, for example, so that they are all discrete, or preorders,
or groupoids, or so that they contain at most one "point". A new variable
X is brought into the schema for this purpose: it stands for a category of
small categories subject to two requirements (i) in order that the right
adjoint of the pair may be restricted to categories of fibrations of type _X,
t h e f o l l o w i n g m u s t h o l d : g i v e n Q a n d Q1 o v e r B s u c h t h a t Q1 i s of t y p e X ,
t h e n (Q, Q1 ) m u s t b e a n o b j e c t of X_; ( i i ) a l e f t a d j o i n t t o t h e r e s t r i c t e d
0 N p , Q e x i s t s i f X i s r e t l e c t i v e i n ( C a t ) s , t h e c a t e g o r y of s m a l l c a t e g o r i e s .
A l l t h e e x a m p l e s p r e v i o u s l y m e n t i o n e d s a t i s f y t h e s e t w o c o n d i t i o n s .
C h o o s i n g X to b e t h e c a t e g o r y o f ( s m a l l ) d i s c r e t e c a t e g o r i e s , b o t h
Kan extensions result with appropriate choices of bifibrations. The bifibra-
f t i o n w h i c h y i e l d s t h e p a i r ~ f - I S e t s , i s , a s i n d i c a t e d b e f o r e , t h e p a i r
(Pf, Qf) over (A, B) . If Pf:(B,f) ) B t o g e t h e r w i t h t h e p r o j e c t i o n
Qf:(_B,f)- ~A_ is chosen as the bifibration (over the pair (]~,A)),the
73
r e s u l t i n g a d j o i n t p a i r i s p r e c i s e l y S e t s f -~ [If. F r o m t h e s w i t c h i n g of t h e
r o l e s of _A and __B in t h i s e x a m p l e i t i s c l e a r t h a t , q u a f u n c t o r of t h e t y p e
0 N p , Q, [If e x i s t s w h e n e v e r A i s s m a l l . T h i s f a c t a g r e e s w i t h t h e u s u a l
r e q u i r e m e n t a n d i m p l i e s t h a t to r e q u i r e t h a t b o t h A and ]5 be s m a l l in t he
g e n e r a l c a s e i s no t t o o r e s t r i c t i v e .
O t h e r c h o i c e s of _X p r o v i d e t h e v a r i o u s e x a m p l e s of h y p e r d o c t r i n e s
c o n s i d e r e d by L a w v e r e ([9--3, [1.03). A l S o , t h i s a p p r o a c h i s p a r t i c u l a r l y
s u i t e d t o a d i s c u s s i o n of t h e c o m p r e h e n s i o n s c h e m a (c f . a l s o G r a y [33) .
T h e c o n t e n t s of t h e p a p e r a r e a s f o l l o w s : i n § l a n d ~3 b a s i c f a c t s a b o u t
f i b r a t i o n s and t h e i r m o r p h i s m s a r e d i s c u s s e d and n o t a t i o n e s t a b l i s h e d ; in
2 t h e f u n c t o r s w h i c h a p p e a r in t h e a d j o i n t n e s s s c h e m a a r e f o r m a l l y i n -
t r o d u c e d ; a d j o i n t n e s s i s e s t a b l i s h e d in.~4 and ~5; t h e c a s e of s p e c i a l t y p e s
of f i b r a t i o n s i s i n v e s t i g a t e d in § 6 ; f i n a l l y t h e l a s t p a r a g r a p h i s d e v o t e d
to e x a m p l e s .
1 • A REVIEW OF FIBRED CATEGORIES
In this section we recall briefly just those portions of the theory of
fibred categories (cf. Gray [2,3];Grothendieck [5]) most needed in this
paper. It is our purpose as well to establish a notation.
Given a category B (locally small), a functor Q:E ~B is called a
0-fibration (or opfibration) if, for any b'B > B' 6B_B - there is a functor
h$:_EB-----*_EB, and a natural transformation eb:J B ~ JB' ° b (here
_EB=Q-I(B ) and JB:_EB )_E i s the i n c l u s i o n f u n c t o r ) , s a t i s f y i n g : Q(eb) =b
and, if e:E ~E' is such that Q(e)=b then there exists e-(b..E) ~E'
u n i q u e w i t h t h e p r o p e r t i e s :
74
(i) e = eo(8 b )E and (ii) Qe= idB,. (For a more elegant description
in terms of comma categories, cf [2, 3].)
Remark: it will be part of the definition of fibration the require-
ment: for each BCB, the category E_ B is small. (This does not imply
that E itself be small, unless B is small.)
A cleavage is a choice of the functors and natural transformations
above. It is called a split if (b'b)~:. = b', ob., and normal if (idB)~.: =
id(EB) •
Let Q and Q be split normal 0-fibrations. Consider a functor T
in a commutative diagram:
Since both Q and Q
for each b:B
T ~E
B
B' , of a natural transformation ~'b:b.~...T
have cleavages one can deduce the existence,
satisfying;
(I.I) rbOSbT = T@ b and
(1.2) Q~'b = idB'
Since the cleavages for Q and Q are split normal one concludes,
furthermore, that
(1.3) rb, b = ~'b,b. ob',.~. ~'b , and
75
(1.4) r(idB ) = id(TB) , where T B is the restriction of T to the
fibre above B.
W e a r e n o w r e a d y t o d e f i n e t h e n o t i o n of a c l e a v a g e p r e s e r v i n g
f u n c t o r . T h e f u n c t o r T , a s a b o v e , i s s a i d t o b e c l e a v a g e p r e s e r v i n g
if each ~'b is the identity natural transformation. If each rb is just
an equivalence, T is called cartesian. The category of split normal
0-fibrations over ]3 with cleavage preserving functors is denoted
S p l i t 0 B . On t h e o t h e r e x t r e m e i f a l l f u n c t o r s T w i t h Q o T = Q a r e
allowed as morphisms, the category of split normal 0-fibrations over
]3 they determine is denoted by (Cat, B) 0. It is a full subcategory of
(Cat, B), the category of objects (of Cat) over _B. There is a canonical
functor from Split0B - to (Cat, B__)0.
Let us recall some very useful properties of fibrations. They are
p r o v e d i n ~2~:
(i) given 0-fibrations QI:E_I >B__ and Q':E_' )El, the
composite QI°Q': E' ) B_ is a 0-fibration.
(2) given any functor f:A_ ) B and a 0-fibration Q:E )B_,
pulling-back along f,
f * E '~ E
f*Q pull-back
A.
f
produces a 0-fibration f~Q:f~E ~ A .
Q
76
t o r s
F ix ing Q1 and f, as above, the above operations determine func-
Q I " - : (Ca t ' - -E l )0 ) (Cat'---B)0 '
f;:" : (Ca t , B )0 > (Cat ' -A)0 "
and
Another useful fact about fibrations is that they correspond to
functors whose values are categories. Since we work onlywith fi-
brations whose fibres are small, the equivalence takes the form
Split0B-- ~ (Cat)---Bs
where (Cat) denotes the category of small categories, itself an object S
of Cat {cf. [7_] and [_8]). We recall that the split normal O-fibration
QG:EG_ )__B which corresponds to a functor G:B_ ;(Cat)s is
given as follows. The objects of E are pairs (B, Y) with BEB and --'G
Y6GB. A morphism (B,Y) )(B',Y') of_E G is any pair (b,g) with
b:B > B ' E B and g:Gb(Y) .. ) Y ' . C o m p o s i t i o n of p a i r s , when
defined, is given by (b',g')o(b,g) = {b'b,g'oGb'(g)). The identity of
(B ,Y) is the p a i r (idB, i d y ) . With the r u l e s (B ,Y) J ) B ; ( b , g ) f >b
a functor QG:__EG ) B results for which there is a canonical cleavage,
split normal because G is a functor.
In the other direction, given a split normal O-fibration Q-E__ ) B,
the functor B ) (Cat) which corresponds to it in this equivalence is -- S
given by B I )E B_ ; bJ ~b,~..~ . Notice that if all fibres of Q are
77
d i s c r e t e , t h i s f u n c t o r f a c t o r s t h r o u g h the i n c l u s i o n of ( s m a l I ) S e t s
( i . e . , d i s c r e t e s m a l l c a t e g o r i e s ) i n t o (Ca t ) . C o n v e r s e l y , to a S
functor G:B ------9Sets c (Cat) corresponds a fibration QG:EG ~B -- S -- --
all of whose fibres are discrete.
Cleavage preserving functors and natural transformations are
equivalent notions under the above correspondence. More explicitly,
if T:E )E is a cleavage preserving functor, the fact that each
fb is the identity says that all diagrams of the form:
T B - -B ~--EB
_E B, UEB , T B ,
b.
a r e c o m m u t a t i v e . C o n v e r s e l y , g i v e n t : G ) G a n a t u r a l t r a n s -
formation, one can define a cleavage preserving functor T:E~ )E G
b y t h e r u l e s (B,~() ! ) (B, tB (Y) ) ; ( b , ~ ) I ){b, t B , ( g ) ) .
We shall omit a discussion of l-fibrations as it is simply a no-
tion dual to that of a 0-fibration. We shall use the following notation:
a cleavage for a l-fibration P:lE ) A w i l l be d e n o t e d by {a.,.,@ }, - - " " a
>IEA and {9 :jAoa. )jA' where, if a:A )A' EA, a,:lE A' - - - - - - a ~'<
S p l i t n o r m a l now m e a n s : (a'a)~.. = a.oa',,,. .-,.- a n d ( i d A ) . = i d ( E A )
Let us now turn to the definition of (i, 0)-bifibrations, as given in
[3].
78
P Q A pair of funcflors A q E ) B is called a (l,0)-bifibration
over A, B if:
(i) P is a l-fibration and Q is a 0-fibration;
(ii) P lIB is a l-fibration and Q I [A is a 0-fibration, for each
B 6 B , A 6 A ;
( i i i ) t he i n c l u s i o n f u n c t o r s j A j B a r e c a r t e s i a n f o r s o m e c h o i c e
of c l e a v a g e s , In f a c t , t h i s c o n d i t i o n i s s u p e r f l u o u s as w e s h a l l a l w a y s
be interested in split normal bifibrations, i.e., such that
(iv) each of P, Q, PIE__ A , Q IE__ B is split normal;
(v) the functors jA, jI3 are cleavage preserving;
(vi) for any a:A ~A'EA b:B )B'6B, the functors a,:
E A' ~ E A and b,: E B )E are cleavage preserving.
There exists (cf [3]) an equivalence of categories
Split(l ,0)(A,B) ~ (Cat)s(A---°P×B)
with a suitable notion of cleavage preserving functor. However, unless
Pv QV A, B are small categories, the bifibration __A ( E V ) ]3 which
corresponds to a functor V:A°P× B ~ (Cat) (even if it factors
through Sets) need not have PV ~ QV with small fibres. In the follow-
ing examples we shall assume bothA and B small.
Given any functor f:A )__B (between small categories) define
Vf:A°Px_ __B )Sets c (Cat)s by Vf(A, B) = Horn B(fA, B), Vf(a,b) =
HOmB(fa, b). By Gray's basic construction (applied to locally discrete
79
2 -ca t egor i e s ) one obtains a bif ibrat ion
diagram:
Pf
A w
B ]3
Qf )B as in the
where the square is a pull-back. We are simply saying that ~_f
comma category (f,___B) and that Qf is the best approximation to f:A
by a 0-fibration: in fact the functor
A_ ~f ~(f,B/
B
is the
)B
which has the required universal property rendering the functor ( -, ]3):
(Cat, B__) )Split0B a left adjoint to the forgetful, is precisely the
right adjoint to the projection Pf:(f, ]3) )A__. From the way the pair
(Pf, Qf) was obtained we now know it is a bifibration, and as such will
the associated fibration construction be of interest to us in this paper.
Dually, the best l-fibration approximating f'A >]3 is the
bif ibra t ion ar i s ing f r o m the functor v f=HomB(- , f-):B__°PxA ) S e t s c ( C a t ) s.
80
It shall be denoted by
Pf E f Qf B ~ > i
T h e s e two e x a m p l e s wi l l be u s e d l a t e r on, when d e r i v i n g Kan
e x t e n s i o n s .
R e m a r k that even for a func to r V:A°P× B }Se t s , the c o r r e s -
ponding b i f i b r a t i o n ( P v ' QV ) need not have PV or QV d i s c r e t e . The
p r e v i o u s e x a m p l e s a r e enough i n d i c a t i o n of t h i s .
2. BIFIBRATION INDUCED PAIRS
From now on, we shall assume that A and B are small categories.
Fix a (1, O)-bifibration A(
(P, Q): A °p > (Cat, ]3) 0
P Q z__ ~B
the func to r : A I
over A , B . Denote by
~OIE A : a~ >a,.
Similarly, one can define a functor (P,--~)'B )(Cat, A) l , given by:
B I ~ PI_EB; b I >b..,,. Recall the definition of bifibration to verify
that these are well defined.
We remark that, in fact, the first one has its image in Split0__B a
(Cat, B)0 , while the second one in SplitlAc(Cat, A) l . Although we shall
81
r e s t r i c t our a t t e n t i o n to c a t e g o r i e s of 0 - f i b r a t i o n s in t h i s p a p e r , i t
should be clear how to dualize in order to obtain analogous results
fo r c a t e g o r i e s of 1 - f i b r a t i o n s .
F o r a g iven s p l i t n o r m a l 0 - f i b r a t i o n QI:G__I } B , l e t
HQI : [ (Ca t , B)0 ] ° p >(Cat ) be t he f u n c t o r wh ich : - - S
(i) to an o b j e c t Q ' E ( C a t , B)0 a s s i g n s the c a t e g o r y HQI(Q ') w h o s e
o b j e c t s a r e t he f u n c t o r s T in
T E '
B
~E__ 1
commutative and whose morphisms c¢:T I
formations with the property Q i ff = idQ, ;
(ii) given a morphism S:Q'
T 2 are natural trans-
Q"E [(Cat, B)0 3°p, i. ) e . ,
S E_" ~_E,
B
commutative, HQI(s):HQI(Q ') j HQl(Q '') is the functor induced
by composition with S. Notice that HQI(s)(ff)= ecS. Thus,
QI(ffS) = (Qlff)S = idQ, S = idQ,,.
Note that we have defined a functor H Q1 with values in (Cat) . S
Let
Q':E__' >_B_B be any fibration (wiL.h small fibres). Since __B has a set of
82
E I o b j e c t s and IE__' I i s a u n i o n of the s e t s I _ I3 I i n d e x e d by 1t3], E__' ha s
a set of objects too. The objects of HQI(Q ') are functors T:E' >E l .
Since both categories are small there is at most a set of them.
Thus, there is a split normal 0-fibration (with small fibres) cor-
r e s p o n d i n g to e a c h H Q1, d e n o t e d
Q(HQ1) : E---(HQ1 ) [(Cat, _B)O ]°P.
(Of course, since (Cat,_B) 0 is not small, (HQI)
The objects ofE__(HQI ) are pairs (Q',T') with Q':E'
and T':Q' ) QIE(Cat, B) 0. A morphism (S, C¢):(Q',T')
is such that S:Q'
transformation,
is not small either. )
. ) B E (Cat , __B) 0
The functor Q is given by the (HQI)
Remark (to be quoted later): if all functors involved are cleavage
preserving, there is a similar fibration over [Split0B] °p, which we
: ) (Spl i t0B) op. shall "also" denote by Q(HQI ) -E_(HQI)
The correspondence Q1
H: (Cat , B_) 0
as f o l l o w s . To any f u n c t o r R
| ) Q(HQ1 ) extends to a functor:
) (Cat, [(Cat, Bl0]°P) 0
in
R _El >_ZZ
given by (S',(Y')o(S, fy) : (S'S, C¢'o(yS').
rules: (Q', T') ! )Q'; (S,(y)I > S.
>(Q", T')
)Q"E[(Cat,__B)0]°P and C~:T'oS >T" is a natural
satisfying QIC~ = idQ,,. Composition, when defined, is
83
c o m m u t a t i v e , t h e r e c o r r e s p o n d s a n a t u r a l t r a n s f o r m a t i o n HR:
H Q1 ) H Q2 w h o s e Q ' - c o m p o n e n t i s d e f i n e d by c o m p o s i t i o n
w i th R. S ince c o m p o s i n g wi th S and c o m p o s i n g w i th R a r e c o m m u t i n g
o p e r a t i o n s (one i s on the le f t , the o t h e r on the r i gh t ) , H R i s i n d e e d
n a t u r a l . We l e t H(R) be the c o r r e s p o n d i n g ( c l e a v a g e p r e s e r v i n g )
m o r p h i s m of f i b r a t i o n s .
(2 .1) We de f ine a f u n c t o r
0 : (Cat , B) 0 Np, Q ) (Cat,A) o
r e q u i r i n g t ha t i t be the c o m p o s i t e :
(Cat , B_) 0 H A *
)(Cat, [Cat,__B)o]°P) 0 (P'QI(cat, A)O,
A .,.
w h e r e (P, Q)" d e n o t e s " p u l l i n g b a c k a long (P, Q)" .
Let us be more explicit as to the definition of N O in ,Q , at least
0 on the o b j e c t s : g i v e n Q1E(Cat ,B_)0, N p , Q(Q1 ) i s the l e f t v e r t i c a l a r r o w
in the pull-back diagram:
pull-back ~" iH Q i)
° I ~ [(Cat, B) 0 ]op
(P,Q)
(2.2) We now define a functor in the opposite direction,
M ° ( C a t , A ) o ~(Cat , B_) o , Q :
by letting itbe the composite
84
p* Q o _ ( C a t , A) 0 ) ( C a t , E__) 0 ) ( C a t , B ) 0 ,
i.e., first pull-back along P and then compose with Q. Explicitly,
for an object QI:EI )A of (Cat, A_)0, M Q(QI ) is depicted in
t h e d i a g r a m b e l o w , w h e r e t h e s q u a r e i s a p u l l - b a c k :
pull-back
Q P
From now on all our efforts will be directed towards showing
that the functor s 0
M (Cat, A)0 ( P' Q
0 Np, Q
> (Cat, 13)0
are adjoint functors.
3. QUASI-NATURAL TRANSFORMATIONS
Similar to the correspondence be£ween cleavage preserving func-
tors and natural transformations, there is a correspondence between
morphisms of (Cat, 13)0 and what we shall call "quasi-natural trans-
formations"
In fact, for functors G,G:B > (Cat) , a quasi-naturaltrans- -- S
formation is precisely what is called a "2-natural transformation"(cf [3])
85
provided one regards 13 as a locally discrete 2-category and (Cat) -- S
as a 2-category. Even so, a direct description without resorting
to 2-dimensional notions is preferable for our purposes and we shall
give it below. We realize, of course, that it is the fact that (Cat) S
is really a 2-category which makes our definition meaningful. However
we do not need to assume the same of B__, and we shall not.
Def. Let G:B ~(Cat) and G:B. )(Cat) be any two func- -- S -- S
tors. A quasi-natural transformation t:G ~G is a pair t=(T,r)
with T=[T B] a family of functors TB:GB ~ GB , indexed by the
objects BEB, and r=(Tb) a family of natural transformations r b
GboT B ,- ~ TB,OGb , one for each morphism b:B B'EB, satisfying
q . n . t . ( 1 ) r b , b = r b , [ G b ] o [ G b ] r b , and
q.n.t.(2) r(idB ) = id(TB).
(These conditions shall be referred to as "the coherence conditions
for a quasi-natural transformation. ")
The following is a law of composition for q.n.t. : given t~T, ~'):
G , ~ G and s=(S,~):~ ~G define ts=(TS, T(~) by (TS)B=TBSB
and (r~)b=TB crbOrbS B. Then, it is not hard to show that ts:~ )G
is a q.n.t.. Composition is clearly associative.
Let us point out that any natural transformationt:G .}G gives
rise to a q.n.t, in the form t=(T,~') with TB=t B and rb=id. In parti-
cular, the identity natural transformation is also qua si-natural and a
unit for composition.
86
Denote by [B,_ (Cat)s]q.n.t. the category of all functors G:
B ~ (Cat) and q. n. transformations. -- S
Proposition . There exists an equivalence of categories
(Cat, B)0 -~ [B__,(Cat)s]q.n . t .
P r o o f . We shal l m a k e use of the s a m e o b j e c t - c o r r e s p o n -
dence as in the p r o o f of the e qu i va l e nce be tween Spl i toB and (Cat}B--s "
It fo l lows f r o m ~1 that given Q : E )B, Q : E >]3 and T wi th
T ~E
B
commutative, the pair (T, I") with T B the restriction
r B : ' --EB
and 1"b:b,T ) T ~ , wi th p r o p e r t i e s (1 .3) and (1 .4) , cons t i t u t e s a
q.n.t, t=(T,I"):G )G, where G,G are the functors B ~(Cat) -- S
corresponding to (~, Q .
Now, let t:G )G be a q.n.t., i.e., t=(T,T) satsifying q.n.t.
(1.2). Define a functor
T E~ -G '>-EG
B
87
as follows:
T(B,Y) : (B, TB(Y));
T(b,{) : (b, TB,(~)°(Tb)~).
Notice that, since Y6GB and TB:GB )GB,
Also, the second component of T(b,g) is the composite
TB(Y) 6 GB.
From q.n.t.(1) follows: T(id(B ' ~)) =T(idB, id~) =
= df(idB, TB(id f)[r(idB)]f ) =
- (idB, idTB(9 )) =
- idT( m ~)
Let us be given composable morphisms
(b, {):(B, Y) ) (B', ~(') and (b', { '):(B', "~') }(B", -'~")
of _E d .
In order to establish that T(b', ~')oT(b, ~)=T(b'b, ~' oGb'(g)), all
we need to show is the validity of the equation
TB,,(~')[(rb,)y,]oGb'[TB,(~)O(rb)~] = TB,,(~'oGb'(g))o(rb,5) ~.
Using, first q.n.t.{2) and then, the naturality of T b we get:
TB,,(g') °TB,,(Gb'({))°(Tb,b)- ~ = TB,,(g') °TB,,(Gb'(g)) ° [(rb,)~b(<f) °Ob'((rb)~)]
= TB,,(g')°[TB,,(Gb'(g))o(rb,)~b(~)_] °Gb'((rb) ~)
(rb) ~ TB,(g) Gb[TB(Y)] ) TB,[Gb(Y)] ) TB,(Y' ) , and thus
defines a morphism (B, TB(Y)) )(B',TB,(Y')) of E G, as required.
88
= TB, , (~ ' )o [ (~ 'b , ) f , ° G b ' ( T B , ( g ) ) ] °Gb'((Vb) ~)
= TB, , (~ ' )O(rb , )~ . ,o Gb ' [TB, (~) O(rb)~] ,
a s we w a n t e d .
T h e r e s t of the p r o o f i s now e a s y and i s l e f t to the r e a d e r .
R e m a r k . The u s u a l e q u i v a l e n c e Spli t0]3 m(Cat)Bs c o m e s out
as a corollary of the above, as can be seen by examining the proof.
In an analogous way, those q.n.t, for which each ~'b is an equi-
v a l e n c e a r e in one to one c o r r e s p o n d e n c e wi th c a r t e s i a n m o r p h i s m s
of fibrations. Let us refer to them as "weakly-natural transformations"
( w . n . t ) .
Another remark before closing the section is the following: for
any functor G:B )Sets C(Cat)s , if G:B >(Cat)s is any functor
and t:G )G a q.n.t., then t is a natural transformation. In the
language of fibrations this remark becomes: if Q:E__------gB is a discrete
s p l i t n o r m a l 0 - f i b r a t i o n and Q:E >B_ i s any s p l i t n o r m a l 0 - f i b r a t i o n ,
t hen any f u n c t o r T : ~ ~ E w i t h QoT=(~ p r e s e r v e s c l e a v a g e s . A n a l o g o u s
s t a t e m e n t s can be m a d e w i t h Gr ( the c a t e g o r y of ( s m a l l ) g r o u p o i d s )
instead of Sets, and by replacing q.n.transformations by w.n. trans-
formations.
The l e m m a we p r o v e in the nex t s e c t i o n i s the key to the d e s i r e d
adjointness M O, Q -4 N O p,Q'
89
4. THE MAIN LE~viMA
F i x a s p l i t n o r m a l (1, 0 ) - b i f i b r a t i o n
A< m E O>B_
o v e r A , B , with__A and_B s m a l l c a t e g o r i e s .
L e m m a . F o r e a c h p a i r of f u n c t o r s
t h e r e i s a b i j e c t i o n b e t w e e n
F:A >(Cat) , G:B >(Cat)s, - - 8 --
(i) the class of all quasi-natural transformations r:(R, ;)):FP )(]Q:
E ) ( C a t ) s , and
(i i) t h e c l a s s of a l l f u n c t o r s @ in a c o m m u t a t i v e d i a g r a m
¢ _E F ) E---(H(QG) )
o p A ) [(Cat,_B) o ]
(P, Q)
where QF and QG are the split normal 0-fibrations corresponding to
the functors F and G, respectively.
]Proof: It is absolutely essential in order to be able to follow
this proof to start by stating in perfect detail what the above conditions
on r and 4~really mean.
(4. i): A quasi-natural transformation r=_(R, p):FP ~GQ is:
a collection of functors RE:FPE ~ GQE indexed by EEE__, and of
natural transformations DE:GQe ORE-----~RE, oFPe {naturality means
90
that, for each f:X1-----~XzEFPE, one has
(~e)xzoGQe[RE(f)] = l<E,[FPe(f)]°(~)e)Xl) ,
satisfying: (i) ~)(idE)=id(RE) and. (ii) the diagrams
GQ(e,e) oR E 'Oe, e ) RE,,°FP(e'e) II II
GQe' o [GQe oR E ] [RE, , oFPe' ] oFPe
GQe' .De ~ ~ F P e
GQe' °RE, oFPe
are all commutative.
(4.z)
of the following;
(i) for each (A,X)£E_F , a functor ~(A,X):E A
i.e., for E6E with PE=A,
~(A,X)(E) = (QE, Y(E,X))
with Y(E,X)6GQE; and for J:EI-----~E26E with Pj=idA,
4~(A,X)(j) = (Qj, g(j,X))
with g(j,X), GQj[Y(EI,X)] }Y(Ez, X) in GQE2, such that
g(idE, X) = idy(E,X) ' and
g(j'j, X) = g(j', X) oGQj'(g(j, X)).
(ii) for each (a, f):(A, X) )(A',X') 6_E F , a morphism
¢(a,f) = (a.:~, 7(a,f))
A functor • satisfying the conditions of the lemma consists
~Ec over B_,
91
with T(a,f):¢(A,X)oa,------- 5 O(A',X') a natural transformation satisfying
Q G ( 7 ( a , f ) ) = idQ]___EA, . Al l t h i s m e a n s t he f o l l o w i n g : f o r e a c h E ' E E A ' ,
[ ~ ( a , f ) ] E , = (Q(Oa)E, , K E , ( a , f ) )
w i t h K E , ( a , f ) : G Q e a [ Y ( a , E ' , x ) ] ,-, ) Y ( E ' , X ' ) in G Q E ' , s a t i s f y i n g
K E { i d A ' i d x ) = i d y ( E , X) ' and
KE,,(a'a ,f'oFa'(f)) = KE,,(a' ,f')oGQoa,[Ka,,E,,(a,f)~,
and s u c h t h a t ( n a t u r a l i t y of y ( a , f ) ) :
g ( j ' , X ' ) o G Q j ' [ K E ~ ( a , f ) ] : K E ~ ( a , f ) o G Q o a [ g ( a , ( j ' ) , x ' ) ] .
Everything here should be self-explanatory to any patient reader.
L e t u s r e c a l l , h o w e v e r , how a is d e f i n e d on m o r p h i s m s :
By the universal property of
( O a ) E ~ ) E ' and no t i ng t h a t a * E ' 2 2 '
(Oa)]E '1 J' a~E' I~ ~ E'I ~E'2 lies above a via P
( s i n c e P j ' * idA, ) one d e d u c e s t h e e x i s t e n c e of
a . ( j ' ) : a , E ' 1 ) a : .E ' 2
u n i q u e wit lx (i) j ' ° ( O a ) E ' l = (Oa)E,2 ° a . ( j ' ) , and {ii) P ( a , ( j ' ) ) = i d A.
We a r e now r e a d y to p r o v e the l e m m a . T h e w a y to do so s h a l l
be a s f o l l o w s : to e a c h r : F P )GQ q . n . t . , we s h a l l a s s i g n a f u n c t o r
~P s a t i s f y i n g the r e q u i r e d c o n d i t i o n s . T h e n we s h a l l s e e t h a t t h i s r
92
c o r r e s p o n d e n c e i s one to one and onto. Le t r = ( R , p ) : F P )GQ be
a q.n.t.
(4.3) For each (A,X)E_E F, let
C r ( A ' X ) ( E ) = (QE, RE(X)), and
¢ (A, X)(]) = (Qj, o.(X)), r j
fo r e a c h E wi th PE=A, j :E1-- - - - -~E 2 E E for wh ich Pj=id A. Le t us
show tha t ~ (A, X) is a m o r p h i s m Q IE A r
>QG in (Cat, B ) 0.
F i r s t , i t is we l l de f ined . Since X E F A = F P E , then RE(X) EGQE;
s ince Pj=idA, F P j = i d F A and so, o j ( X ) : G Q j [ R E I ( X ) ] ~ RE2(X) ,
as r e q u i r e d .
Next , i t i s a f u n c t o r . It i s c l e a r tha t ~ r ( A ' X ) ( i d E ) = ( O ( i d E ) ' D i d ( X ) ) =
( i d Q E , i d R E ( X ) ) = i d ( Q E , R E ( X )) = i dCr (A ,X) . But a l so , ~ r ( A ' X ) ( j ' J ) =
(Q(j" ' j) ,Dj,j(X)). On the o the r hand, ~ r ( A , X ) j ' o L(A,X)j=(Qj ' ,Dj , (X))o(Qj ,D.(X)) = J
(Qj ' °Qj , 0 j , (X)°GQJ ' [Dj (X) ] ) -
The first components are equal since Q is a functor. The second
components are equal since [De ~ is coherent (cf (4. i))
Pj =id A •
F ina l l y , the d i a g r a m ¢ (A, x)
E A r ) E G
B
and because
is commutative. This is clear.
93
(4 .4 ) F o r e a c h (a, f ) : (A, X)
r ( a ' f) = (a,.,.,. y-f) and d e f i n e
y'f: %(A, X)oa, } %(A', X')
for each E'6E A' let
[Y-fIE' : (Q(Oa)E" RE'(f)°D(Sa)E, (X))"
) (A ' , X ' )C E__F, l e t
in the following way:
In order to see that this is well defined all we have to observe
(X)] }RE,[FPSa(X )] with
is that, since P(@ ):a, we can compose a
(X): GQ0 [R E' P(@a)E, a a
RE,(f):RE,(Fa(X)) ) RE,(X') -
denote this composite by (Kf)E,.
(Kf) E ,:RE, (f) o p(8 a)E ,(X). )
(For simplicity we shall sometimes
I.e., [y-fiE =(Q(Oa)E ,,(Kf)E,), with
(4.5) In order to prove that y-f is a natural transformation we
we must use all the information of (4.Jl) ,i.e., the fact that r=(R,D)
is quasi-natural. Notice that naturality is expressed by the commutativity
of all diagrams like this:
(Q(ea~ l, ;~f]~, 1 (Q(a-",-'E'l)'Ra E' (X)) ,, }{QE'I,RE{(X'))
! 1 [{Oj', %, (X'))
' [~f]E'z ) (Q(<)E'Z (~Z, z, RE, (x,)) (Q(a~:'~E'2)' Ra~:-'E ' Z (X)} ) 2
for each j' :E' 1 ) E' 2 EE__ with Pj'=idA, (Notice that the left vertical
arrow is, by definition, er(A'X)(a-':-'J')')
94
The c o m m u t a t i v i t y of this d i a g r a m can, t h e r e f o r e , be e x p r e s s e d
by m e a n s of two equa t ions :
(1) Qj'oQ8 = Q0 oQ(a j ' ) , and
(2) D j , (X ' )oGQj ' [RE , I(f)oo8 (X ' ) ] : RE, (f) o0 e (X ' ) °GQ0a[D a j , (X)] . a 2 a *
The first one follows from the very way a.j' was defined (see (4.2)).In
o r d e r to p r o v e the s econd we shal l d raw a big d i a g r a m :
GQj ' [D o (X)] G Q j ' [ R E , (f)] a 1
: o )
GQ8 a [ 0 a . j , (X) ] II I
IV
• ~ to
o 8 (x) RE, (f) a Z
0j,(x')
made up from smaller diagrams which we now discuss. The diagram
CQj' [RE, (f)] GQj,[RE,I(Fa(X)) ] l )GQj'[RE' I(X')]
~j,(Fa(X)) I I I pj,(x')
(Fa(X)) > (X') RE' 2 (f) RE' 2
RE, 2
is c o m m u t a t i v e b e c a u s e Dj, is a n a t u r a l t r a n s f o r m a t i o n . The d i a g r a m
GQj'(p o (X))
GQj' [GQe a [R a,E';,~ I(X)]] a .)OQj, IRE, I(Fa(X)) ]
I I ~ )
%j,.e ) a
RE,2(Fa(X))
is commutative by the coherence conditions (see (4. i)).
95
The d i a g r a m
GQ@ [GQ(a., . j ' )[R_ ~, (X)]
GQSa[P a j,(X)]l IV ~ D(@ a j,)(X)
GQOa [R a E,2(X)] - (Fa(X)) -", -~ 08 (X) >RE'2
a
is c o m m u t a t i v e , again by c o h e r e n c e of the {p }. e
Finally, Ill says that
Pj' o8 = P0 ° a.,.j' , which follows f r o m the definition a a 4.
of a,j' (cf (4.z)).
Our t a s k is next to v e r i f y tha t ¢ is a func to r . r
(4.6) %(id(A ' X)) = %(idA, idx) = df((idA);.., ~(idx)).
One m u s t then check only that [T( idx)] E = id(QE, RE(X)) , for each
E EGG_. But, by definition
[Y( idx) ]E = (Q0(idA)' R E ( i d x ) ° ~ 0 (X)) = (Q(idE), idRE(X) ° PidE(X)) • (id A )
Using that Pid =idR , the desired conclusion follows since P is a normal E E
f ib ra t ion , i . e . , %( id(A ,X))=(idE___A , id (QE, RE(X)) = i d % ( A , X ) .
(4.7) Given maps (a, f):(A,X) ~(A',X') and (a',f'):(A',X') }
(A",X") of EF, we want • (a,f) o~ (a',f')=~ (a'a,f'oFa'(f)). Since P r r r
is split, (a'a) = a~ oa',. An examination of the diagram
96
EA,, a',;,~ A' a,¢, A _ , , , E_ ~E_
x,'l ¢r/~',X'~ ~r~A,X~
- - G
i n d i c a t e s t h a t w e m u s t h a v e
o ! Tf, Tf a ;,. = Tf, ° F a ' ( f ) "
L e t E " E E A ' ' . W e s t a r t f r o m the l e f t h a n d s i d e of t h e e q u a t i o n
w e w a n t to e s t a b l i s h :
[Tf']E"'[Tf']a' E" = af(Oea '' RE"(f')°D@ (X))(Q8 ,Ra, E,,(f) oD@ (X)) ~:~ a ' a ;:. a
: ~(%,.%, tRE,,(f').%,(x)3.o%, ~h, E,~f).%(x~)
U s i n g , on t h e one h a n d t h a t @a ,a=ea , o8 a ( t h i s f o l l o w s by u n i v e r s a l i t y ,
a n d s a y s , m o r e e x a c t l y t h a t ( 8 a , a ) E , , = ( O a , ) E , , ° ( O a ) a , . ] E , , ) a n d on t h e o t h e r
t h e n a t u r a l i t y of DO , we g e t t h e l a s t e x p r e s s i o n a ~
= ( O ( e a ' a ) ' R E " ( f ' ) ° [ R E " ( F a ' ( f ) ) ° D e a ' ( F a ( X ) ) ] ° G Q e a ' ( D S a ( X ) ) )
= ( Q ( 0 a , a ) , R E , , ( f ' oFa ' ( f )} OPea , (Fa (X) ) o G Q e a , ( p c a ( X ) ) ) .
B y c o h e r e n c e of t he [De] r e l a t i v e to t h e m o r p h i s m s Oa, Oa,, t h e
l a s t e x p r e s s i o n b e c o m e s
= (O(ea,a), RE,,(f' oFa'(f))°Pe , (X)) = df )~(f' °Fa(fD (x)" a a
97
By (4.__6) and (4, 7), ¢ i s a f u n c t o r a n d i t i s c l e a r f r o m the r
d e f i n i t i o n s t h a t i t f i t s in to t h e r e q u i r e d c o m m u t a t i v e d i a g r a m .
W e h a v e , t h e r e f o r e , d e f i n e d a c o r r e s p o n d e n c e rl .) 4~ . L e t r
u s now s e e t h i s i s a b i j e c t i o n .
(4: 8) r I > ~ i s a n i n j e c t i o n . L e t r = ( R , O) a n d ~=(R, D) r
be q : n . t . F P - )GQ. A s s u m e ~ = ~_ . T h e n , r r
(I) for any E6E__ and X6FPE, RE(X)=RE(X). T h i s f o l l o w s
f r o m :
(QE, RE(X)) = df ~r ( PE, X)(E) : ~_(PE,r X)(E) --dr (QE, f~E(X)};
(2) for any E6E and f:Xl------gX 26FPE, RE(f)=[<F(f).
T h i s f o l l o w s f r o m :
((idA),:., Tf) = r i f t ( i d A , f) = ~ ( i d A, f) = df((idA)x., ~f) , i . e . , y f=~f ,
e q u i v a l e n t l y , RE( f ) o~) (X) = RE( f ) oDo (X). S i n c e -" O(id A ) ( id A ) 0 idA -1dE
and b o t h D ( i % ) ( X ) = i d R E ( X ) = idf~E(X) = 0 ( i dE) ( X ) , t h e c o n d i t i o n
reduces t o RE(f) = ME(f);
(3) for any e:E )E', pe=De , In order to prove it decompose
e=(OPe)E,oe (notice that Pis a l-fibration) as in the diagram
E
( Pe),E' ,,,
(8) Pe E'
)E '
where ~ is unique
for which the diagram
commutes and P~=idpE
98
The equation pe=Pe is a consequence of the equations (i) p~=p~ and
(ii) pOpe pope, for if we had these, we would also have
. . . . GQe pe(p ~ ) Pe: P(epeoe) (~)Spe) oFPe oGQOpe(p~ ) pepe
: 0%pe°GQ0Pe( a) : (0poOe) <
Let us show (i) first. Let X6FPE. Since e:E . )(Be) E' is such
that P~ = idpE , ~ 6E_ PE. Thus,
(Q~,p_(X)) = %(PE,X)(e) : • (PE,X)(6) = (Q~,p-(X)). e df 9 e
We now show (ii): for any XEFPE we can construct in E F a morphisrn
(Pe,idFPE,): (PE,X) ~ (PE', FPe(X)).
Then, %(Pe, idFPE,) = ~9(Pe, idFPE, ) reduces to Y(idFpE,) =~idFPE,)
which, in turn, since both RE, ,f{E, of the identity are identities, reduces
to the required equation (X) (X). P0 Pe : 0(9 Pe
(4.9) r i ) ~ is a surjection. r
Let ~be given as in (4.2). We define r:(R,p) as follows so as
to have • =4~. r
For any E and X6FPE, let RE(X)=Y(E,X), and given f:Xl------gXz 6
FPE, let RE(f):KE(idpE , f). (This is defined since (idpE, f):(PE,Xl) )
(PE, X2) is a morphism of E F.)
Now, RE(f): Y(E,XI) ~Y(E, X2) , as canbe seen fromthe fact
that, although RE(f ) is really a map
99
GQ8(i d ) [Y((idpE);:.-E, XI)] • PE
since P is normal and @(id )=id . PE E
RE:FPE ~GQE is a functor;
> Y(E, XZ),
It is immediate to see that
RE(idx)=dfKE(idpE , id X) =idy(E, x)~idRE(X) ,
by (4.2). Also by the conditions given in (4.2) one has
RE(f ')°RE(f) = K(idpE, f')°K(idpE, f) = K(idpE,f'oF(idpE)(f))
Then,
= K(idpE, f'f) =
Let us define, for each e:E )E'6E_,
Pe=K(Pe, idFPe(X) ) oGOepe(g(~ , X)).
RE(f 'f).
(4.10) The pair r=(R,p):FP )GQ is a quasi-natural trans-
formation.
The proof will be broken up into several parts.
(i) The composite makes sense and establishes
0e(X) :GQe(RE(X)) ~ RE,(F Re(X))
as required. (Write down each map and notice that GQe=GQ(epe)OGQ(~).)
(_2) An alternative way of defining Pe is as follows: let DOpe(X) =
K(Pe, idFPe(X)) and p_(X)=g(~,e X). Then force coherence for the de-
composition e=OpeOa , by letting Oe = D0 oGQOpe(D~). Indeed, Pe
for e=epeOe such that ~ is the identity, it follows from the original
I00
definition of ~e that p@pe(X)=K{Pe, idFPe(X)) , this, since for any X,
g(id, Xl=id. Similarly, let @Pe be the identity in the decomposition
e=@peO~. It then follows that ~(X)=g{~,X) since P~=id.
(3) Trivially, it is the case that PidE{X)=idRE(X ). Indeed, then
~=id E and @pe=idE in the decomposition e=@peOe. Now,
PidE(X) = K(P{idE) , idFpe(X) ) oGQ(@idE)[g(idE, X)]
= K(idpE, idFPe(X))°GQ(idE)[id R (X) ] E
= idRE(X ) °GQ(idE)(idRE(X )) = idRE(X ) •
The hard point is to establish the equation
~e,e(X)=(~e,)FPe(X) °GQe' (Pe(X)),
necessary to complete the proof of (4.10). Notice that, for composites
of the form e=@peO~, the above formula holds, by (2). And it also holds
easily for composites of the form @p(e,e)=@pe ,o@Pe, as we show below:
for each X,
(~6pe,~ Pe(X)[ GOepe,(~ep~ X)) ] =K{ Pe',idF p e ,(F Pe(X)) ) "GOepe,(K( Pe,idF Pe(X) ))
=K(IOe ' ° l °e , idFPe , (F Pe(X)) °FPe ' ( idFpe(X) ~
=K(P(e'e),idFlo(e,e}(X )) = Dep(e,e)(X)" That is,
101
(4) #ep(e'e)(X) = (D e Pe' )FPe(X) ° GQOpe'(#Ope (x))'
The following observation is a consequence of the fact that P
is a split normal l-fibration over A:
(5) Given e:E )E'and e':E' }E", and letting e'e=
@lO(e,e)Oe'e be the canonical decomposition, it follows that e'e =
[ (Pe ) ,~G]oe .
The following diagram illustrates this fact, using the uniqueness
in the def in i t ion of e'e :
e'e
( P e ) , [ ( P e ' ) , t C " ]
e e l IT
" E > E' )E"
e e'
Let us show, finally, that
(I) De,e=De,(FPe) o(GQe')De holds.
Using the definition of De,e , this reduces to
(If) Dep(e,e)°GQOp(e,e)(D~e ) = De,(Fl°e)o(GQe')De.
Using the definition of De ' in the right hand side and coherence
(4___) on the left, we get
102
(III) DO Pe ' °GQOpe'(D8 Pe) oGQ8 Pe' (GQ0 Pe(De-~-e) ) :
[D o °GQOpe,(De,)] o(GQe')D , Pe ' e
from which it follows that it is enough to establish
(IV) GQOpe,(DOpe ) oGQOpe,(GQOpe(De--7-ee)) :
give s
GQOpe,(D ~) °(GQe')D e.
Letting e':Ope ,De -~ on the right hand side and collecting terms
(V) GQOpe, [DOpe oGOOpe(De--Tee ) ] = GQOpe,[D J °(GQJ)(De)] •
In other words, (I) holds iff
(VI) DepeOGQOpe(De-~e) = D~°GQe-~(D e) does.
Recall that we have shown in (5) that e'---e = [(Pe),~] De . It is
clear that coherence holds for morphisms on a fibre for
g(j'j,X) = g(j',X)oGQj'(g(j,X)).
I.e., De--~e = D[(pe).~ 3De = D(Pe),(~)° [GQ(Pe),(J)3 D e .
in (VI) we get
Sub stituting
(VII) De ° [GQepe(D(pe) ' (~))3 °GQepe[GQOpe [ G Q ( P e ) , ( ~ ) ° D - ] = Pe ',-" e
D~ oGQ~ (De).
It is clear from the diagram of (5) that ~o(@Pe)E, = (Ope)(Pe ,) E"°(Pe)*(e-~)"
Notice now that the left hand side of (VII) is , in fact , equal to
103
~[epe o(Pe),(e ')] °[GQ(OPe o(Pe) , (~) ) ] Og
Thus, using the above remark, and substituting twice into this ex-
pression we get
(VIII) 0 [ ~ oe ]oGQ[~ OOpe](O e) = p~ o [ (GQ~)pe] . Pe
Let us now substitute the equivalent of D in the right hand e
side while using the fact that GQ~ is a functor. This gives
(Ix) p [~ ,°epe]OGO[e ,°epe] (0~ ) = [p~ °GQe,(p0p)] oGOe,[GQSpe(Pe) ].
Recal l that par t of the data for • is that, for each (a , f}EEF,
T{a,f) is natural , this, meaning that, for each j' such that Pj '=idA,.
g(j',X')°GOj'[K(a,f)] = K(a,f) oGO0a[g(a....j' ,X')] .
Apply this now with a=Pe, f=idFp e and j'=e'~. It says, precisely:
Pe' ° GQ~(~O Pe) = DOPe °GQOpe [D(Pe)"(~)] " - . -
Since the express ion on the r ight hand side of this equation is, by
definition, ~ , it follows that (IX) holds iff o8 Pe
(X) GQ[~'°OPe]/9-e : GQe-] [GQOl°e(Oe)] '
which is trivially true. The conclusion is that (1) holds anc~therefore ,
r=(R,D):FP )GQ is a q.n.t.
In order to complete the proof of the surjectivity, and thus, of
the Lemrna, we must only verify that ¢ = • . This is easy: r
(l) by definition, %(A,X)(E)=(QE, RE(X))=(QE , Y(E,X))=¢(A,X){E);
(2) also, ~r(A'X)(J)=(QJ'Dj(X))=(QJ'g(j'X))=~(A'X)(J);
104
(3) @r(a, f) = (a~s, yf) w i t h
(Yf)E' = (QOa'RE'(f)'°0 (X)) = (QOa, KE,(idA, ,f) oKE,(a ,idFa(X))) a
= ( Q e a , K E , ( i d A , on, f o F ( i d F a ( X ) ) ) = (QOa, K E , ( a , f)) = y ( a , f ) .
N o t i c e t h a t in t h e l a s t bu t one s t e p t h e c o n d i t i o n on K E , of ( 4 . 2 ) h a s
been used with the composable pair of maps a:A ~A' and idA,:
A ' IA'. We c o n c l u d e t h a t 45 (a, f)= 45(a, f) and t h u s , 45 =45. T h i s r r
is the end of our proof.
5. ADJOINTNESS
With the help of the Lemma of the last section, we can now prove
e a s i l y t h e f o l l o w i n g
T h e o r e m . F o r a n y s p l i t n o r m a l ( 1 , 0 ) - b i f i b r a t i o n A_( P.._E
0 N o M p, Q -{ P , Q .
Proof. The functors M 0 and N O • p,Q P,Q are defined in }2.
Q
W e s h a l l e s t a b l i s h a d j o i n t n e s s v i a a s e q u e n c e of e a s y l e m m a s , m a k i n g
u s e f r e e l y of t h e m a i n l e m m a p r o v e d in §4.
L e t u s be g i v e n s p l i t n o r m a l 0 - f i b r a t i c m s over A and o v e r B.
W e m a y a s s u m e (cf ~1) t h a t t h e y a r e of t h e f o r m Q F : E F }A__ and
GQ:E G )B, for some functors F:A }(Cat) G:B )(Cat} -- -- - - S ~ -- S '
r e s p e c t i v e l y . T h e f o l l o w i n g l e m m a i s d e s c r i b e d in m o r e g e n e r a l i t y
f o r w e s h a l l n e e d i t l a t e r in a n o t h e r c o n t e x t as w e l l :
1 0 5
(5.__~1) Let H:C )(Cat) be any functor , and U:D.. )C also - - S
a functor . Then, the square
QH,U
EHo U } E---H
U D ~ C
with the top map U defined by the rules :
(D, Z) ) (UD, Z) ; (d, h) } (Ud, h) ,
is a pull-back. (This is easy to show.)
(5.2) Morphisrns S.DI 0 • " P,Q(QF ) ' ~ QG of (Cat, 13)0 are in
one-one co r r e spondence with quas i -na tu ra l t r a n s fo rma t i o n s r= (R,D):
FP ) GQ.
F o r the proof of this observat ion , notice that a mo rp h i sm S:
M 0 P, Q(QF ) ) QG is no more than a functor which fits into acom-
mutat ive d iagram
S pul l -back (P, QF) • ) .E G
o L Mp, Q(
B
and this , by (5.1), is nothing e lse than a commuta t ive square
S EFp >E G
Q F P [ I Q G E >t3 - - Q
106
Again, by (5.1), the pull-back of Q and QG is given below, together
with the unique R which exists by the commutativity of the diagram
above :
EFp
GQ pull-back ~EI G
QG
Q >B
To such an R, commuting with QFP and QGQ' corresponds, in turn
(by the proposition proven in ~3), a quasi-natural transformation
r:(R, D):FP -)GQ.
Conversely, to each such r=(R, r~):FP }GQ q.n.t, corresponds
(again by ~3) a functor R:EFp-------~_GGQ for which QGQOR=QFp.
This RI composed with Q, as in the diagram
R --EF p ~ --EGO )_E C
E_ ~B Q
gives a morphism (~oR : M 0 P,Q(QF ) ) QG'Pr°perties of pull-back
diagrarns say that these two processes are inverse to one another. This
proves (5.2).
107
( 5 . 3 ) M o r p h i s m s T : Q F ) N O .. p , Q ( Q G ) of (Cat ,_A) 0 a r e i n
o n e - o n e c o r r e s p o n d e n c e w i t h f u n c t o r s • i n a c o m m u t a t i v e d i a g r a m
QF
¢ E F ~_EHQ G
A I-- ~ [(Cat, B)0 ]°p (P, Q)
T h i s i s e v e n m o r e i m m e d i a t e , i f o n e r e c a l l s t h a t t h e f u n c t o r
0 Np, Q(QG) i s d e f i n e d p r e c i s e l y a s a p u l l - b a c k of ( P , Q) a n d Q
(HQG)
F r o m ( 5 . 2 ) , ( 5 . 3 ) a n d a n d t h e m a i n l e m m a ( ~ 4 ) , f o l l o w s t h e
existence of a bijection
Horn ~ ( Q )-~ (QF' NO (Cat, Bl 0 P,Q Q~ G H°m(cat, A) 0 P,Q(QG ))"
In v i e w of t h e g e n e r a l t y p e of m o r p h i s m s i n t h e s e c a t e g o r i e s ,
n a t u r a l i t y i s i m m e d i a t e . 0 0
Thus Mp, Q -t Np, Q. The proof of the
t h e o r e m i s c o m p l e t e d .
6. SPECIAL FIBRATIONS
I t i s o u r a i m i n t h i s s e c t i o n t o o b t a i n f r o m t h e a d j o i n t n e s s of
0 0 M p , Q a n d N p , Q , a w h o l e c l a s s of a d j o i n t p a i r s b e t w e e n v a r i o u s
categories of special fibrations.
Def. Let X be any full subcategory of (Cat) . A fibration - - S
Q : E B i s s a i d t o b e of t y p e X o r a n X - f i b r a t i o n i f , f o r e a c h
B ~ B, EBEX.
108
Denote by X-(cat, 15)0 the full subcategory of (Cat, B__)0
whose objects are split normal 0-fibrations of type X, over 15.
It is clear that the equivalence set up in §3, now becomes
X(cat,_B)0 ~ [B,X]q.n.t.. This is easily established.
In order to derive some useful consequences of the adjoint-
hess of §5, it seems that two assumptions are needed on X. Before
mentioning them, let us point out that the following holds for any
X c (Cat) : -- S
(6.1) Pulling back along f:A
f*: ~X(cat' 13)0
)B restricts to a functor
> X(Cat, A) O.
The proof is immediate if we recall that, for a fibration of the
form QG:EG_ --------)B, f*(QG)=QGf. If G:B_ -}(Cat)s factors through
XC(Cat) , so does Gf:A ,~Cat) . -- S -- S
The first assumption on X shall be:
[1] X is a reflective subcategory of (Cat) . -- S
In o r d e r to f i x i d e a s , w e l e t ~ : ( C a t ) ~ X be t h e r e f l e c t i o n S
f u n c t o r , w i t h r e f l e c t i o n m a p s ~ = Cec:C ) ~ . (C) , f o r e a c h C E
(Cat) . I.e., ~:(Cat) ~X_X_ is left adjoint to the inclusion I: S S
X >(Cat) ; moreover, ~I=id X. -- S
Denote by IB: X(cat, B) 0 > (Cat, B) 0 the inclusion functor,
f o r B E ( C a t ) . A f i r s t c o n s e q u e n c e of a s s u m p t i o n I l l i s t h a t X ( c a t , B) 0 - - S - -
i s a r e f l e c t i v e s u b c a t e g o r y of (Ca t , 13)0, i . e . ,
109
(6.2) The functor [~]9:(Cat, B) 0 )X(cat, ]9)0 , defined
fibrewise via ~ , is left adjoint to the inclusion I B- . (We omit
the proof. )
[Z] We shall say that XC(Cat) satisfies property (H) (for -- S
" h e r e d i t a r y " ) if: g iven Q1 E X ( c a t , B)O and Q E (Cat, B_.)O, it fo l lows
that HQI(Q) 6 XC(Cat) . -- S
It is c l e a r that , w h e n e v e r X is de f ined v i a a p r o p e r t y on m o r p h i s m s
of the kind which a natural transformation inherits from its components,
thenXhas property (H). For example, X could be the category of
all small categories such that all of their morphisms are: monic, epic,
invertible or identity morph~sms. However, we wish to include Z as an
example of a category with property (H) although we cannot think of
Z (the arrow category or, rather, its image in Cat) in these terms.
Theorem (Adjointness for categories of special fibrations_)
(H).
Let X__ be any full reflective subcategory of (Cat)s, with property
Let A, B be any small categories and _A< P _E Q- ~__B any
split normal (i, 0)-bifibration.
Then there exists a restriction of the functor N O P,Q
N O X X p,Q : --(Cat']9) 0 > --(Cat,A) 0
and, as such, is right adjoint to the functor M 0 defined to be the P,Q
to a functor
composite, as in the commutative diagram below:
110
x_( c at,___A) 0
(Cat, A_) 0
Proof.
M 0 P,Q
>~(Cat, ~)0
) (Cat , B_) 0 M 0
P , Q
It f o l l o w s i m m e d i a t e l y f r o m the t h e o r e m of § 5.
0 Q : (Ca t , B)0 >(Cat , A__)0 i s t he c o m p o s i t i o n F i r s t , r e c a l l t h a t N p ,
of two f u n c t o r s, one of w h i c h is p u l l i n g b a c k a l o n g a f u n c t o r
A
(P, Q):_A )[(Cat, B)O ]°p
By (6.1), this restricts to the subcategory of _XX-fibrations, this , without
any assumptions on X. Property (H) has the virtue of insuring that also
t h e f u n c t o r
H: (Cat,_B) 0 op
}(Cat, [(Cat,_B) O] )0 '
when restricted to the suhcategory of X-fibrations over B_, have only X-
fibrations over [(Cat, ]3)0]°P as values.
0 From the above follows that we can define a restriction of Np, Q
as a functor N O p, Q in a commutative diagram
0 Np, Q
X(cat, B) 0 ~X(cat, A_) 0
I I - -
#
(Cat , .__B) 0 2 (Cat , A) 0 0
N P,Q
From (6.2), the above and the fact that I A-- is fully faithful for
111
each A E(Cat)s ,
below, c lear ly natural .
Let QI:E_I , , )_A, QZ:_E2
0-fibrations. Then,
Hom X(cat,~)o
the general theorem of §5 delivers the isomorphisms
)B~ be arbitrary split normal
(~ B--MOp, QIA--(Q 1 ), Q2 )
0 QIA(Q1 ), IB(Q2)) Horn(Cat, _B) 0 ('v[ p,
0 Q(IB__(Qz )) _~ Horn( C at,A)0 (IA (Q 1 ) , . Np,
H°m(cat'---A)O (IA (Q i ) ' I_A_ 0 (Np, Q(Q2)))
0 H°mx (Q i' (Q2))"
--(Cat, A_) 0 Np, Q
This completes the proof.
We remark that, in view of the proposition proven in §3, an
equivalent formulation of the above theorem is: for each bifibration
over A, B, there is induced a pair of adjoint functors
M o [A, X] P, 0 ' [B X]
q'n't¢" --'O- ' q.n.t. Np, Q
If, furthermore all objects of X_ are discrete categories then,
the adjoint pair becomes
x _A M0
P'Q ~X B , an adjoint pair
0 between functor
Np, Q categorSes.
112
7.EXAMPLES AND APPLICATIONS
Useful applications of the adjointness schema of 36 depend on
appropriate choices of the variables X and (P,Q) involved.
Recall that X is any full reflective subcategory of (Cat) which -- S
satisfies property (H).
The following remark is useful in the sequel: the category Q
H l(Q,) consists of functors T:E' ~E 1 commmuting with the
projections, with morphisms ~:S ) T natural transformations
for which Ql~=idQ,. The latter condition quarantees that each
component of ~belongs to some fibre of QI" More precisely, if
E' EE', ~E' is inthefibre of Q l above Q'E' E B. IfQ 1 is a fibration
of type_Z, this says that ~E' is a morphism in a category which is an
object of X.
Examples.
(7.1)_X = (Cat) . The smallness of the fibres of the fibrations S
implies that always HQI(Q ') is a small category (cf. 52).
(7.2)_X = Groupoids. A groupoid is a small category each of
whose morphisms is invertible. A natural transformation whose com-
ponents are all invertible is itself invertible: this proves property (H)
by the above remark. The category of all groupoids as a full sub-
category of (Cat)s is also reflective: making arrows invertible in the
sense of Gabriel-Zisman [I] is left adjoint to the inclusion functor.
(7.3)X = Sets. Denote by Sets the full subcategory of (Cat) s whose
objects are the discrete categories. A natural transformation is the
113
identity natural transformation if and only if each component is the
identity morphism. Also, taking connected components is a functor,
usually denoted by [I 0, and is a reflection of (Cat} into Sets (cf.[11). S
(7.4) X = Z. By Zwe mean the full subcategory of (Cat) con- s
sisting of the categories @ and ~. A fibration of type Z is thus a
fibration for which each fibre is either empty or has exactly one
Q HQI(Q, ) po in t . A s s u m e the c a t e g o r y H I(Q,) to be n o n - e m p t y , let T 6
be any object. It follows from~3 that T is completely determined by
a family of functors TB:(E_') B )(El) B indexed by the objects of_B
and satisfying certain conditions. If Ql is a fibration of type 2_ and Q'
is arbitrary, there is at most one such family. Thus, property (H)
holds. Also, a reflection of (Cat)s into ~is given by the functor
which sends 0 into e and any non-empty small category into ~ .
Now that we have some illustrations of categories X let us
c o n c e n t r a t e on spec i f i c b i f i b r a t i o n s . A n a t u r a l cho i ce is to c o n s i d e r
the two c a n o n i c a l b i f i b r a t i o n s a r i s i n g f r o m any f:A ~ B E (Cat ) s ,
as described in ~l.
functor s
X(C at, _A )0 (
The bifibration (Pf, Qf) induces a pair of adjoint
0 Mpf , Qf
0
Npf, Qf
_X(cat ' B_) 0 .
In turn, the bifibration (Pf, Qf) induces the pair of adjoint
functor s
-X(cat, B_)O~ 0 Mpq Qf
0 Npf , Qf
> -X(Cat,_A)o
114
Theorem. Let X be a category of discrete categories in the
context of the adjointness schema Then, the functors N O and • pf, Qf
0 Mpf Qf are naturally equivalent to the functor induced by pulling
back along f.
C o r o l l a r y (Kan e x t e n s i o n s )
Under the canonical equivalence between categories of functors
0 0 and categories of fibrations, the functors Mpf, Qf and Npf, Qf cor-
respond, respectively, to ~f and l-If, the left and right Kan extensions
along f of X-valued functors.
Before proving the theorem let us remark that the value of the
Corollary should not be taken to be yet another proof of the existence
of Kan extensions; indeed, these functors exist under more general
circumstances. What should be surprising is the fact that both adjoint
pairs ~f-4X f and xf-~l-[f come out as instances of a single type of
adjoint pair. Moreover, whereas ~f is a functor of type Mp, Q
X f is an example of both l'[f a functor of type Np, Q, _
We shall prove the theorem as a sequence of lemmas.
and
First, denote by
Yon: B °p ) Split0B__
the functor which assigns to an object B of B__the fibration (~:(B,B)---->B_
which best approximates the functor rB~ : I ~ B. To a map b 6 B, Yon
assigns the corresponding map between comma categories, clearly
cleavage preserving.
Next, define a functor
%: E G ~ _E(HQG) for e a c h G:__B ) ( C a t ) s
115
by the following rules:
G (B,Y) ! ) (Yon B, Ty),
Ty (C,h) l
with Ty given by:
> (C, Gh(Y))
Ty
( k , i d ) | ) (k, id).
Clearly, Ty:Q~B ~ ------>QG and is cleavage preserving. On the
morphisms,
~G (b, g) ! ) (Yon b, Tg), with Tg given by:
(Tg)h, = (id, Gh'(g)).
The verification that % is well defined and a functor is left to the
reader.
It follows from the definition of % that the diagram
G E G >E -(uQG)
B_ .,, ) [SplitoB_] °p
Yon
is commutative.
Lemma 1 (Yoneda) The above diagram is a pull-back.
Proof. Denote by ~o the unique functor which fits into the diagram
below, where the inner square is a pull-back:
116
E
B
G
pull-back
Y on
E--( HQG )
I Q(H QG
> [Splito__B ] °P
Define a functor X candidate for an inverse to ~0, as follows:
to each pair (B,T) with B E B and T a cleavage preserving functor
as in
T (B, B) E
- - - - G
B
for X(B,T) = (B, TB(idB) ). Next, given a morphism b:B-------~B'
and a natural transformation Y:TBoyonb >TB, let X(b,Y) =
= (b, Yid ). With these definitions, it is straightforward to verify B'
that X is a functor and that the composite X~ is the identity. The
corresponding assertion for the composite ~X depends on the fact
that only cleavage preserving functors are brought into the picture.
Indeed, by definition,
¢~X(B,T) = ~0(B, TB(idB)) = (B,T(TB(idB))).
117
Now, for any h:B )C, T(TB(idB))(h) = Gh(TB(idB)). Since T is
cleavage preserving,
Gh(TB(idB)) = Tc(h). Thus, T(TB(idB) ) = T.
Also, ~X(b,9')=qg(b,~,idB,)= (b,9'(y )). Since [Y(9' )]h' = id B , id B ,
= Gh'[9'idB,] = ~'h"f°r all h', the proof is now finished, i.e. @X is
the identity.
Lemma 2. The functors NOpf, Of and f;'" are naturally equivalent.
Proof.
Observe first that the diagram
Pf' Qf X op A ~ [--Split0B]
~ ~ n B
is commutative. (I.e., notice that Yon(fA) = QfI(fA,B) , while
Yon(fa) = a .) This implies that the pull-back diagram which defines .-,~
the value of N O Qf at an object QG' is computable in two stages, as P f,
indicated below:
NOf,Qf(QG )
o > ~ E--(HQG)
A ) B_ ~ [_X(Split0B__ )]op. f You
(Remark that we are justified in writing Split0B in lieu of (Cat, ]3)0,
this since_X has only discrete categories as objects.) It follows
118
immediately from Lemma 1 that
NOpf, Qf (QG) --- f*(QG).
0 L e m m a 3. The functors MI:~ Qf and f* are natural ly equivalent .
0 is ob- Proof. Recall that the value of Mpf, Qf at an object QG
rained by applying the reflector ~:(Cat) >X fibrewise to the S
0 f ibrat ion M ~ , Q f ( % > over A defined as in the diagram:
E-(G.pf )
A < (~,f) Qf pf
~B
QG
We now claim that pulling back along f has the same effect on QG"
To do so, we define functors ~,X in a diagram
~_A [E(G p f ) ] --EGf < - °
A
such that both ~X and X~ are identity maps. Let (A, Y) 6 E G f, i.e.,
A EA and Y E GfA. Consider the objects (fA,idfA,A) E (_B,f) and
{fA,Y) E ~G" Since FJ(fA,idfA ,A) = fA = QG(fA,Y), the pair of
119
objects is an object of E__(GoPf ). We let ¢p(A,Y) = ((fA,idfA,A);(fA ,Y)).
Similarly, given (a, g):(A, Y) ~(A',Y') 6 E_Gf, i.e.,a:A ~A'
and g:Gfa(Y) ~ Y' E GfA' (which must therefore be the identity
since G has values in_X), the assignment qg(a, g) = ((fa, a);(fa, g))
makes sense.
Define X by the rules:
X((B,~ ,A) ; (B , Y)) : (A,G~(Y)) and
X((b, a);(b, id)) = (a, id) .
Not ice tha t a l l m o r p h i s m s wi th in a s ing le f i b r e a re sent in to
the i d e n t i t y u n d e r X so tha t X can be ex t ended to a f unc t o r on
~A--[]~(GoPf)]. It is immediate that X~is the identity.
To see that ~X is also the identity the assumption that X has
discrete categories as objects becomes essential. Let us compute
q~X((B,/~,A);(B, Y)) = {p{A,G~Y) = ( ( fA, idfA,A) ; ( fA,G/~V)) .
C o n s i d e r the m o r p h i s m
(~ , idA) : (B , ~ ,A) ------> (fA, id fA,A) of (B_,f)
and the m o r p h i s m
(B, i d G ~ y ) : ( B , Y ) ~(fA,G/~Y) of _E G.
P r o j e c t i n g o n t o A the p a i r ((/~,idA) , (/~,idG/3y)) g ives idA:A >A.
T h e r e f o r e , s ince ~ m a k e s f i b r e s at l e a s t d i s c r e t e , both ¢OX((B,~,A);(B,Y))
and ((B,~.A);(B, Y)) a r e i d e n t i f i e d .
T h e s e l e m m a s c o n s t i t u t e a p roof of the t h e o r e m . The c o r o l l a r y
fo l lows e a s i l y f r o m the o b s e r v a t i o n that , u n d e r the eqvLivalence of
120
--X(Split0B) and X_ B- for any B_, p u l l i n g - b a c k a long f:A_ ) B c o r r e s p o n d s
to the func to r x f : x B- > _X A-, induced by c o m p o s i t i o n wi th f .
We c l o s e th i s s e c t i o n wi th a b r i e f c o m m e n t on the c o m p r e h e n s i o n
s c h e m a of L a w v e r e (cf . L a w v e r e [9, 10]: a l so G r a y [3, 4 ] ) . Le t a
h y p e r d o c t r i n e (or p a r t of it) be g iven wi th s o m e ~r ~ (Cat )s and,
fo r e a c h B 6 ~r, P(B) = X--(Cat, B_) 0 fo r s o m e cho ice of X_ wh ich is
adequa t e (we m u s t have c a r t e s i a n c l o s e d ) . Then we le t the p a i r
~f-lf.(-) consist ofthe functors MOpf, -INOf, Qf Qf. This will be
enough to discuss the comprehension schema, which says, in this
context, that the functor
(~r, B_)
given by the ru le :
fa
> X-(Cat, B__) o
0 Mpf , Of ( idA) '
has a right adjoint. Observe next that the diagram below gives the value
0 of Mpf, Qf(xdA_):
Pf
A (f, B_)
[ id(f, B)
> (f, B_) Qf
B >A - - P f --
id A
and that, therefore, this is precisely Qf, the best approximation to
f by a O-fibration.
This does not yet ensure the right adjoint: it depends on ]r as
well as onX.
121
If :It = ( C a t ) s , t h e n t h e a b o v e f u n c t o r h a s a r i g h t a d j o i n t f o r a n y
X o n l y r e l a t i v e ( in U l m e r ' s s e n s e , c f . [ ~ ] } to t h e i n c l u s i o n
J: X S p l i t 0 B , X--(Cat,B)0 .
T h u s , t h e c o m p r e h e n s i o n s c h e m a h o l d s f o r c h o i c e s s u c h a s
P(B_B) S e t s B- o r P(B) I B = = -- a s c a t e g o r i e s of p r o p e r t i e s of t y p e B .
R e s t r i c t i n g ]r s u i t a b l y , e . g . , l e t t i n g ]Y be t h e c a t e g o r y of
g r o u p o i d s , t h e a d j o i n t e x i s t s r e l a t i v e to t h e i n c l u s i o n of t h e c a t e g o r y
of f i b r a t i o n s o v e r B and c l e a v a g e p r e s e r v i n g m o r p h i s m s i n t o t h e c o r -
r e s p o n d i n g c a t e g o r y of f i b r a t i o n s and c a r t e s i a n m o r p h i s m s . A g a i n ,
t h i s s a y s t h a t , f o r e x a m p l e , t h e h y p e r d o c t r i n e w i t h ~lr = G r o u p o i d s
and P(B_) = S e t s B h a s t h e c o m p r e h e n s i o n s c h e m a . F i n a l l y , one c a n l e t
] r = S e t s . T h e n , c o m p r e h e n s i o n h o l d s f o r any c h o i c e of X; h o w e v e r ,
f r o m t h e e x a m p l e s a v a i l a b l e t h e o n l y s u i t a b l e c h o i c e w o u l d be X=Z ;
t h u s , t h e h y p e r d o c t r i n e w i t h ~ = S e t s and P(B)=Z B a l s o s a t i s f i e s
t h e c o m p r e h e n s i o n s c h e m a .
W e h a v e t h u s r e c o v e r e d a l l of t h e ( s e m a n t i c a l e x a m p l e s of
h y p e r d o c t r i n e s w h i c h o c c u r in [9 , 101 and h a v e a l s o p r o v i d e d a u n i f i e d
p r o o f t h a t t h e c o m p r e h e n s i o n s c h e m a h o l d s f o r e a c h of t h e m .
M c G i t l U n i v e r s i t y
M o n t r 4 a l
122
REFERENCES
i.
2. m
3. m
4. w
5.
6. w
7.
8.
9.
10.
Gabriel, P., and Zisman, M., Calculus of fractions and Homotopy Theory. Springer-Verlag, Berlin-Heidelberg-New York 1961.
Gray, J.W., Fibred and cofibred categories. Proceedings of the Conference on Categorical Algebra - La Jolla 1965,pp.21-83. Springer-Verlag, Berlin-Heidelberg-New York 1966.
Gray, J.W., The Categorical Comprehension Scheme. Category Theory, Homology Theory and their Applications III. Lecture Notes 99, pp.242-312. Springer-Verlag, Berlin-Heidelberg- New York 1969.
Gray, J.W., The 2-/%djointness of the Fibred Category Cons- truction. MS. 1969.
Grothendieck,/%., Catdgories fibrdes et descente. Sdminaire de g4om4trie alg4brique de l'Institut des Hautes Etudes Scien- tifiques, Paris 1961.
Kan, D.M., /%djoint Functors. Trans./%mer. Math. Soc.87, pp. 295-329 (1958).
Lawvere, F.W., Functorial Semantics of Algebraic Theories. Thesis. Columbia University. New York 1963.
Lawvere, F.W., The Category of Categories as a Foundation for Mathematics. proceedings of the Conference on Categorical _Algebra - Za Jolla 1965, pp.l-20. Springer-Verlag. Berlin- Heidelberg-New York 1966.
Lawvere, F.W., /%djointness in Foundations. (to appear in Dialectica).
Lawvere, F.W., Equality in Hyperdoctrines and Comprehension Schema as an Adjoint Functor, in "Proceedings of Symposia in Pure Mathematics" volume 17, Applications of categorical algebra, AMS (1970).
l__!l. Tierney, M., Lecture McGill University, February 25, 1970.
12. Ulmer,F., Properties of Denseand Relative Adjoint Functors. Journal of Algebra 8, pp.77-95 (1968).
THE DOUBLE CATEGORY OF ADJOINT SQUARES*
Paul H. Palmquist
Received October 17, 1970
0. I n t r o d u c t i o n
This p a p e r g e n e r a l i z e s the dua l i t y of the c o r r e s p o n d e n c e b e t w e e n lef t and
r i g h t ad jo in t s by i n t r o d u c i n g ad jo in t s q u a r e s and t h e i r c o n t e x t of s i m p l e r s q u a r e s .
S e c t i o n 1 s e t s f o r t h f ac t s about and r e l a t i o n s b e t w e e n double c a t e g o r i e s and Z - c a t e -
g o r i e s . In p a r t i c u l a r , e v e r y double c a t e g o r y c o n t a i n s a c a n o n i c a l Z - c a t e g o r y . S e c -
t ion 2 i n t r o d u c e s a double c a t e g o r y of ~ s q u a r e s bui l t f r o m a Z - c a t e g o r y ~, and
con ta in ing C~ and i ts s y m m e t r i c C~ s. S e c t i o n 3 e x p o s e s a d j u n c t i o n s as s p e c i a l
s q u a r e s , d e f i n e s ad jo in t s q u a r e s and the r e l a t i o n s b e t w e e n t h e i r fou r c o m p o n e n t s ,
and g ive s u n i v e r s a l c h a r a c t e r i z a t i o n s of the double c a t e g o r y of ad jo in t s q u a r e s and
the Z - c a t e g o r y of ad jo in t Z - c e l l s .
S e c t i o n 4 de f ines a Z - d i m e n s i o n a l h o m - f u n c t o r fo r Z - c a t e g o r i e s and a r e l a t -
ed Z - f u n c t o r taking ad jo in t s as v a l u e s . Sec t ion 5 e x p o s e s the c a t e g o r y of ad jo in t
Z - c e i l s as a r e l a t i v e c a t e g o r y . S e c t i o n 6 c o n s t r u c t s a Z - c a t e g o r y of c y l i n d e r s f r o m
a double c a t e g o r y . F i n a l l y , Sec t i on 7 r e p r o v e s s o m e b a s i c f ac t s about a u t o n o m o u s
c a t e g o r i e s (which a l m o s t a r e m o n o i d a l c l o s e d c a t e g o r i e s [5] ) b y a d i s p l a y of ad-
j o in t s q u a r e s and c y l i n d e r s .
1. Double C a t e g o r i e s and Z - C a t e g o r i e s
We r e c a l l the fo l l owing d e f i n i t i o n due to E h r e s m a n [41.
This p a p e r is a s u m m a r y of a d i s s e r t a t i o n s u b m i t t e d to the Uni~ ,e rs i ty of C h i c a g o , f o r the P h . D . d e g r e e , unde r the s u p e r v i s i o n of P r o f e s s o r S a u n d e r s Mac Lane .
124
Def . l . l A d o u b l e c a t e $ o r y 29 is a c o l l e c t i o n 1291 of d o u b l e m a p s t oge th -
e r w i t h two r u l e s of c o m p o s i t i o n : v e r t i c a l c o m p o s i t i o n # a n d h o r i z o n t a l c o m p o s i -
t i on ;:% s u c h tha t :
1) o n d e r #, 12>] is a c a t e g o r y 29#; f o r # - c o m p o s a b l e m a p s S,S '6[2>[ we
w r i t e S # S ' f o r the # - p r o d u c t of S w i t h S ' ( p i c t u r e d a s S o v e r SI);
2) u n d e r ;',-~, ]2>1 is a c a t e g o r y 2> ; f o r ; ' , - ' - composab le m a p s S , T E ]2>] we
w r i t e S~".-'T f o r the , : - ' -product of S w i t h T ( p i c t u r e d as S to the
l e f t of T ) ;
3) i f we d e n o t e the c o l l e c t i o n of i d e n t i t y m a p s of 29# ( i . e . , " o b j e c t s " of
29#) by (29#) 0 , we require (29#% to be a ,:-"subcategory of 29~";
4) the collection (29) of identity maps of 29~" is a #-subcategory of 2>"; 0
and
5) w h e n both sides a r e defined, we require the equality
(S # S') ~:-" (r # T') : (S ;:-" T) # (S'):-" T').
T h e r e l a t i o n s h i p of S , S ' , T and T ' i s i n d i c a t e d in t he d i a g r a m b e l o w
(1) S T
S' T'
We w r i t e 2}oo f o r ( N (2>)o' t he c o l l e c t i o n of d o u b l e i d e n t i t i e s of 2>, i . e . ,
~naps w h i c h a r e i d e n t i t i e s w i t h r e s p e c t to b o t h # and ~:-" c o m p o s i t i o n .
29t N o t a t i o n : F o r S E ]291 we w r i t e b(S) f o r t he d o m a i n of S a s a m a p in
i.e., b(S)E(29#)o and S#b(S)= S, and t(S) for the codomain of S in 29#. These
are the bottom and top identities for vertical composition, r(S) and £(S) denote
the domain and codomain of S in ~"~ i.e. , right and left identities, respectively.
Def. I.Z A m a p M of the d o u b l e c a t e g o r y 29 i n t o the d o u b l e c a t e g o r y
125
is an assignment from I.~I to I~ I which respects # and ,'.-" composition, i.e.,
yields functors ~# ~ ~# and .~:" • ~". These maps form the (meta) category of
double categories Dble.
There are eight dualities for double categories, which give automorphisms
of Dble corresponding to the eight symmetries in the automorphlsm group of the
square. We give notation for the four basic automorphisms of period Z on ~ where
S is a double map in .~.
l) #-duality. #2> has elements #S with (%)#--(~#)op and (#/~i:"--='~':".
Z) ;:,-duality. '~.~ has elements ;S with C:~)#-~/~# and (':>);:~-----(.~':-')op
3) 6-duality. 6~ has elements % with (~)# ~-(/~#)op and (6~)-':'-_=. (.~':-')op,
4)
Recall that the assignments S~-~b(S) and S: ~ t(S) give functors of
to itself, e.g. , b(S ;'.-" T) = b(S) ;:-'b(T), each with image (~#)o" And a-dually, the
assignments SF >r(S) and S, , g(S) give functors of .~# into itself.
i.e., as double categories ~ .
(Y-duality. u~ has elements ~S with (~)#-=~:" and (~.~)~-~--~, l.e.,
interchange ~:-" with #, b with r and t with ~. The reader is in-
vited to draw the diagrams which result from applying these dualities
to Diagram (i).
in-
Def. 1.3 An (abstract) (#) Z-category G is a double category G in
which each ;:-" identity is a # identity, therefore, a double identity, i.e. , ('~)o ='~OO"
In a Z-category we denote horizontal ('-:-') composition of elements S and T
by juxtaposing them, e.g. , ST, and use the symbol o as in Sos ' to denote verti-
cal (#) composition. Double identities are called 0-cells, #-identities are called
I-cells and arbitrary double maps are called Z-cells. Thus if s is a Z-cell with
l-cells, F = t(s) and G = b(s), and 0-cells,
B = r(s) = r(F) = r(G) and A = ~(s) = ~(F) = ~G),
126
w e h a v e t h e d i a g r a m b e l o w
F
N o t e : 1 - c e i l s a r e s i n g l e a r r o w s a n d Z - c e l l s a r e d o u b l e a r r o w s . F o r 0 - c e l l s A
a n d B w e h a v e t he " h o r n s e t " ~ (B ,A) w h i c h i s a c a t e g o r y u n d e r # - c o m p o s i t i o n ,
indeed a #-subcategory of /~#!
P r o p e s i L i o n 1 .4 T h e (#) Z - c a t e g o r i e s f o r m a r e f l e c t i v e f u l l s u b c a t e g o r y
D b l e # (='--Z-Cat) of D b l e . M o r e p r e c i s e l y , e v e r y d o u b l e c a t e g o r y 2 in Db[e c o n -
t a i n s a s u b d o u b i e c a t e g o r y .~' i n D b l e # w i t h t he f o l l o w i n g u n i v e r s a l p r o p e r t y : If
3" is in D b l e # a n d M : ~ - - - ~ i s a m a p of D b l e , t h e n M f a c t o r s a s a u n i q u e m a p
M t : 3 " - - ~ f f f o l l o w e d b y t h e i n c l u s i o n ~q" ~ ~, a s i n t h e c o m m u t a t i v e d i a g r a m
P r o o f . T h e m a p s of .~' a r e t h o s e of the f u l l ' ~ - s u b c a t e g o r y of .0 ¢ d e -
t e r m i n e d b y t h e d o u b l e i d e n t i t i e s ./~ O O
We r e m a r k t h a t t h e t h r e e a u t o m o r p h i s m s of D b l e w h i c h d o n ' t i n v o l v e
c r - d u a l i t y g i v e B ~ n a b o ~ u ' s d u a l i t i e s f o r Z - c a t e g o r i e s [3, pp . 2 6 - 2 7 ] :
# - d u a l i t y in D b l e g i v e s c o n j u g a t i o n ( )c in Z - C a t ,
T ':' - duality gives transposition ( ) , and
6 _ duality gives syrnmetrization ( )s
Conjugation abstracts the familiar notion of "op" duality in the Z-category
of categories Cat, since for Z-cells (natural transformations) f and g (g°f) °p =
fopo gop we actually have the equivalence ( )op: ~atc__~Cat, involving the conjugate.
By <Y-duality we can define the subcategory Dble also isomorphic to Z-Cat,
J.
since Dble" and Dble # are isomorphic by restriction of the autcrnorphJslTl ¢; on Dble.
127
Z. The Double C a t e g o r y 9{ ~ S q u a r e s
We now d e v e l o p the doub le c a t e g o r y of s q u a r e s . G i v e n a Z - c a t e g o r y ~, we
c o n s t r u c t a doub le c a t e g o r y ~ S q w h o s e doub le m a p s c o n s i s t of fou r t y p e s of
squares.
Def. 2.1
B' and two l-ceils
I G i v e n the fo l lowing f i x e d d a t a f r o m ~ : f o u r 0 - c e l l s A,A ,B, and
¢ : A - - -~A' , : B - - - ~ B ' , we add fou r o t h e r s e t s of d a t a to con-
s t r u c t four typical ~ squares. First, construct a -~ square S from l-celts
g :A ( B and jr:At B and the Z-cell f : @J_-~- J_~¢, as diagrammed below
A( J- B
A'( B' j'_
We i n d i c a t e the da t a fo r S_ by the 6 - t u p l e d(S_) = (-; q~ ~; J_, J_~;f-) of da t a
f r o m ~,, i . e . , ( s i g n a t u r e ; lef t , r i gh t ; top, b o t t o m ; i n s i d e ) . When the s i g n a t u r e
is obv ious f r o m the con t ex t we o m i t i t .
The le f t i d e n t i t y £(S_) f o r S has da ta (¢, ¢ ; A , A ' ; ¢ ) , r(S) has da t a
(~ ,~; B , B ' ; ~ ) ) t ( S ) has da ta (A,B; J_, J_; J_), and b(S_) h a s da ta ( A ' , B ' ; J , j_t; j_~).
Second , c o n s t r u c t a + ~ s q u a r e S+ f r o m 1 -c e l l s J + : A ( B and J ~ : A ' ( B ' and
the Z - c e l l f+: J~¢-.~----- ~J+, a s d i a g r a m m e d b e l o w
A,. J+ )B
s+ ¢
It ha s d a t a (+; ¢, ~/ ; J+, J ; ; f+) and i d e n t i t i e s I(S+) wi th da t a (¢ ,¢ ;A ,A ' ;~b ) ) r ( s )
wi th da t a ( ~ , J ; B , B ' ; ~ ) , t(S) wi th da t a ( A , B ; J ÷ , J+; J÷) and b(S} wi th da t a
# I # # I (A ,B ;J~, J~; J+). T h i r d , c o n s t r u c t a + / - ~ s q u a r e S+/_ f r o m 1 - c e l l s J÷:A ; B
and J: : A ' ( B ' and the 2 - c e i l f+/_: ! b ~ J: #J+, a s d i a g r a m m e d b e l o w
128
_
_~ J+ , B
¢ ~ 4
s _ B s
J i
The data and identities are similar to the preceding two types. ~(S) has data
(+/- ;¢, ¢;A,A';$), r(S) has data (+/- ;*,4; B,B';4), t(S) has data (+;A,B;J+,J+;J+)
_ A ' B I ; ' j t ; j,_ a n d b(S) h a s d a t a ( ; J_ , _ ).
I g $ L a s t , c o n s t r u c t a - / + ~ s q u a r e S _ / + f r o m l - c e l l s J : A * B a n d J + : A - - ~ B
%
a n d the 2 - c e l l f /+ : J ~ S J - ~ 4. The r e a d e r is i n v i t e d to c o n s t r u c t the d i a g r a m
a n d r e l a t e d d a t a . We a l w a y s h a v e 1 - c e l l s w i t h p l u s s i g n a t u r e f r o m l e f t to r i g h t a n d
w i t h m i n u s s i g n a t u r e f r o m r i g h t to lef t ; and the two c e l l s a l w a y s go f r o m r i g h t t_~o
le f t , i . e . , f r o m 4 - c o m p o n e n t to S - c o m p o n e n t . L e t i, and i ' be + o r and a b -
b r e v i a t e i]i = i f o r s i g n a t u r e s .
Def. 2.2 We define vertical #-composition and horizontal *-composition
in the only way possible. Consider two ~ squares S and S' with arbitrary signa-
tures a / b and c / d s u c h tha t b(S) e q u a l s t (S ' ) . T h i s f o r c e s the e q u a l i t y b = c
a n d the d a t a h a v e t h e f o r m d(S) = ( a ] b ; ¢ , 9 ; J , J ' ; f ) a n d d(S ' ) = (b/d;¢S, %,; j i j a ; f l ) .
We e x a m i n e the d a t a d ( S # S ' ) f o r the c o m p o s i t e S o v e r S e and s e e a l l bu t the
l a s t i t e m is c l e a r a n d e v e n the l a s t i t e m is u n i q u e l y d e t e r m i n e d :
d ( S # S ' ) = ( a / d ; ¢ ' ¢ , * ' , ; J , J ' ; f") .
To d i s c o v e r the i n s i d e fu we m u s t e x a m i n e d e t a i l s . S i n c e t h e r e a r e e i g h t
p o s s i b i l i t i e s f o r o r i g i n a l s i g n a t u r e s ( the t r i p l e (a ,b ,d ) h a s e i g h t p o s s i b l e v a l u e s )
t h e r e a r e e i g h t c a s e s of v e r t i c a l c o m p o s i t i o n . We n o t e tha t the (+,+,+) and ( - , - , - )
c a s e s a r e s t r a i g h t f o r w a r d , so we w i l l d e r i v e the
a l l of t h e m . G i v e n # - c o m p o s a b l e ~ s q u a r e s S.~_
d e n t f r o m the d i a g r a m s b e l o w :
(+ , - ,+) c a s e and t h e n j u s t l i s t
--/SI/+ the d e r i v a t i o n is e v i - o v e r
129
A J+ ~B
A', J' '3' l ~l[ J-I+ ,,
a" ~-B" #
J +
¢'¢
J+
N J+
J+
.f%J
fH Thus, in the (+,-,+) case equals /"¢'f o f'~j. We list a[[ cases in a table be-
low, and invite the reader to construct the corresponding diagrams.
(a,b,d) f"
(-,-,-)
( - , - , + )
( - , + , - )
(-,+,+)
(+,-,-)
(+,-,+)
(+,+,-)
(+,+,+)
~'f o f'~
u j+~'f o f'~
f'#j o j"%'f
f' ~j o ~'f
O'f o f'~ j
j"~'f o f'? j
f'~ o j"?'f
f'~ o ~'f
s q u a r e s f o r m a c a t e g o r y ( ~ S q ) # u n d e r # c o m p o s i t i o n . T h e r e a d e r i s i n v i t e d
to v e r i f y t h e a s s o c i a t i v i t y a x i o m : t h e r e a r e 16 (=2 4 ) c a s e s .
We n o w d e f i n e h o r i z o n t a l "~ c o m p o s i t i o n in a n a t u r a l f a s h i o n . C o n s i d e r two
C - s q u a r e s S a n d T of a r b i t r a r y s i g n a t u r e a / b a n d c / d s u c h t h a t r (S) e q u a l s
L ( T ) . T h i s f o r c e s t he e q u a l i t y of s i g n a t u r e s a = c a n d b = d . ( T h u s t h e r e w i l l b e
f o u r d i s j o i n t - ' : - ' - s u b c a t e g o r i e s - - o n e f o r e a c h s i g n a t u r e ) . T h e d a t a h a v e t he f o r m
130
d(S) = ( a / b ; ¢ , ~ ; J , J ~ ; f ) a n d d (T) = ( a / b ; ¢ , X ; K , K ' ; g ) . We e x a m i n e t he d a t a d ( S * T )
f o r t h e c o m p o s i t e S n e x t to T a n d s e e the f i r s t t h r e e i t e m s c l e a r l y : d ( S * T ) =
(a/b;¢, x;L,L';h), The last three items vary with the signature. Bgnabou [3] has
treated the + and cases, so we will limit ourselves to a diagram of the +/-
case and a tabulation of all four cases.
A J+ ,B ~ , c A %J+ ,B
¢ ~ - ' ~ " - - g - - x ¢
! l ! B l A'~ j , B'~ KL c ' ~ = = C'¢J+ ~ ~KxZ+5+ 'J'K'_ f ggJ+
signature
+
+/-
-/+
top
K+J
K+J
J K
J K
bottom
Y_ K'_ l !
K+J+
J' i<'_
inside
K+f o g j+
f o jig j+
K+f K o g
fK_o jig
Thus @ squares form a category (@Sq) under horizontal composition.
Example. Given a category ~, Linton's [9] 'r~wisted morphism" category
A is just (-/+@ Sqi:', if we take ~ to be the degenerate 2-category L determines.
We are now ready to prove
Theorem 2.3
~:' composition.
The ~ squares form a double category @Sq under # and
P r o o f .
b e e n s a t i s f i e d .
F o r P a r t
We verify the definition. Parts i) and 2) of the definition have
3 ) , ( ~ Sq% # i s c l o s e d u n d e r ~'.-" c o m p o s i t i o n . A ~ s q u a r e i s a
131
#-identity if and only if its left side and right side are degenerate l-ceLLs, its in-
side is a degenerate Z-cell, and its signature is + or - . These conditions are
preserved under ,*--composition. Similarly for 4), (C~Sq) ° is closed under #
composition, since a C~ square is a ".'-identity if and only if the top and bottom are
degenerate l-ceils and the inside is a degenerate Z-celL
It only remains to verify 5), the permutabifity axiom
(S*T)#(S'*T') = (S#S')*(T#T')
' T' for composable squares S,S,T and . The outsides are equaL, e.g., the left
identities are
The equality of the insides can be shown by a "path analysis". There are eight
c a s e s v a r y i n g w i t h t he s i g n a t u r e s a s i n s i m p l e # - c o m p o s i t i o n . We d i a g r a m the
( + , - , + ) c a s e b e l o w :
S'
J K A , B •
J' K' A'~ B'*
1 [ f' ~, g
n
j# K ~I
X T
.I
k' T'
C"
The first 0-cell is A and the last 0-cell is C I'. There are six paths from
A to C u, a n d t h e y a r e c o n n e c t e d b y e x p a n s i o n s of t h e i n s i d e s of t h e ~ - s q u a r e s a s
d i a g r a m m e d b e t o w , w h e r e w e h a v e l a b e l e d t he d o u b l e a r r o w s w i t h t he s q u a r e s t h a t
i n d u c e t h e m .
132
> I~','~,, J _ _ > . ,> > ... )
I~'J'¢'¢ < I~'J'¢'J"~J > , > I~','K'XKJ <T' X'XKJ
"4 "
I~'S¢'J'K'XK J
T h e d i a g r a m s i t s in the c a t e g o r y ~ ( A , C a ) . T h e d i a m o n d c o m m u t e s by the b i f u n c -
t o r a l i t y of j u x t a p o s i t i o n . Thus the top c o m p o s i t i o n , w h i c h is the i n s i d e of
(S # S ' ) * (T # T ' ) e q u a l s the b o t t o m c o m p o s i t i o n , w h i c h is the i n s i d e of ( S ' T ) # (SQ-'T').
T h e o t h e r c a s e s a r e t r e a t e d s i m i l a r l y .
T h u s ~ s q u a r e s f o r m a d o u b l e c a t e g o r y and we a r e in a p o s i t i o n to a p p l y
the a n a l y s i s of d o u b l e c a t e g o r i e s d i s c u s s e d e a r l i e r , i . e . , we e x a m i n e the 2 - c a t e -
g o r i e s i n s i d e ~ Sq.
P r o p o s i t i o n 2.4 G i v e n the 2 - c a t e g o r y ~ and the d o u b l e c a t e g o r y ~ S q , t he
2 - c a t e g o r y ( ~ S q ) ' g i v e n by P r o p o s i t i o n 1.4 is i s o m o r p h i c to the d i s j o i n t u n i o n of
~, w i t h ~ s . In d e t a i l we h a v e the i s o m o r p h i s m s :
(C~ Sq)' --- (-C~ Sq)' U (4C~ Sq)', (-C~Sq)' -~C., and (+C.Sq)' ='- C- s .
P r o o f . Let the s q u a r e S h a v e i n s i d e i(S) and c o n s i d e r t he a s s i g n m e n t
S I---~i(S). F o r S in ( - ~ S q ) ' th i s i s a 1 - 1 c o r r e s p o n d e n c e on to if- w h i c h p r e -
s e r v e s the o r d e r of b o t h c o m p o s i t i o n s
133
(-~ S q ) '
F A, B
A ~, Ft B
F
F'
For S in (+~ Sq)' this is a i-i correspondence onto C that inverts the
order of both compositions, i.e., an isomorphism (of Z-categories) onto ~s,
(+~ Sq) ' =" C "s
F A
A F '
~B
1 B
~ B
A~~fs B s
H e n c e f o r t h , we w i l l i d e n t i f y the two Z - c a t e g o r i e s C~ a n d ~s w i t h s u b d o u b l e c a t e - I
gories of ~ Sq, when convenient . The reader is invited to examine ( % Sq/) ( it
is e q u i v a l e n t to C~').
N o t a t i o n . G i v e n S a d o u b l e m a p in /~, we d i s p l a y S a s a m a p in 2~ # b y
the f o l l o w i n g # - a r r o w n o t a t i o n S : t(S)~--g- b(S) .
3. T h e D o u b l e C a t e g o r y ~ S q a n d the Z - C a t e g o r y fl(C~)
We now g e n e r a l i z e the n o t i o n of a d j o i n t f u n c t o r to a d j u n c t i o n s of 2 - c e l l s a n d
to a d j o i n t s q u a r e s .
Def. 3.1 A ~ - a d j u n c t i o n I c o n s i s t s of two # - i n v e r s e ~ s q u a r e s _J+/_ a n d
l _ / + w i t h d a t a i n c l u d i n g d e g e n e r a t e 1 - c e l l s on the l e f t a n d r i g h t , i . e . , d(J+/_) =
( + / - ; A , B ; J + , J ; j + / _ ) , d(J_/+) = ( - / + ; A , B ; J _ , J + ; j / + ) , a n d s u c h t h a t J + / _ # J _ / + e q u a l s
the # identity J/ and _J_l+#_J+l_ equals a_. We say £ is an adjunction from
B to A.
if C~ e q u a l s ~ a t , J+ is r i g h t a d j o i n t to J w i t h un i t j_/+ a n d c o u n i t j+/_.
134
Conversely, adjoints with unit or counit specified, yield ~at
O-cell A in ~, we have a degenerate @-adjunction A with
a d j u n c t i o n s . F o r e a c h
d(~+/_ ) = 0 / - ; A , A ~ , . ~ ) .
Def. 3.2 A ~ s q u a r e S is s p e c i f i e d b y the f o l l o w i n g da t a f r o m ~ S q
j i d(S) = (J+/_ , S_/+, _+/_ ) s u c h tha t the f i r s t a n d l a s t s q u a r e s c o m e f r o m a d j u n c t i o n s J
a n d J ' a n d the # c o m p o s i t e J+/_ #__8_/+#J~+/_ is d e f i n e d in ~ Sq. V e r t i c a l # - c o r n -
position is defined by
d(S#S') = (J÷ ,S +#J~+ S' * - - /_ ---/ - - t_ # _ _ /+ ' - -J +/_)"
And horizontal *-composition is defined by
d (S# ~T) = (J+/-- - * I K ~ / - ' S - / + * T - / + ' - 4 - J ' *K+] ) '
w h e r e the c o m p o s i t i o n s on the r i g h t a r e in ~ S q .
c a l l S a n a d j o i n t s q u a r e .
We s t a t e the o b v i o u s
If we can o m i t m e n t i o n of ~ , we
T h e o r e m 3 . 3 . ~ s q u a r e s f o r m a d o u b l e c a t e g o r y ~ ¢ S q u n d e r ~ a n d
~:~ i compositions.
P r o o f . Th e v e r i f i c a t i o n of the f ive a x i o m s f o l l o w s c o m p o n e n t w i s e f r o m
t h e i r t r u t h in ~ Sq.
A n a d j u n c t i o n ~ = (J+/_, J_/+) d e t e r m i n e s a ff~- s q u a r e S(J ) w i t h da t a
d(S(_J)) = (_J /_ , J _/+, J+/_ ). W h e n w o r k i n g in the c o n t e x t of ~ s q u a r e s we le t
n o t e S(J) . _J is a # i d e n t i t y in ~ S q .
J de -
Def. 3 .4 The m a t r i x m(S) for a ~ s q u a r e S is the f o l l o w i n g Z by Z
a r r a y of G s q u a r e s
( ,J ,i denotedby ('') s . # Y , / s_/+ s
~ - / + - 4 + - ~ - ! - -
135
Proposition 3.5
compatibility relations:
The entries of the matrix m(_S) satisfy the following
( c ) S l ~ =_Jl~#SBl~#!~/5 (~,B,~,8 =+ o r -) .
I Also, if S # S' is defined, -- !Sc~'B #S~/.f_ (S # S')(~/. 8
Lemma 3.6 Given two ~ adjunctions J and jl any of the following
sets of data is sufficient to uniquely determine a ~ square S with top J and
bottom J' .
a)
b)
c)
Moreover, if the data comes from a
A matrix of squares satisfying Equation C.
Any cd~O-square s/~:i-~7-_J B (a,e = + or -).
The inside %/B = i(S~/B) of such a square together with the left side,
i.e., the l-cell ¢:A --~A', the right side, i.e. , ~ : B--~B', and the
signature o~/B.
square S , we recover exactly this square-S.
Theorem 3.7 Given S in ~ Sq the assignment S ~--~S (resp., SI • S )
gives a d o u b l e functor
F : ~ Sq * - ~ Sq ( r e s p . , ~ + : f l~ Sq ~ + ~ Sq).
T h e s e functors are faithful since t h e i r # c o m p o n e n t s ~ # ( / r : ) a r e faithful. A l s o
the assignment s~-~S_l+ (resp., s~-.s+/_) gives a functor
TT_/+ : ( ~ Sq) ~ (- /+ ~ Sq);:" r e s p . , ~+/_ : ( ~ Sq) ) (+/- ~ Sq ; .
Proposition 3.8 Given the outside data of a ~C~ square S, i.e. , l-cells
¢:A-----~-A t, ~ : B - - - ~ B ' a n d a d j u n c t i o n s J : A - ~ - - - B a n d J ' :A~--~ --- B , w e h a v e the
following isomorphisms of classes of Z-cells
136
,+ ¢(B,B )(~, J CJ_)
• ~(A,A'I(J:¢ J+, ~I
I I , e . ( B , a ) ( J , , e J )
(This fact, observed by Linton [9], started my development of the theory of squares.)
Proof. Given a ~C~ square S, call the matrix of Z-ceils
i(s+) i(s+/_) 1 i(s_/+l i(s_) /
the inside of __S, i(S). Let __S have the outside data given in the Proposition, then
the components of the matrix i(S) sit in the appropriate classes of the theorem.
Thus we have a map from the class of squares S with fixed outside into each cor-
ner of the diagram. These maps are isomorphisms by Lemma 3.6, and they give
the maps of the diagram.
As a special case we have the well known fact that given two C~at adjunc-
tions and a natural transformation j : J~----- J'_ between the left adjoints, there
exists a unique natural transformation j+: J+====,~J~, called the conjugate, in the
opposite direction between right adjoints. In our notation
S j+ is the +C~ Square _J+/_ # j_ #__/jr '+ .
A useful setting for the further study of adjunctions is given by the following
Definition 3.9. Given a Z-category C~, let 2(C~) be the canonical Z-category
(ZC~Sq) t derived from ~Sq by Proposition 1.4.
Note. Given a Z-category C~ Maranda defines a Z-category ~# in [13, pp.
763-?64], having the same 0-cells as ~. ~#(A,B) is the category whose objects J
are the same as those of ~(C)(A,B), i.e., adjunctions, and whose maps J ~-J'
I are pairs (~: J+ ----~J+, B: J_ ====~j_t) of Z-cells satisfying the equation
137
J-/+ - ' J J or8 ~ '-J ' J ' ) e q u a l s J ' 1 B + - + - -/+
and the dual equation
e q u a l s J~_
P r o p o s i t i o n 3.10. ~#(A,B) is i s o m o r p h i c to the s u b c a t e 6 o r y of ~(~) (A,B)
c o n s i s t i n g of i n v e r t i b l e Z - c e i l s .
Proof. The 2-cell B thought of as a -~ square lifts uniquely to a 2~
s q u a r e S ' , i . e . , i(S~): ft. S i m i l a r l y , c~ c o r r e s p o n d s to a + ~ s q u a r e s and
~ s q u a r e S, i . e . , i(S+) = ~. The M a r a n d a r e l a t i o n s s a y tha t the i n s i d e s of the
~ s q u a r e s S ' # J + # S + and J ' S ' S ' J ' .... / _ __/+ are equal. Hence, #J#S(= #S) and _ are
equal by Lemma 3.6. Similarly, S#S' equals the # identity ~, and thus S and
s, a r e # i n v e r s e . Th i s g i v e s i( ) as the i n v e r s e of c~
of 8, and J ' is i s o m o r p h i c to J v ia S ' .
S $ $ : J - - ~ and i t s i n v e r s e S
and i(S_) a s the i n v e r s e
m a p
C o n v e r s e l y , g iven in ~(C~(A,B) we h a v e t h e
( i ndeed i s o m o r p h i s m ) , - . ( i ( S + ) ' i ( s t ) ~ in ~ # ( A , B ) . R e c a l l i n g T h e o r e m 3.7 we
obtain the fo l l owing
C o r o l l a r y 3.11 The i n d u c e d f u n c t o r s
~T : ~(C,)(A,B)--~(A,B) and ~+: ~C,)(A,B) --~S(A,B)
a r e fu l l and f a i t h fu l , i . e . ,
~(A,B)IJ, Y) --~ ~(¢)(A,B)(J, Y) ~(A,B)(J+, ~).
Hence , g iven a c a t e g o r y X a, and a f u n c t o r
P :X---~(A,B), P lifts to P:X ~(~)(A,B), i.e., IT P = P ,
if and on ly if fo r e a c h o b j e c t X in X, P_(X) i s J_ f o r s o m e a d j u n c t i o n _J in
138
X --~ ~ s (A, B) . ~(~)(A,B). The ana logous s t a t e m e n t holds for P + :
Proof. These categories are # subcategories of those in Theorem 3.7.
T h e o r e m 3.12 Let e+:+~SqC-- -~-~Sq and e : - ~ S q ~ -~ -~Sq be the i n c l u -
s i on functors. Let J be an adjunction and 0(J) : _J+/_. Then the two maps (e+1"[+) #
and (e_T[_) # of (2~Sq) # into (~Sq) # are naturally isomorphic via 0, as diagram-
m e d below
(_~ sql# o > (+eSql #
oroovor fO:
to c a t e g o r i e s ~X and f u n c t o r s P _ : X - - - ~ ( - ~ S q ) # ~ and P + : X - - ~ ( + ~ S q ) # equipped
with a n a t u r a l i s o m o r p h i s m { f r o m e_D_ to e+P+ .
de f ine
P roof . Given the f u n c t o r s P_ and P+ and the n a t u r a l i s o m o r p h i s m
P : X ~ ( ~ S q ) # as fo l lows . F o r each ob jec t X in X let P{X) be the ad-
j u n c t i o n with m a t r i x
p+(x) i(x) ) ~-t(x) e_(x)
This is an ad junc t i on , s i nce P+(X) and P (X) a r e i d e n t i t i e s in (~ Sq) #, and
~(X) : P_(X) ~*P+(X) is an i s o m o r p h i s m . Thus we can apply T h e o r e m 3.7 to ob ta in
the Hfting of P+ to P, and we have the required commutative diagram
139
X
P , P+
(-C. Sq) (+C Sq) #
!
(#c. Sq)
C o r o l l a r y 3.13 Let e : g ( A , B ) ,(-O-Sq) ~" • ( ~ S q and
4 : C]S(A,B)---~(+C- Sq) #1" ~-(~ Sq) # be the n a t u r a l e m b e d d i n g s . The c a t e g o r y
~(~)(A,B) and the pa i r of f u n c t o r s
1T : #(C-.)(A,B)---~C-(A,B) and Tr+:~(C.)(A,B) , ~ s ( a , B )
equ ipped with the n a t u r a l i s o m o r p h i s m e ' : e'lT_ _ .~- e;l~+ a r e u n i v e r s a l with r e -
spec t to c a t e g o r i e s X and p a i r s of f u n c t o r s P : X - - - ~ ( A , B ) and P + : X - " ~ S ( A , B )
t equipped with n a t u r a l i s o m o r p h i s m s }: e'P_ _ ..~ e+P+ , i .e . , we have a un ique
l i f ted func to r P : X ~$(~)(A,B) s a t i s f y i n g the e q u a l i t i e s r r p = P _ , i T + p = p + , and
@'p=~.
An a p p l i c a t i o n of this c o r o l l a r y is g iven in the next s ec t ion .
4. The 2- F u n c t o r s Jgom~ and I~'om#c "
In this s ec t ion we g e n e r a l i z e the no t ion of horn f unc t o r [1Z] to Z-categories.
Let ~ be a 2 - c a t e g o r y and (A,A') a p a i r of 0 - c e l l s . ]Then ~ ( A , A ' ) is a
c a t e g o r y , i . e . , a 0 - c e l l in Cat. We ex tend this p a t t e r n by the fo l lowing
Def. 4.1
define the functor
Let F :A ~ B and F ; A' B' be a p a i r of 1 - c e l l s in O..
U-(F_, F;) : ~ (A,A' * C-(B,B')
Similarly, let (~: F -~G_ and #: F ; :=:==::a~G; be a pa i r of 2 - c e l l s in ~. We
We
140
define the natur al tr ans formation
C(o, ~'):c(F_, ~ ) --~--~(G_, %)
by the equality C-(c~, c~')(a) = c/a¢ for each a in ~(A,A'). Hence, ~(F_, F4~ ) is a
l-cell in ~t and ~(~,c~ p) is a Z-cell in ~at.
In Definition 4.1 everything was transformed covariantly except F_ ; hence,
we have t he following
Theorem 4.Z. For ~ any Z-category,
JVorr~ : c--txC- .-
(A,A')
_ F' G '
l i ( B, B' )~
~at
.~A,A')
"~ C,(B, B')"
is a 2-functor.
This new functor is closely related to the ordinary horn functor,
h o m c : C ° P x c - - - - ~ S e t s , fo r a c a t e g o r y C, w h e r e Se t s is the c a t e g o r y of s e t s [12].
The r e I a t i o n s h i p is g i v e n by the fo i i owing
Corollary 4.3. For ~ any Z-category, ~orr~ is an extension of the ord-
inary horn functor for ~[1], i.e. , the following diagram commutes up to the natural
inc lus ion
~IZ'omo. C t XC. C.at
[1] X~[1. ] Sets horn[ I ] .,~.~',v,',~,
141
where for a set X we let D(X) be the category whose only maps are the identities
I for each element x in X, and ~[i] is the l-skeleton of ~ [3], i.e., the ordi- x
nary category consisting of l-cells from ~J.
Corollary 4.4.
g -functor
For C~ any Z-category, we have induced a unique
~'om~.: ; (e . ) c x 2(e.) , ~e .a t )
w h o s e v a l u e s a r e a d j o i n t s , s u c h t h a t t h e f o l l o w i n g d i a g r a m c o m m u t e s
= [ c - txc - ] I- ~ a t
~(~/c x ~(~) . . . . . . . _~;o~_~_ - -> ;( t/
1 l, = ~ C a t s
Proof. The proof is essentially an application of Corollary 3.13.
F i x i n g v a r i a b l e s we h a v e e x p o n e n t i a t i o n , i . e . , t h e p a r t i a l
X (-)~ = ~om~ (X, -) ~(G) , ~(Cat) and
"horn functors"
are 2-functors. In particular exponentiation preserves adjointness.
5. ~(C.)(A~B ) as a Relative Category
In this section we examine a different kind of "horn functor" on the category
142
~(¢)(A,B), i . e . , the functor
o p r
I : ~(~)(A,B) × L${C.)(A,B)J
(,J', D I . 1 / . . I •
• ,L j _ = ~ 3 _
If j : J1 "~== Jz is in ~{~)(/K,B), then j_: 5--1 ~ J--Z- deno te s the ins ide of
the -~ s q u a r e j _ and j+ : J l + = = ~ J z + d e n o t e s the ins ide of j + . S i m i l a r l y , let
J+/_ : J J+ ==~1A denote the ins ide of J_+/_ and J_/+: % ~ J+J_ denote the ins ide
of J / + . And ~(B,B} is a m u l t i p l i c a t i v e c a t e g o r y [1,3] in the u s u a l way. We r e -
f e r to B~nabou [2] fo r the not ion of a r e l a t i v e c a t e g o r y o v e r a m u l t i p l i c a t i v e
c a t e g o r y .
Theorem 5.1. Let ~ be a 2-category and A,B be a pair of 0-ceils in ~.
Then the category ~(~)(A,B) induces a category ~(A,B) relative to the multiplica-
tire category ~(B,B) . The objects in ~(A,B) are ~ adjunctions, and the under- m
lying category of ~ is just ~(~)(A,B). Similarly, we have a category ~I(A,B)
{with the s a m e objec ts ) r e l a t i v e to ~ S ( A , A . . m
Proof.
i)
2)
3)
4)
The data for
We summarize the data for £= £(A,B).
The objects of ~ are the objects of ~(¢)(A,B), i.e., adjunctions
j,j1,.., from B to A.
Let jt/j equal {J;)(J ) an object in ~(B,B).
Let c(J,J__~,J"):(J__"/J~)(J1/J) ~ (j1~/j) equal
Let i(J) : 1B • J/J equal the 2-cell J_/+.
~t(A,B) is dual.
[3]
Remark. An object A in a category ~ relative to 9/ gives the monoid m
( A / A , i ( A ) , c { A , A , A ) ~ in 9 /m. Le t t ing C1 equal C.at, we r e c o v e r the well
143
known fact that a C~at-adjunction J determines a monad (triple)
( T = J + J _ , J / + : I B - - ' ~ - T , J ÷ J + / _ J _ : T T ~T) .
6. Cylinders.
In Section 7 we will recall some of B6nabou's [1,3] treatment of multiplica-
tive categories and their maps and lift these notions to autonomous categories. But
first we need to generalize B6nabou's notion of cylinder [3].
Def. 6. i.
double maps in
Z-category /~i contained in ~.
we call the 4-tuple
Let l) be a double category, and let UI, Uz,V I, and V 2 be
I~I. Moreover, let U 1 and U 2 be Z-cells in the canonical
If UI#V 1 and Vz#U 2 are defined and equal in I>#,
(commutative) ~ cylinder.
Taking components of Q
cylinders to I/~I, satisfying
We examine
c~ = ( u I , u Z, v I , v z)
gives four projections top, bog fr, and b__kk, from
Q = (top(Q), bot(Q), fr(Q), bk(Q)) .
two ways to paste such cylinders together in
Def. 6.2. Le t Q, % , and R be 2~ cy l inde r s . F i r s t , Q and %
composable if fr(%) equals bk(Q); and their composite %oQ is
(top(Ql)#top(Q) , bot(O.l)#bot(Q), fr(Q), bk(Q1) ).
Second,
are o
~'~ composition is componentwise. Q~',-'R is defined and equal to the 4-tuple
(top(Q) ~ ~ top(R), bot(Q) ~',-~ hot(R), fr (Q) ~:~ fr (R), bk(Q) ~ bk(R)1 '
144
if each component on the right is defined in /}
We may picture o composition as back to front pasting, and ~-~ composition
as horizontal pasting.
Theorem 6.3. For a double category ~, ~ cylinders form a 2-category
/~Cyl. The projections top and hot give 2-functors from ~Cy[to ~' and are
natural in ~.
For l}= -C~ Sq, we have B~nabou's cylinders based on C~ with projections
into ~ = (-C~ Sq)' [3, p. 73].
We will abbreviate (C~Sq) Cyl to C~ Cyl, (~Sq) Cyl to ~I~ Cyl, (-C~ Sq) Cyl
to -C~ Cyl, etc.
Proposition 6,4. The projections F+:~C~Sq--~+C~ Sq and
F_ : 2GSq--~ -C~ Sq induce projection 2-functors
T[+ : ~I~ Cyl --~ ~C~ Cyl and K : ~ Cyl --~ -<5 Cyl .
7. Maps of Multiplicative and Autonomous Categories.
We are now in a position to use lifting to describe the relation between mul-
tiplicative categories and autonomous categories. Our purpose is to illustrate the
viewpoint of adjoint squares and cylinders and their use in proving theorems. For
a different treatment of Theorems 7.7and 7.9 the reader is invited to read about the
"basic situation ~' in Eilenberg-Kelly [5, p. 477-489].
We start with the notion that a multiplicative category 9/ is a 6-tuple rn
(~,®,l,a,~,r), satisfying the coherence conditions that the following diagrams
c omrnute.
145
M1
(A ®B) ®(C ®D) (A® (B ®C)) ® D
MZ ~®(A®B) a ) (ZeA)® B
' ~ ' ~ ' ~ A ® B ~ B
M3 A®(~®B) a ) (A®I)®B
B A®B
Mac Lane [i i] and Kelly [8] have shown that M1 and M3 imply coherence.
A well known example of a multiplicative category is
: juxtaposit ion, 1., id, id, id) .
Def. 7.1. Let 9/ and 9/i = (At,®t, II, a1,~l,r l) be multiplicative categor- m m
ies. A multiplicative functor F from 9] to 9/I is a triple (F, fo f2) where m m m
F : A ~ A' is a functor, fo : I' ~ F(1) is a map in A', and
fZ = If Z ®, ~A,B :F(A) F(B) ~F(A ®B)j ~
is a family of natural transformations. These data must satisfy the relations given
by the following three commutative diagrams:
146
F 1
F 2
F 3
FA®(FB®FC)
~ FA®f 2 FA®F(BeC)
F(A ®(B®C))
I'®' FA -
FA <
FA ®I'
F A <
a
F(a)
fo® FA )
> (FA®FB) eFC
F ( A ® B ) ® F C
FI®' FA
-F(I®A)
FA ®' FI
-. F(A ®I)
F is strictly multiplicative if fo and f2 are identities. m
Examples. For a multiplicative category 9/ , the left regular m
representation
Lm = \(L:A-~C~at(A'A)' ~°= ~ - i : , ~ ~ ~ 1A=--~-I ®-' (A,B)- aA, B ;A® (B~--) =~-(A®B)®-)
AI ~A®-
is a multiplicative functor from 9/ to ~at(A,A) . Also, ~ : ~)(A,A) -+~(A,A~ m ~ m m rn
~+:~($)(A,A)m--~c~s(A,A)m are strictly multiplicative. and
The data of the left regular representation suggests the
A Def. 7.2. For a multiplicative category 9/ define the functor -~J :A~----A
m -,~
by the equality, % (x) = A®x for x any map in A. For a multiplicative functor
Fm : (F, f°,f2): 9/m * 9/'m' define the -C~at square AS : AS(Frn )_ by the diagram
147
8< A NA
• A ' ( FAj.
We now translate F2 and F1 into squares and cylinders. Condition F2
on 1 ? is equivalent to the commutativity, for each A in A of the -Cat cylinder m
h
1
with front the -C.at square (-;F,F;A,A';F), back ~ , top~° :IJ~ m= 1 and ~ - A
bottom (foj_)(~,)o : F%,~c = Ijt~_ IA '
S i m i l a r l y , c o n d i t i o n F1 i s e q u i v a l e n t to t h e c o m m u t a t i v i t y , f o r e a c h A
and B in A of the -C~at cylinder
A,B~Z
148
with front (As_)-",-" (BS), back A®Bs_, top ~2: A®Bj.~.== Aj#j.~, and bottom
Theorem 7.3. Given a map of multiplicative categories F : ~ ~" ~' , m m m
the triple
Qm(Fm) : ( Q : A . ~ AS(Frn) , ,QO(~rn), [A~BQ2_])
is a map of multiplicative categories from m
(top) Qrn equals L m
in the diagram
and (bot) Qrn equals
to -Cat Cyl(F,F)rn. Moreover,
Z ~ F, which are both multiplicative, as m
~a m m
% m
/ /
/
Lr- r n
O- at(A, A) tta t~*,\ rrl
~ top
-) -C.at CyI(F, t;') m
~ bot
Def. 7.4. Let ~ be a multiplicative category. If for each A in A, we m
have a @at adjunction A~ with Aj equal to A®- , then we say the collection
{Aj_} gives ~m a left autonomous structure. We call the pair ~ = (~rn' [Aj_])
a left autonomous category.
An application of lifting using Corollary 3.11 gives an automatic proof of
the following
Theorem 7.5.
omous structure {A j]
Let ~ be a multiplicative category. ~ has a left auton- m m
if and only if the left regular representation
Lm : ~m - ' ~ Cat(A, A) m lifts along
149
Jr_m: ~ (6at)(A, A) m --~-~ at(A, A)m
to a multiplicative functor
__( , Bz) ( )j ( )j,oj :A ~ J__,j : (Aj)(Bj)_.~ A@ , - - m -- tt~ -- --
i ' e ' ' L =(/Tm - m ) ( ( ) $ m ) "
We recall a notion slightly more general than autonomous category in
Def. 7.6.
\ : A °p X A - - ~ A ttt~ ~ tt~
(A,B), , A\B
L e t 9/ b e a mul t ip l i ca t ive c a t e g o r y equipped with a bifunctor m
and natural isomorphisms
PA, B,C :(A®B)\C , B\(A\C) ~nd iA:A ,I\A.
We call the bifunctor \ an internal horn functor and call the 4-tuple
(%,\,p,i) a category with internal homfunctor, if the data satisfy the commu-
tativity conditions in the following diagrams:
A1
X X
A2 A\ ( I \X) * P ( I®A)\X A3 I \ ( A \ X ) , P (A®I)\X
A\X A\X
150
Mac Donald [I0] has defined coherence for a,~,r,p, and i and has shown
t h a t t h e s e t h r e e conditions t o g e t h e r with c o h e r e n c e in 9/ imply coherence. (Tech- m
nlcally, he deals with the transpose of 9/ • ) m
Coherence for autonomous categories is simply disposed of by symmetric
duality of adjoints in the following
Theorem 7.7. Each left autonomous category ~/~ uniquely determines a
category with internal horn functor i-9/~.
A Proof. Let \ be the bifunctor with values A\B = ('%r+)(B). Let PA,B,C
equal (C) and let i equal where we use the data from Theorem 7.5. ~B
By the symmetric duality of adjoints, the commutativity of A1 is equivalent to MI,
AZ to M2, and A3 to M3. In detail, for the second case consider the following
diagram in ~ (C~at)(A, A):
L2 Ij Aj ,I ~ I@Aj
Aj
It is the unique lifting of I%42 by IT_ or of A2 by [[+. Thus it commutes if either
of them do, hence they are equivalent.
Def. 7.8. Let \,p, ij and \',p',i "\I , , , ] be categories with internal
horn functor s. Let F be a multiplicative functor from ~/ to ~/' and let m m m
g -- B : FA\'FB • r(A\B)j be a natural transformation. We call the pair (%,g)
a map of categories with internal horn if the following two compatibility conditions
are satisfied. For all objects A, B, and C in .A the following two diagrams
c or r~rnut e
151
HI
I, !
F(A ® B)\' F(C)
~ fz ~FIC) F(B)\' F(A\C)
I F(B)~g
H2
f\'F(A) < f°~F(A} F(I}\'~(A)
FA > F(I\A) F(i)
We use the full power of the lifting technique, not just symmetric duality,
to prove
Theorem 7.9. Let 9/~ and 9/' be left autonomous categories and let i-~
- ;
and i-9/~ be their induced categories with internal horn functors. Then every map
F : 9/ ~ 9/' of the underlying multiplicative categories uniquely determines a m m rn
map (Fin,g) : i-9/ ---~i-9/~ of the induced categories with internal horn.
Proof. We define gA,- to be the inside of the +C~at square
FAjI \ Aj_+/_#%_# __/+/, where we use data from Definitions 7.4 and 7.2. Thus
iN,-: F(A~-)~ (l~A~) ~i (F-) determines gA,-: F~\-) --~-FA\'F-, and we con-
sider the unique 2Oat square % which is the lifting of AS .
To prove the hexagonal condition, i.e. , the commutativity of the diagram
HI for all objects A, B, and C in A, consider the ~C/at cylinder A'BQz=Q ob-
tained uniquely by lifting (via 17_) the -C.at cylinder A,BQZ_ = Q_ pictured in
i 5 2
Definition 7.2. Thus the front of Q_Q_ is the horizontal composite square %;'.-" BS ,
the back is A®Bs,_ the top is Z.j: (%)(Bj_) _> A®Bj,_ which is the lifting (via ~_)
of ~2:A®(B®-) ~(A®B)®-, and the bottom is
(FA~FB)j, -0- F(A®B)j,, -
which i s the lifting of
FA®' (FB ®' -) :===~ (FA®'FB) ®'---~- F(A®B) ®'- .
The hexagon HI and the hexagon F1 are "dual" precisely in the sense that they
are the ~ and ~_ projections of (their common lifting) the ~at cylinder A~B~!
Similarly, HZ and FZ are dual via the
Definition 7.2. In fact the multiplicative functor
~(~at) Cyl (~'~)m"
~at cylinder lifted from Qo_ in
Q of Theorem 7.3 lifts to m
This "duality" is more subtle than symmetric duality and requires us to
picture squares, but it reduces to symmetric duality when the squares "are
Z -cells"
University of California,
Irvine, California
153
[i]
[z]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[ i o]
[i i]
[ i z ]
[13]
[14]
[15]
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B~nabou, J. Introduction to Bicategories. "Reports of the Midwest Category Seminar," Lecture Notes in Mathematics, 47 (1967), 1-77, Springer, Berlin.
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Eilenberg, S. and Kelly, G. M. Closed categories. "Proceedings of the Conference on Categorical Algebra, La Jolla, 1965". Springer-Verla~ New York, 1966, 4ZI-56Z.
Freyd, P. Abelian Categories. Harper & Row, New York 1964.
Gray, J. W. "Sheaves with Values in a Category, " Topology, 1 (i 965), 1-18.
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Mac Donald, J. L. "Coherence of Adjoints, Associativities, and Identities," Arch. der Math., 19 (1968), 398-401.
Mac Lane, S. "Natural Associativity and Cornmutativity, " Rice University Studies, 49 (1963), 28-46.
Mac Lane, S. "Categorical Algebra," Bull. Am. Math. Soc., 71 (1965), 40-106.
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Mitchell, B.
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STRUCTURE ET SEMANTIQUE ABSTRAITES :
EXTENSION A DES CATEGORIES DE MORPHISMES
D'UNE PAIRE DE FONCTEURS ADJOINTS
Pierre Leroux
Received Oct., 1970
Introduction
Les th~or~mes, maintenant classiques, d'mdjonction des foncteurs Structure
et S6mantique (voir, par exemple, F.W. Lawvere [9], J. Benabou [i], F.E.J. einton
[Ii] et [12], et J. eambeck [8]) d~pendent essentiellement de l'adjonction des
foncteurs "exponentiation" et "hom interne" dans la "cat6gorie" Cat des cat6gories
et foncteurs.
Nous montrons, darts ce travail, qu'il s'agit d'un processus d'extension
des categories "comma" de morphismes d'une paire de foncteurs adjoints qui peut
s'effectuer dans un cadre beaucoup plus g~n~ral. Ace niveau d'abstraction, le
processus est tr~s simple et, de plus, s'applique ~ d'autres situations, con~ne les
structures quasi-quotients de C. Ehresmann [3] et les transferts de structures
d'effscement [iO].
Pour cela, nous sommes amends ~ ~tablir une th~orie g~n6rale, dans l'esprit
du calcul des cat6gories "comma" (J.W. Gray [4]), dont le r~sultat central est le
suivant : si U : C ) ~ est un foncteur et h : D ---> U(C), oO C ~ ICI, est un
morphisme de ~, le foncteur compos~
(C, C) (C~ U)> (U(C), ~) (h; ~)> (D, ~)
U (voir § I pour la notation), not~ S h , poss&de un adjoint ~ gauche
: (D, > (C, C) d&s que U poss~de un adjoint & gauche et que C admet des ~)
sommes fibr6es finies. D'autres th6or~mes d'adjonction sont obtenus en s'inspirant
construction de ~ , dont ouelques-uns sont religs ~ la notion de "locally de la
adjunctable functors" de J.J. Kaput [7], et m~nent ~ des thgor~mes d'existence de
structures U-quasi-quotients et de U-sous-morphismes engendr~s.
155
Utilisant la notion de coTmage relative, nous pouvons alors formuler une
th4orie abstraite de Structure et S4mantique ; les th4or~mes classiques d'adjonction
en sont un cas particulier.
Nous avons entrepris dans [1o3 une ~tude des structures d'effacement,
r~cemment d~finies par W. Zimmerman [15] et g4n~ralisant les structures injectives
de Maranda [13]. Nous mentionnons ici le processus de transfert (inverse) des struc-
tures d'effacement que la th~orie g~n4rale nous a inspir4 et qui est essentiellement
different et, en un certain sens, adjoint ~ gauche du processus connu [13], [15]
de transfert (direct) de ces structures.
Nous ne nous sommes pas pr4occup4s des probl~mes de fondement soulev4s
par l'utilisation de l'exponentiation dans les cat4gories. Nous laissons au lecteur
le soin de faire lui-m~me les restrictions qui s'imposent ~ l'occasion sur la
"grandeur" des cat4gories et d'interpr4ter dans le cadre ensembliste de son choix
les "categories" Cat, Catd, C G, etc.
Ce travail constitue, avec [i0], is majeure partie de notre th~se de
doctorat pr~sent~e ~ la Facult4 des Sciences de l'Universit4 de Montreal. Nous
voulons exprimer route notre reconnaissance ~ M. Jean Maranda, dont les conseils
judicieux et l'encouragement constant furent essentiels. Nous svons aussi b4n~fici4
du support financier du Consiel national de recherches du Canada pendant l'~laboration
de ce travail.
§i. Foncteurs induits entre cat4$ories de morphismes
Si C est une cat4gorie, [C[ d~signe la classe des objets de C ; si C et C'
sont des objets de C, C(C, C') d~signe l'ensemble des morphismes de C ~ C' dans C ;
1C d~note le morphisme unit~ de C ~ C, et I C, le foncteur identit4 de C ~ C. Si
U : C > ~ est un foncteur, t U est la transformation naturelle identit~ de U g U ;
U ~ ; ~ > ~ est le foncteur induit entre les cat4gories duales.
Soient F : ~ > C et G : ~ > C, deux foncteurs ayant m~me codomaine.
Rappelons que is cat~gorie ("comma" de F.W. Lawvere [9]) de morphismes (F, G) d4ter-
min4e par F et G est obtenue en demandant que le diagramme suivant soit une limite
156
projective dans la cmt~gorie Cat des categories et foncteurs, oh Do et D I
foncteurs "domaine" et "codomaine", respectivement.
sont les
(F, G)
C C
En particulier, si G = I C : C > C, on ohtient la cat6gorie (F, IC) ,
notre plus simplement (F, C), en prenant le produit fibr4 suivant :
(F, C)
C
C 2
Par exemple, si F = C : ~ > C est le foncteur d~termin~ par l'objet C
de C, la cat~gorie (C, C) obtenue est appel~e la cat~$orie des obiets au-dessous de
C [5]. Explicitement, les objets de (C, C) sont les morphismes u : C > X de C
de dommine C ; si u' : C > X' est un autre objet, un morphisme de u ~ u' dans
(C, C) est un morphisme x : X > X' dans C pour lequel xu = u' ; la composition
est alors induite par celle de C et on a lu dans (C, C) ~gal ~ i X dmns C .
C
X ) X" X
Remarquons l'existence du foncteur oubli ~vident
I C = ~c : (c, C) > C : DI°P2 = @C
L ~C
u l > Dl(U)
xl >x
est fiddle mais n'est toutefois pas un plongement en g~n~ral. @C d~finit un
157
"diagramme" de C et C = lim @ C .
i.I. Proposition. @C cr4e les limites projectives et les co4galisateurs. Les sommes
directes dans (C, C) sont des sommes fibr~es dsns C . Ainsi, si C est bicompl&te,
il en est de m~me de (C, C).
Si f : C' > C est un morphisme de C, on a un foncteur
(f; C) : (C, C) ---------> (C', C) : I u l > uf
[ x~-->x
f C' >C
S X
x
X'
1.2. Proposition. (f; C) est un foncteur fiddle et commute aux co4galisateurs. Si
C poss~de des limites projectives ou des sommes fibr4es finies, (f; C) commute aux
limites projectives= Si f est un ~pimorphisme, (f; C) commute aux sommes directes
et est un plongement plein. Finalement on a @c,O(f; C) = @C "
1.3. Proposition.
De plus,
(C, C) (f; C) > (C', C)
C
Si f : C' ----> C et f' : C" > C' sont des morphismes de C , on s
(f'; C)o(f; C) = (ff'; C)
(Ic; C) = l(c ' C)
En d'autres termes, on obtient un foncteur r
(-; C) : C ~ > Cat :
[ C l > (C, C)
f l > (f; C)
C et ~C = " " L J{~C]C6'C' = - J [[@C}CE'C' est une transformation naturelle de (-;C) ~ ~ o~
158
d~signe le foncteur constant
C: C* > Cat : I C l > C
[ f l > I C
On remarque que, via ~C C = l~m (-; C). Par ailleurs, le foncteur
(-; C) : C * > Cat d~finit un scindage pour le foncteur D : C ~ > C qui est ainsi o
une fibration scind~e sur C (voir A. Grothendieck, [5]).
Soit U : C > ~, un foncteur. Pour chaque objet C de C, on d~finit
un foncteur
(c; u): (c, C) > (u(c), m) U
X
> U(u)
> u(x)
/ X
x
X'
u(u)/...~ u(x)
u(c) / U(x)
u(u~~,)u(x , )
1.4. Proposition. La famille (-; U) = [(C; U)}CEICI est une transformation naturelle
de (-; C) ~ (-;~)oU*, deux foncteurs du type C * > Cat. De plus, on ales relations
siV : ~)
(-; IC) : ~(-;C)' et
> g est un autre foncteur.
(-; you) : ((-; v).u*)o(-; u) ,
U~ C* > ~*
(-; u) /~. (-;C) ~-----v/ (-; ,
Cat
Suivant D.M. Kan [63, si ~ est une cat~gorie, nous noterons par G d la
"cat~gorie" des diagrar~ne ~-valu~s dont voici une br~ve description : Les objets
de ~d sont les foncteurs D : g > ~ ; si D' : ~' > ~ est un autre objet de ~d '
un morphisme de D ~ D' est un couple (F,~) o~ F : ~----> ~' est un foncteur, et &
est une transformation naturelle de D ~ D'oF ; la composition est alors donn4e par
la r~gle (G,~)o(F,~) = (GoF, (~F)o&).
159
' ~ GoF > 3""
G G
Ainsi is proposition 1.4 montre que nous svons construit un foncteur
Comm : Cat > Cat d
:I C--->(-; C) u > (u~,(-; u)) .
Remarquons ~galement qUe la famille ~ = [~C}CEIC [ o~ ~C = U, est une
transformation naturelle de C ~ ~ oU ~ et que
~C = ~ et VoU = (~wU~)o~
Nous svons donc aussi un foncteur
Cons : Cat > Cat d
[ c > E
l u > (u ~, ~)
et il est facile de montrer que la famille ~ = [(Icw,@C)}CEICat[ est une transfor-
mation naturelle de Cormn ~ Cons.
Soient U', un autre foncteur du type C > ~, et 5, une transformation
naturelle de U A U'. Pour C E ICI et u : C > X E I(C, C) I, ross posons
(m~)u = %1 (u) = ~X
1.5. Proposition. La famille m C~ = {(m~)uEI(C, C) Iest use transformation naturelle
de (C; U) a (~C;@)o(C; U'), deux foneteurs du type (C, C) ~ (U(C), ~). Si
f : C' > C est un morphisme de C ,
(U(f); ~) ~ m C = m C' W (f; C) .
Si 5' : U' > U" et ~ : V > V' sont des transformations naturelles, oh Vet V'
sont des foncteurs du type ~ > g,
160
; m C C C = [(~C ~) w ,3 o m m(I ' o~ ~
C ~ U(C) C m~ = Lm~ ~ ((~C ; ~)o(C; U'))] o [(U(C); V) ~ m ] .
Utilisant la notion de "modification" de J. B4nabou [2] (en fait, 18
structure de 3-cat4gorie de 2-Cat), il est possible (mais tr~s long!) de munir
Cat d d'une structure de 2-cat4gorie telle que le foncteur Co=~ : Cat > Cat d soit
un 2-foncteur. C'est essentiellement ce qu'affirme la proposition pr4c4dente ; par
C}cE exemple, la famille m = [mc~ IC Iest une modification de (-;U) ~ ((-; ~)~glW)o(-; U').
De plus Cons : Cat > Cat d devient aussi un 2-foncteur et ~ : Comm > Cons, une
2-transformation naturelle.
Signalons finalement que la cat4gorie "Comma" (C, C), (not4e C/c dans
[5]) dite des objets au-dessus de C permettrait une construction analogue & celle
du foncteur Comm. Cependant, tenant compte de l'isomorphisme de cat4gories
(-)~ : Cat > Cat : I C I > C ~
[ U l ~ U ~
et de la relation
(C, C) = (C, C*) ~ ,
on constate que ces deux constructions donnent des foncteurs 4quivslents,
§ 2. Le Lemme fondamental
Soient U : C > ~ et F : ~ > C, des foncteurs pour lesquels F est
adjoint ~ gauche de U. II existe alors, pour chaque C 6 ICI at D 6 I~I, une bijection
aD, C : C(F(D), C) > ~(D, U(C))
naturelle en C et D. Darts un tel cas, nous 4crivons (F--~ U ; ~).
2.1. Lemme fondamental. Soient u : C > X et d : D > T, des morphismes de C
et ~ respectivement. Alors la bijeetion ~D,C X ST, x se restreint ~ une bijection
de l'ensemble des couple (h,g) ~ celui des couples (h,g) rendant respectivement
commutatifs les diagrammes (~) de C et (~) de ~ suivsnts :
161
F(T) ~ > X T g > U(X)
F(D) > C D > U(C) h
La d4monstration, dans la section suivante (§ 3), de l'adjonction des
u foncteurs S h et mentionn4s dans l'introduetion est bas4e sur le lemme fondamen-
tal ;dans ce but nous en donnons le raffinement suivant, d'abord 4nonc4 par Linton
dans un cas particulier [12].
Laissant fixes u : C > X E I(C, e) l, d : 4 > T E I(D, 4) I , at
h : D > U(C) (at donc aussi ~ = ~e(h) : F(D) > C), la bijection ~T,X se
restreint, par le lemme fondamental, ~ une bijection ~d,u de l'ensemble ~(d, u)
des morphismes g rendant le disgrsmme (~) commutatif ~ l'ensemble g~(d, u) des mor-
phismes g rendant le diagramme (w~) commutatif.
~ et ~ s'4tendent ~ des foncteurs naturellement 4quivslents 2.2. Proposition.
du type
(D, 4)* × (C, C) > ~ns.
En effet, six : u > u', o~ u' : C > X' 6 l(c, C) I ; et t : d' > d, o~
d' : D > T' 6 I(D, 4) I sont des morphismes de (C, C) et (D, 4) respectivement,
le diagrarmme commutatif
C(F(T), X) ~T~X
C(F(T'), X')
C(F(t), x)
se restreint au diagramme cormuutatif
~T',X'
> ~(T, U(X))
1 4(t, U(x))
> 4(T', u(x'))
Nous mentionnons ici les deux resultats suivants, hgalement tires de la
theorie elementaire des foncteurs adjoints, et qui apparaissent de fason naturelle
B la lumiere du lemme fondemental.
2.3. Corollaire. F est adjoint j. gauche de U si et seulernent si F P est adjoint B
2 gauche de U .
2.4. Corollaire. F est adjoint h gauche de U si et seulement si il existe un iso-
morphisme de categories au-dessus de B x @ :
J 3. Theoremes d'adjonction
Soient U : @ j 8, un foncteur, et h : D + U(C), un morphisme de a9 . Nous posons
S: = (h; B)o(C; U) : (C, C ) (u(c); 8) (D, 8) . u u
Explicitement, S (u) = U(u)h et Sh(x) = U(x). h
163
3.1. Remarque. On a une 4galit~
g~(d, U(u)) = (D, ~) (d, S~(u))
naturelle end 6 I(D, ~)[ et u E I(C, C)[. En effet, pour t : d' ----> d dams (D, ~)
et x : u - > u' dans (C, C) ,
g~(t, U(x)) = (D, ~1 (t, S~(xl)
U Pour cela nous Notre but est de construire un adjoint ~ gauche de S h.
supposons que U poss~de un adjoint ~ gauche, F : ~ > C, et que C admet des sommes
fibr4es finies (pushouts). Alors si h : F(D)
h, le foncteur
~ : (D, ~)
est construit de la faqon suivante : Si d : D
> C est le morphisme correspondant
>(c, C)
> Test un objet de (D, ~), M~(d)
est le morphisme obtenu en prenant la somme fibr4e suivante dans C :
d F(T) > P(d)
F(d)
F(D)
~(d)
> C
sit : d' ----> d est un morphisme dans (D, ~), o~ d' : D
morphisme rendant commutatif le diagramme suivant :
> T', ~(t) est le seul
F(T) d >p(d)
g F(D) > C
3.2. Remarque. Posant ~d,u(V) = vd si v E C(P(d), X), on obtient une bijection
~d,u : (C, C) (~(d), u) >~U(d, u)
naturelle end E [(D, ~)I et u E I(C, C) I.
164
D~monstration
F(T)
F(d) I F(D)
J i
> P(d) / /,~ d;/u
h ) C
Si v E (C, C) (~(d), u), v~(d) = u et alors
vdF(d) = v~(d)~ = u~ ; donc ~d,u(V) = vd E ~U(d, u). R~ciproquement, si f E ~U(d, u),
fF(d) = uh ; il existe donc un unique v : P(d) ~.- X dans C tel que u = v~(d), i.e.,
v E (C, C) (~(d), u), et f = vd, i.e., f = ~d,u(V). Ainsi, ~d,u est bien une bijec-
tion du type voulu.
Pour d~montrer la naturalit~, soit t : d' > d dans (D, ~) et x : u --> u'
dans (C, C). Ii s'agit de voir que le diagrmmme suiv~nt est commutatif.
u> u)
(C, C) <~(d'), u') ~d',u' > ahU(d, u')
Or si v E (C, C) (~(d), u),
~h(t' x)(~d ,u (v ) ) = xvdF(t)
= xv~(t)d'
= ~d,,u,((C, C) (~(t), x)(v)) .
165
3.3. Th~or~me. Soient U : C > 8, un foncteur, et h : D > U(C), un morphisme de
. Alors si U poss~de un adjoint ~ gauche et si C admet des sommes fibr~es finies,
le foncteur Sh U poss~de aussi un adjoint ~ gauche, ~
D~monstration. Combinant la proposition 2.2 et les deux remarques pr~c~dentes, on
obtient une bijection compos~e
naturelle end E l(D, 8) Iet u E I(C, C) I.
U 3.4. Corollaire. Sous les hypotheses du th~or~me, S h commute aux limites projectives
et ~ , sux limites inductives.
3.5. Corollaire. (Grothendieck [5]). Si C admet des sommes fibr~es finies et si
f : C' > C est un morphisme de C, le foncteur
(f; C) : (C, C) > (C', C)
poss~de un adjoint ~ gauche. Cons~quemment, D : C 2 > C est une bifibration sur C . o
I C D~monstration. Prenant U = F = I C , on a (f; C) = (f; C)o(C; I C) = Sf ; (f; C)
a donc un a d j o i n t ~ gauche Mf ( n o t r e M ) : (C', C) ) (C, C) . A i n s i , r e m a r q u a n t
qu'alors ~ = f, si u' : C' ) X' est un objet de (C', C), le morphisme fi': u' ----> uf
dans (C', C) est une fl~che universelle d~finisssnt u : C ~ X comme objet
(f; C)-libre associ~ ~ u' si et seulement si le diagramme suivant est une somme fibr~e :
u ! X' >X
u I lu u C' > C
f
C Remarquons que si f est un ~pimorphisme, le compos~ M~o(f; C) est naturel-
lement ~quivalent ~ I(C ' C)" En effet, dans ce cas le diagramme suivant est une
somme fibr~e.
166
X
uT C
C
I X >X
>C
3.6. Corollaire. Si C poss~de des sommes directes finies, le foncteur
C ~C : (C, C) > C
a un adjoint g gauche.
C D~monstration. C poss~de un objet initial Net alors on peut identifier @C ~
(C; ~C ) : (C, C) > (N, C), o~ ~C est le seul morphisme N > C. Alors si
(C • C', il, i 2) est une somme directe de C et C', le diagramme suivant est une
somme fibr~e.
C !
~C' I
N
i 2 >C@C'
i I
Par exemple, si ~nc d~signe la cat~gorie des anneaux associatifs, commu-
tatifs et avec ~lSment unit~ et des homomorphismes, et si C E [~nc[, on sait aue
Im cmt~gorie des C-alg~bres unitaires et homomorphismes est isomorphe ~ (C, ~nc).
GnC admettant des sommes directes, ~ savoir ®~, le foncteur oubli
(C, ~nc) > ~nc
poss~de un adjoint ~ gauche. Explicitement, la structure de C-alggbre libre sur
un anneau A est donn~e par l'injection i I : C ) C ®zA . II est ~galement intSres-
sant d'interpr~ter les autres r~sultats de cette section en prenant pour U, l'oubli
~nc - > gns.
3.7. Corollaire. Si (F --~ U ; ~) et si C admet des sommes fibr~es finies, pour
tout objet C E ICI, le foncteur
167
(c; u) : (c, C) > (u(c), $)
a un adjoint ~ gauche.
D4monstration. Prenant h = IU(C) : U(C) > U(C) dans ~, dans le th4or~me 3.3., on
a ~ -i s u = ~U(C),C(Iu(c) ) = ~C : FU(C) > C dans C et (C; U) = (Iu(c); ~)o(C; U) = 1u(c)
(C; U) a donc un a d j o i n t & gauche M U : (U(C) , $) > (C, C) , no t~ p l u s s i m p l e - 1U(C)
U ment M C .
En fait, si k : U(C) ----> K est un objet de (U(C), ~), M~(k) est le mor-
phisme indiqu4 dans la somme fibr4e suivante :
F(K) > P
FU(C) > C OC
Un autre cms particulier int4ressmnt du th4or~me 3.3 est celui o~ l'on
prend C = F(D) et h = ~D,F(D)(IF(D)) = OD : D ----> UF(D). Alors,
h = PD = IF(D) : F(D) -----> F(D). On a
S U = PD (PD; ~)o(F(D); U) : (F(D), C)
dont l'adjoint g gauche, M U , est tout simplement PD
> (D, ~) ,
(D; F) : (D, ~) > (F(D), C) .
En effet, si d : D
ment une somme fibr~e.
> Test un objet de (D, ~), le diagramme suivant est triviale-
F(T) IF(T) > F(T)
l F(D) 1F(D) > F(D)
168
3.8. Corollaire. (J.J. Kaput [7]). Si F : • > C a un adjoint h droite, pour chsque
objet D 6 I~I, le fonctenr
(D; F) : (D, $) > (F (D) , C)
a a u s s i un s d j o i n t ~ d r o i t e . E x p l i c i t e m e n t s i ( F - ~ U ; a , p , O), l t s d j o i n t
droite de (D; F) est S U : (F(D), C) > (D, ~) . PD
3.9. Proposition. Si (F -~ U ; ~ , p , O) et si C et ~ sdmettent des sommes fibr4es
finies, ~ chaque morphisme h : D > U(C) correspondent un disgramme commutatif de
foncteurs
(C, C)
(F(D), C)
(c;
U S h
D
> (D, $)
(u(c), ~)
et un diagramme,commutatif ~ ~quivalences naturelles pros, de foncteurs, adjoints
gauche des pr4c4dents :
(F(D), C)
~ / ~ (D; F)
(c, C) < (D, ~)
(u(c), ~)
D~monstrmtion. Le triangle inf~rieur du premier diagramme commute par d~finition
U de S h et le triangle correspondant du second, par transitivit4 des adjoints
U (~(d)) fait intervenir deux sommes fibr4es qui, gauche ; d'ailleurs le calcul de M C
mises bout h bout, donnent la somme fibr4e d4finissant ~(d). La commutativit6 des
triangles sup~rieurs suit de la relation U(~)pD = h ou encore, pour ce qui est du
deuxi~me diagramme, de la d~finition m~me de ~U .
169
Remarque. Cette proposition sugg~re deux sutres f~ons de d~montrer l'adjonction
de ~ et S h U . En effet, d'une part, I~ relation ( ~-~ (h; ~) ) Ctant scquise
d~s que ~ sdmet des sommes fibr~es finies, on peut dsns ce css montrer directement
que ( McU_~ (C; U) ) et prendre ~ = McU o ~ . Cependent l'autre possibilitY,
i.e. par les triangles sup~rieurs, est be~ucoup plus simple (suggestion de J. B~nebou):
il est en effet tr~s facile de v~rifier directement que ( ( D; F) --~ S U ) PD
(J.J. Kaput) et puisque l'on s ( ~-~ (h; C) ), le th~or~me 3.3 suit de Is
transitivit~ des adjoints ~ gauche.
F(T) > F(Q) > P
ffC F(D) F(h) > FU(C) ~ C
La commutetivit~ des triangles sup~rieurs suit de le relation U(~)pD = h ou encore,
pour ce qui est du deuxi~me diagramme, de la d~finition m~me de ~ .
Consid~rons maintenant le diagramme de foncteurs suivsnt, o~ le losange
de droite est le produit fibr~ d~finissant (U~ ~).
/J
c 2 ~ > (u, ~)
c
est le seul foncteur pour lequel P2o~ = U~ et PlO~ = Do , qui existe car D oU ~ = UoD . o o
Rappelons que les objets de (U, ~) sont des couples (C,d), o~ C E ICI et d : U(C) --> T
est un morphisme de ~, et que les morphismes (C,d)> (C',d') dans (U, ~) sont des
couples (f,t), o~ f : C > C' E C ett : T ----> T' E ~, pour lesquels le diagrsmme
suivsnt est commutatif. t
T >T'
U(C)-- U(f) U(C') Alors P2 : (U, ~)
P2(f,t) = (U(f),t) et ~ : C ~
> ~ est donn~ explicitement par P2(C,d) = d et
> (U, ~), par ~(u) = (C,U(u)) et ~(f,x) = (f,U(x)).
170
X
uI C
x u(x) >X'
f U(f) >C'
u(x) > u(x' )
u u r i u(c) > u(c' )
La d4monstration du th4or~me suivant est laiss4e au lecteur.
3.10. Th4or~me. Le foncteur
~: c ~ >(u, ~)
poss~de un adjoint ~ gauche si et seulement si pour chaque objet C de C, le foncteur
(C; U) : (C, C) > (U(C), ~)
poss~de un adjoint & gauche.
Remarquons que ceci peut se produite sans que U poss&de lui-m~me un adjoint
gauche comme le montre l'exemple obtenu en prenant pour C , la sous-cat~gorie de
la cat4gorie des groupes constitu4e de tousles groupes et de leurs 4pimorphismes
et pour U, le foncteur oubliant & la cat4gorie des ensembles et surjections.
3.11. Corollaire. Si U poss&de un adjoint ~ gauche et si C admet des sommes
fibr4es finies, le foncteur
~: c A > (u, ~)
poss~de un adjoint ~ gauche
: (u, $) > C 2
D~monstration. Ceci est une cons4quence imm4diate du corollaire 3.7 et du th@or~me
3.10. Explicitement, si (F -~ U ; ~ , p , ~ ) et si (C,t) est un objet de (U, ~),
o~ t : U(C) > D, ~(C,t) E IC~I est le morphisme indiqu4 dans la somme fibr~e
suivante de C :
F(D) > X
F( t)I I~( C,
FU(C) > C O C
t)
171
3.12. Proposition.
fibr4es finies, la projection
Si U poss~de un sdjoint & gauche et si ~ poss&de des sommes
poss~de un adjoint ~ gauche
P2 : (U, ~))
~2 N : > (U, ~)
Preuve. Si (F -~ U ; ~ , p , O ), soit d : D > T, un objet de ~2 et posons
N(d) = (F(D),k), o~ k = M ~ (d) est le morphisme indiqu~ dans Is somme fibr4e sui- PD
vante de ~ .
T >K
D > UF(D) PD
Ii est alors facile de v4rifier que le morphisme (PD,t) : d > k = P2N(d) de ~-
est une fl~che universelle d4finisssnt N(d) 6 I(U, ~)[ comme objet P2-1ibre asso-
ci4 ~ d .
Puisque P2oU~ = U 2, on remsrque alors que le disgrarmne suivant est con~au-
tstif, ~ 4quivslence naturelle pr~s.
< F 2 ~2
(u, ~)
La dualisstion des r4sultats de cette section m4rite quelque peu d'stten-
tion. Les relations de base sont
(C, C ~) = (C, C) W , (f; O r) = (C; f)w
et (F-~ U ; 6, 0, O) -$ ~- (U ~
de C
, (C; u*) = (u; C)*
I F~ ; a-l, P, O)
Ainsi, partant d'un foncteur F : ~ > C et d'un morphisme k : F(D) --> C
(k : C > F~(D) dans C~), et posant
172
S~[ F~ ~ = (S k ) ,
il est facile de se rendre compte que S~[ est le compos4
(~, D) (F; D) > (C, F(D)) (C; k) > (C, C)
Si F poss~de un adjoint A droite U : C > ~ et si ~ admet des produits
fibr4es finis (pullbacks), on obtient alors le foncteur
M{< M F{~ ~ = ( k ) : (C, C) > (~, D)
Explicitement, si u : X > C est un objet de (C, C), M~[(u) 6 I(~, D) Iest le
morphisme indiqu4 dans le produit fibr4 suivsnt de ~, oh k : D ---> U(C) est le mor-
phisme correspondsnt ~ k par l'adjonction.
P M*F(u)[ D
> u(x)
U(u)
> u(c)
3.13. Th4or~me 3.3 W. M~ k est un adjoint ~ droite de S ~F .
Comme cas particuliers, supposant que (F -~ U ; ~ , p , ~) et que C et
sdmettent des produits fibr4s finis, on obtient les relations suivantes :
__C°r" 3.5 ~. ((C ; f) -~ M'~Cf ) . On 4crit aussi
Co___[r. 3.7 ~. ( S ~F -~ M~D F )
Co r. 3.8 ~. ( S{,~F C -~ (U; C) )
A titre dVillustration, nous donnons une d4monstration d'un r~sultat tir4
de Is th~orie des topos mbstrmits.
3.14. Proposition. (Lawvere-Tierney). Soit g une cmt4gorie avec limites projectives
finies pour laquelle, V X E Igl, le foncteur - xX : g > g poss~de un adjoint
droite (_)X, et le plongement K : ~ > gp, o0 g d4note la cat~gorie des appli- P
cations partielles de g, poss~de sussi un adjoint ~ droite (~). Alors pour chaque
f : X ----> Y dans g, le foncteur
173
f* = M*7 : (8, Y) > (g, x)
poss~de un adjoint ~ droite, ~f .
D~monstration. Nous remarquons d'abord que le plongement
admet comme adjoint ~ gauche la projection P : (g , X) P
l'application partielle (i,v) : Z .... > X E l(gp, X) I, o~ i : Z' > Zest un
monomorphisme, le morphisme v : Z' > X E l(g, X) I.
Supposons, pour fixer la notation, que (K--~ (N) ; ~ , ~ , ¢ ), o~
e X = (~X,Ix), et soit f : X
correspond un unique q0 : YXX
X
YXX -
(K; X) : (g, X) --> (gp, X)
> (£, X) qui associe
> Y. A l'application partielle ({f,Ix},l X) : YXX --> X
>~ et le produit fibr4 suivant de g .
i X >X
l, Nous constatons alors que le foncteur fw est naturellement ~quivalent au
compos4 des foncteurs suivants :
S ~-XX S~X (g, Y) ~ > (g, ~) > (g X) - P > (g, X) p,
En effet, cela r4sulte, au moins pour un objet y : T > Y de (g, Y), de ce que dans
le diagramme suivant,
w Z >X
{r'w} 1 {f' ix] 1
TXX > YXX Y × I X
I X 1 x >X !, > X
>X
ITXX
TxX
174
le rectangle, juxtapos4 des deux carr4s, est un produit fibr4 si et seulement si le
carr4 suivant est un produit fibr4.
W Z >X
!
T Y >Y
Par consequent, l'sdjoint ~ droite [If : (g, X) > (g, Y) de fw est le
compos4 des adjoints ~ droite des foncteurs pr4c4dents, i.e.
[If = M ~-XX o ((_~); X) o (K; X)
Ainsi, six : Z > X 6 [(g, X)[, [If(x) est le morphisme indiqu4 dens le produit
fibr4 suivant, o0 ; est le morphisme correspondent g q) par (-×X --[ (_)X).
p
x I y > X ~'X
Dens [7], J.J. Kaput d4montre le r4sultat suivsnt, analogue au th4or~me
3.10, ms is qui n'en est pas le dual.
3.14. Proposition (Kaput). Le foncteur
U # C ~ : > (~, U)
induit par U : C > ~ poss~de un adjoint ~ gauche si et seulement si, pour cheque
objet C de C , le foneteur
(u; c) : (C, c) > (~, u(c))
poss~de un adjoint ~ gauche. On dit alors que U poss~de un adjoint local ~ gauche.
Utilisant cette terminologie, le dual du corollaire 3.8 s'4nonce alors de
le faqon suivante : Si U poss~de un adjoint ~ gauche, il poss~de aussi un adjoint
local ~ gauche.
175
La pr~ciproque, cependant, est fausse : les notions de coimsges et d'imsges
relatives, pr~sent~es dans la section suivsnte, fournissent des exemples de foncteurs
aysnt un adjoint local sans avoir d'adjoint.
§ 4. Coimages relatives.
Soit 4 une cmt~gorie. Une sous-cat~gorie ff et 4 pour laquelle
I ~ I = I 4 I sers mite coextensive ; ~ peut slors ~tre consid~r~e comme une sous-
ferm~e sous is composition et pour Isquelle V D E 141 , classe de morphismes de 4
IDE ~ •
4.1. D4finition. Soit ~ une sous-cat4gorie coextensive de 4 ; d6signons par P
le plongement canonique ~ > 4 . Nous dirons que 4 admet des ~-coimases si pour
tout objet D de 4 , le plongement
(D; P) : (D, if) > (D, 4)
poss~de un adjoint ~ droite
JD = J : (D, 4) > (D, ~)
Ceci revient ~ dire que tout morphisme d : D > T de 4 poss~de une
d~composition d = tJ(d) svec J(d) E ~ telle que pour toute autre d~composition
d = t'j avec j E if, il existe un unique morphisme j' E ~ tel que j'j = J(d)
et tj' = t'.
d D >T
T"
4.2. D4finition. Une sous-clssse g de morphisme de 4 est dite ouverte ~ droite
si ts E ~ et s 6 g entralne t E S .
176
ts D )T'
T
Si P :
coextensive ~ dans
plongement
> ~ d~signe le plongement canonique d'une sous-cat#gorie
, alors ~ est ouverte ~ droite si et seulement si le
(D; P) : (D, ~) > (D, ~)
est un foncteur plein pour tout objet D de ~ .
4.3. D~finition. Une sous-classe g de morphismes de
si pour toute somme fibr~e de ~ du type suivant,
s t D' ) T'
s D > T
s E g e n t r a S n e s f E g •
est dite S-fib-ferm4e
4.4. D4finition. Si ~ est une sous-cat4gorie coextensive ouverte ~ droite et
S-fib-ferm~e de ~ et si ~ admet des ~-coimages, on dira que ~ est une sous-
cat4gorie parfaite ~ droite de ~ .
Dualement, on obtient les notions de ~ -images, de sous-classe de morphis-
mes ouverte g gauche et P-fib-ferm4e et de sous-cat4gorie parfaite g gauche de ~ ;
Ii suffit de consid4rer la cat4gorie ~ duale de ~ .
4.5. Exemples
I) Si ~ denote la classe de tousles monomorphismes de ~ , ~ est
avec images si et seulement si ~ admet des ~ -images et alors ~ est une
sous-cat4gorie parfaite ~ gauche de ~ .
177
2) La classe ~ des foncteurs fid&les d~termine une sous-cat#gorie ouverte
gauche et P-fib-ferm~e de Cat.
3) La classe @J~ de tousles ~pimorphismes r~guliers (i.e. les coegali-
smteurs) de ~ est ouverte ~ droite et S-fib-ferm~e. Si g~ est ferm~e sous Im
composition et ~ est mvec co~gmlisateurs, g~ est une sous-cat~gorie parfaite
droite de ~ si et seulement si tout morphisme de ~ se dgcompose de fmqon unique
(& isomorphisme pros) en un 6pimorphisme r~gulier suivi d'un monomorphisme ; cela
se produit, par exemple, si, de plus, ~ admet des produits fibres finis.
4) La classe ~ des foncteurs F : ~ > C pour les~uels l'application
IFI : I~I ----> ICIest une bijection est une sous-cat6gorie psrfaite ~ droite de Cat
Le lecteur pourrm comparer le th~or~me suivant, dit d'adjonction de Struc-
ture et S~mmntique mbstrmite, mvec le th~or~me 7.2.
4.6. Thgor&me. Soient U : C > ~ et F : ~ > C des foncteurs pour lesquels
(F -~ U ; ~, p, a) et o0 C est avec sommes fibr~es finies. Soit ~ , une sous-
cat~gorie de ~ telle que ~ admette des ~-coimages et soit M une sous-catggorie
coextensive de C , ouverte & droite et S-fib-ferm~e. Supposons finalement que
F(~) ~ ~ . Alors pour tout morphisme h : D > U(C) dans ~, le foncteur compos@
(C; P~)
(c, ~) > (c, C)
o~ P~ dgnote le plongement canonique
: (D, ~)
D~monstration. Dans la somme fibr~e de C
U Sh JD
(D, ~) > (D, ~) ,
) C , poss~de un adjoint & gauche, not~
>(C, ~)
suivante, on constate que si j E ~,
F(j) E ~ et donc ~(j) E M car M est S-fib-ferm~e.
F(T)
F(j)]
F(D)
>X
~(j)
Par cons~quent, le foncteur compos~
178
(D; P~) (D, 3) > (D, ~) > (C, C) ,
U adjoint ~ gauche de JD o S h passe par (C, ~)
de (C, C). On obtient ainsi l'adjoint cherch~.
(D, ~) > (C, C)
I I (D, 3) ~ > (C, ~)
4.7. Corollaire. Si ~ poss~de des sommes fibr4es finies et si ~ est une sous-
qui est une sous-cat4gorie pleine
cat4gorie parfaite ~ droite de ~ , pour tout morphisme g : D > D' de ~ , le
foncteur compos4
(D'; P) (g; ~) JD (D', 3) > (D', ~) > (D, $) > (D, 3)
poss~de un adjoint ~ gauche, ^ M ~ : (D, 3) g
> (D', 3), restriction de M ~ ~ (D, 3) g
§ 5. Structures quasi-quotients et sous-morphismes engendr4s
D~notons par G : C > C ~ le foncteur induit par le seul foncteur de
2 ~ I . Explicitement, G(C) = i C et G(f) = (f, f).
Si U : C > ~ est un foncteur, (U ; ~) d~signe la sous-cat4gorie
pleine de (U, ~) dont les objets sont les couples (C,p), o0 p : U(C) > Test
un ~l~ment de ~, i.e. un ~pimerphisme r4gulier de ~ . Alors le foncteur compos~
o G (voir § 3) passe par (U ; £A%~). On obtient ainsi un foncteur R et un dia-
gramme commutatif :
C G > C ~ ~ >(U, g)
(U ; 8R~)
179
La d4finition suivante est essentiellement doe ~ C. Ehresmsnn. Elle est
donn4e dans [3] pour le cas o~ ~ est is eat~gorie des ensembles et des applica-
tions d'un univers donn~.
5.1. D~finition. Soit (C,p) un objet de (U ; g~). Si (u,t) est un morphisme dsns
(U ; ~) de (C,p) ~ (X,Iu(x)) = R(X) qui est une fl~che universelle d~finisssnt
X E ICI comme objet R-libre associ4 ~ (C,p), on dit que u d6finit X comme le
U-quasi-quotient de C par p.
t T > U(X
1u(x)
u(c) U(u) > u(x
5.2. Th~or~me. Si C poss~de des sommes fibr~es finies et si le foncteur U : C--> ~
poss~de un adjoint ~ gauche F : ~ > C , pour tout ~pimorphisme r~gulier
p : U(C) > T off C E ICI, il existe un morphisme u : C > X d~finissant X
comme le U-quasi-quotient de C par p. De plus (X,u) est un quotient de C, i.e.,
u est un eo~galisateur.
D~monstration. Pour l'existence des U-quasi-quotients, il suffit de montrer que R
c 2 a un sdjoint ~ gauche. Or le foncteur "codomaine" D I : > C est sdjoint ~ gauche
de G. Par ailleurs, si (F-~ U ; ~, p, ~) et si C poss~de des sommes fibr~es
finies, ~ poss~de un adjoint ~ gauche M (Corollaire 3.11.). Puisque (U ; g~)
est une sous-cat~gorie pleine de (U, ~), is restriction de D 1 o M, adjoint
gauche de ~ o G, ~ (U ; g~) est l'sdjoint ~ gauche de R.
D1 C~ < c < (u, ~)
(u ; ~)
Explicitement, si (C,p) est un objet de (U ; g~), le morphisme
180
u : C > X d4finissant X comme le U-quasi-quotient de C par p est le morphis-
me indiqu4 dans le diagramme de gauche suivant qui est une somme fibr4e de C .
7 F(T) > X
FU(C) > c qC
~(c,p)
T
P
u(c)
u(x)
Alors, puisque p est un co4galisateur, il en est de m~me de F(p) car F
commute aux limites inductives. ~C ~tant S-fib-ferm4e (§ 4), u est done aussi
un co4galisateur. (X,u) est alors le quotient de C le plus fin pour lequel U(C)
passe par p.
La notion de sous-objet d'un objet C de C engendr4 par un morphisme
g : D > U(C) de ~ , o~ U = C > ~ est un foncteur, est utilis4e par P. Freyd
dans la d4monstration du "Adjoint Functor Theorem". D'autre part, C. Ehresmann [3]
4tudie une notion analogue, o~ des classes particuli~res de monomorphismes sont con-
sid4r4es. La d4finition suivante est ~ mi-chemin entre les deux.
~C d4note la classe de tousles monomorphismes de C . ~C est ouverte
gauche (§ 4) de sorte que pour C 6 ICI, (~C' C) est une seus-cat4gorie pleine
de (C, C). D4signons par H
u : C---> ~, ~ (~C' c).
(c, c)
T (~c' c)
5.3. D4finition. Soient g : D
un monomorphisme de C . Si m
m est un U-sous-morphisme engendr4 par
U-engendr~ par g.
la restriction de (U; C), induit par le foncteur
(u; c) > (& u(c))
> U(C), nn objet de (~, U(C)), et m : B > C,
est un objet H-fibre associ4 ~ g, nous dirons que
g et que (B,m) est le sous-ob.iet de C
181
u(c)
> U(B)
Si U pr4serve les monomorphismes, il revient alors au m~me de dire que
B est le plus petit sous-objet de C pour lequel g passe par U(B). Neus dirons
que g (ou quelquefeis D par abus) U-en$endre C si le sous-objet de C
U-engendr~ par g est C lui-m~me.
5.4. Th4or~me. Si C est avec images et si le foncteur U poss~de un adjoint
gauche F : ~ > C, tout morphisme g : D > U(C) de $ U-engendre un
sous-objet (B,m) de C.
D4monstration. II suffit de montrer que le foncteur H
Or si C est avec images, le plongement
(~C' C) > (C, C)
poss~de un adjoint ~ gauche. Par ailleurs, U ayant un adjoint ~ gauche, le foncteur
(C, C) ) (~, U(C))
poss~de un adjoint ~ gauche (corollaire 3.8~.). La conclusion suit donc de la
transitivit~ des adjoints ~ gauche.
Explicitement, dans la situation du th~or~me, si g : F(D) ) C est
poss~de un adjoint g gauche.
le morphisme de C correspondant ~ g, m : B > C est l'image de ~ .
C U(C)
F(D) > B D > U(B) q
182
La cat~gorie C est dite r~$uli~re ~ droite si tout morphisme de C se
d~compose en un ~pimorphisme r~gulier suivi d'un monomorphisme. Dans un tel cas,
C est ~videmment avec images, et si U : C >
g : D -----> U(C) U-engendre C d~s que ~ : F(D)
lier ; ainsi les objets de C U-engendr~s par D E
et (F--I U), un morphisme
> C est un ~pimorphisme r~gu-
~I sont les quotients de F(D).
Combinant alors les notions de U-quasi-quotient et de U-sous-morphisme engendr~,
on obtient Is possibilit~ de d~finir un objet de C par "g~n~rateurs et relations
dans ~", comme le montre is proposition suivante.
5.5. Proposition. Si C est r~guli~re ~ droite et poss~de des sommes fibr~es
finies, et si U : C > ~ est un foncteur aymnt un sdjoint ~ gauche F, pour tout
objet D de ~ (les "g~n~rateurs"), tout ~pimophisme r~gulier p : UF(D) > T
(les "relations") d~termine uniquement un objet C de C , U-quasi-quotient de
F(D) par p et U-engendr~ par D.
§ 6. Cas des bifoncteurs.
Consid~rons deux foncteurs ~ deux variables,
U : ~ X C > ~ et F : ~ X ' > C ,
pour lesquels F est adjoint ~ gauche de U. II existe alors, pour chaque
D E 141, B E I~I, et C E ICI, une bijection
B ~D,C : C(F(D,B), C) > ~(D, U(B,C))
naturelle en ses trois composantes.
Soient k : D > D' B' , g : > B, et f : C > C', des morphismes
de ~ , ~ , et C respectivement. S'inspirant du Lemme Fondamental (2.1), on
obtient Is proposition suivsnte :
B B' 6.1. Proposition. La bijection ~D,C × ~D',C' se restreint ~ une bijection entre
les couples (h,h') pour lesquels le disgramme (~) ci-bas est commutatif et les couples
(h,h') pour lesquels le diagramme (~) est commutatif.
183
(~)
F(D
F(k,g]
,B)
F(D',B')
~C
>C'
(~) k
D' h'
> U(B,C)
~ ( B , f )
U(B,C')
///~(g,C'/~)
) U(B',C')
On reconnait dans le diagramme (~), un morphisme (k,g,f) :
(D,B,h) > (D',B',~) de la cmt~gorie (comma) (F(-,-), C) ; notons le foncteur
( F ( - , - ) , C ) . . . > ~ x s ~ x c
D'autre part, le diagramme (~) sugg~re l'existence d'une cat~gorie, que
nous noterons (~, U(¢~,-)), dont les objets sont les triplets (h,B,C), ob
h : D ~ U(B,C) est un morphisme de ~ , et dont les morphismes sont les triplets
= (k,g,f) pour lesquels le diagramme (~) est commutatif ; si
~' = (k',g',f') :(h',B',C') > (h",B",C") est un morphisme de (~, U(*~,-)), il
est facile de voir que le compos~ ~r o ~ = (k'k,gg',f'f) est un morphisme dans
(~ , U(~,-)) de (h,B,C) b (h",B",C"), d'o~ le foncteur
(~, u ( ~ , - ) ) ~ ~ x ~ x C
La proposition 6.1 exprime alors l'isomorphisme des categories (~, U(~,-)) et
(F(-,-), C) au-dessus de ~ X ~ X C .
1 8 4
($, u(.,-)) ~ (F(-,-), C)
.it N x g x C
Chaque objet B de ~ d4termine des foncteurs
U B = U(B,-) : C > ~ et F B = F(-,B) : ~ > C
pour lesquels F B est adjoint & gauche de U B, et chsque morphisme g : B' > B
de ~ d4termine des transformations naturelles
U g : U B'
donn4es par
uIB
> U B
(ug) C = U(g,C) et
et Fg : F B > FB,
(Fg) D = F(D,g). On a alors
= et = %U B FIB ~F B
U g'g = U g' o U g et = F o Fg, Fg,g g
Nous rappelant la proposition 1.5., la transformation naturelle
U g : U B' > U B induit, pour chaque C 6 ICI, une transformation naturelle
C C m U , que nous noterons m de (C; U B')- ~ (U(g,C); ~) o (C; U B) dont io g ~
g composante (m C) en un objet u : C > X de (C, C) est donn~e par gu
= (Ug)DI (u) (m~) u = U(g,X)
A un objet h : D-----> uB(c) de (~, U(~,-)) correspond un foncteur
U B S h (not4 S~) : (C, C) > (D, ~)
ayant un adjoint ~ gauche
~B (not4 ~) : (D, ~) > (c, c)
d~s que la cat~gorie C admet des sommes fibr~es finies (c.f. § 3). De plus, si
h' : D' ------> uB'(c ') est un autre objet de (~, U(~,-)), et si ~ = (k,g,f) est
un morphisme dans (~, U(~,-)) de (h,B,C) ~ (h',B',C') (c.f. diagramme (~)),
185
on obtient le diagramme suivant, o~ tout ce qui doit commuter commute.
B S h
(C, C)
(f; C)
(C; U B) > (uB(c), ~) (h; ~)
f); ~)
(uB(c'), ~)
/
> (D, $)
!
(C', C) > (U B (C'), ~) > (D', ~) (C'; U B') (h'; ~)
(k; ~)
m ! S h ,
yg c' yg 6.2. Proposition. Posant = (h'k; ~) ~ m , est une transformation nsturelle g
B' B de (k; ~) o Sh, & S h o (f; C). Si ~I = (k'IB'f)' y~l estl'4galit~
B B (k; ~) o Sh, = S h o (f; C). Pinalement, si ~' = (k',g',f') est un morphisme de
(h',B',C') & (h",B",C") dsns (~, U(~,-)), on a
y~'g = (y~ ~ ( f ' ; C)) o ((k; ~) ~ yg')
(C, C)
(f; C) I
(c', C)
(f'; C) I
(C", C)
B S h
B ! S h ,
B vr
Sh,,
> (D, ~)
(k; ~)
> (D', ~9)
(k'; ~)
> (D", ~)
186
Supposons maintenant que C admet des sommes fibr~es finies, et soit
= (k,g,f) : (D,B,~) > (D',B',~'), un morphisme de (F(-,-), C). Si
d : D' > T est un objet de (D', 4), on obtient le diagrsrmne suivant, o~ les
rectangles int~rieurs et ext4rieurs sont les sommes fibr4es d4finissant ~(dk)
et Mh,(d) respectivement, et ~ est le seul morphisme rendant le tout commutatif.
FB,(d)
dk
FB,(T)
FB(T)
FB(dk)' I
FB(D)
(D')
I f
I f
/ /
/ /
f /
> P(dk)
) c
>
i T J
FB, C' >
P'(d)
B T
Mh, (d)
6.3. Proposition. La famille ~t ~ = {bt~} d E I(D', 4) I est une transformation nsturelle
de ~ o (k; 4) ~ (f; C) o ~',. De plus, si
~' = (k',g',f') : (D',B',h') > (D",B",h") est un morphisme de (F(-,-), C), on a
~' ~ ~' (~ = ((f; c) ~ ) o ~ (k'; 4))
187
(D, ~)
(D', ~)
~,, (D", ~)
) (c, c)
I (f; C)
> (C', C)
I (f'; C)
> (c", c)
6.4. Corollaire. Nous supposons que les ~l~ments suivants sont donn4s.
a) Un diagramme commutatif dans ~ :
dl / / / ~ T1
D I T.
D 2
T 2
>
> T 2
b) Un morphisme g : B' > B dans B .
c) Un diagramme commutatif dans C :
188
F(D, ,B)
F(k,B) hl
F(DI'g) ~
F(k,g) ~ F(D2,B)
F(D2,B')
~2
1,i\ , > C' ~f
> C 1
"> C 2
hl
f2
\ > c:~
Alors, posant ~ = (k, iB,f) , ~' = (k,IB,,f') ,
~2 = (ID2'g'f2)' ~" = (k,g,f"), on a ~" = ~2 ~ = ~' ~I
obtient le diagramme ¢ommutatif suivant dans C :
~I = (IDl'g'fl) '
dans (F(-,-), C) et on
B (tl)
p ~. Pl(d{)
~/d I
P{(d i)
(d I )
~ i (&)
Pl(d2 k)
~d2k
B (t2)
P2(d2 )
~2 ~d 2
Pi(d2k) - -
P~(d 2)
l(t')
> Pl(dlk)
>
Mhi(t 2 )
t
M~(t 2)
~'dlk
~2 ~d~
<d?.
189
§ 7. Structure et S#mantique
Nous montrons dans cette section comment la relation d'sdjonction
"Structure" et "S~mantique" apparaSt comme un cas particulier de la th~orie g~n~rale
~tablie pr~c~demment.
L'exponentiation dans Cat peut-~tre interpr~t~e de deux fsgons diff~rentes,
donnant, d'une part, un foncteur U : Cat X Cat ~ > Cat, oN U(C,G) = C ~ et,
d'autre part, un foncteur F : Cat X Cat~* > Cat ~, o~ F(~,C) = C ~. Alors la
bijection (qui est en fait un isomorphisme de cat@gories)
Cat (~, C ~) ~ Cat (~, C ~)
naturelle en ses trois variables ~, ~ et C E ICatl, peut aussi s'~crire
Cat ~ (F(~,C), ~) N Cat (~, U(C,~))
et exprime simplement le fait que F est adjoint ~ gauche de U.
Alors ~ chaque foncteur H : ~ > C ~ = uC(~) correspondent deux
foncteurs
u C C > (~, Cat) S H = S H : (~, Cat~) = (Cat, ~)~
et
Explicitement, SHC(V)
: (~, Cat) > (~, Cat~) = (Cat, ~)~
= cVoH et SHC(X) = C X,
Z'
C V J J
H ~ C ~
C Z
ex
C ~'
190
f, et si T : H > ~ est un objet de (~, Cat) , ~(T) est le foncteur indiqu~
dmns le produit fibr~ suivant de Cat (som~ne fibr~e dans Cat~), o~ H est le
foncteur correspondant canoniquement ~ H (~ est un morphisme de F(8, C)
dans Cat ~) ;
M~(T)
P > C ~
g G >
de plus, si @ : T - > T' est un morphisme de (~, Cat), o~ T' : ~ > ~',
est le seul foncteur rendant commutatif le diagramme suivant de Cat :
M~(e)
~Dt
~ \\ \\
G
> C ~'
> e g
Une application immediate du Th4or~me 3.3. nous donne le r4sultat suivmnt
que l'on pourrait appeler Th~or~me pr~liminaire d'adjonction de structure et s~man-
tique.
4 7 . 1 . Th4or~me. e s t a d j o i n t ~ g a u c h e de S H .
R a p p e l o n s (§ 4) que s i g d 4 s i g n e l a s o u s - c a t 4 g o r i e e o - e n t e n s i v e de
d o n t l e s m o r p h i s m e s s o n t i e s f o n c t e u r s T : 8
induite ITI : l~l > I~l est une bijection,
Cat
~ pour lesquels l'application
est une sous-cmt~gorie pmrfaite
191
droite de Cat . En particulier, posant gh4o(~) = (8, g), ~h@o(~),
objets seront appel4s des ~-th~ories, est une sous-cat~gorie pleine de
et le plongement canonique poss~de un adjoint ~ droite
dont les
(~, Cat)
J : (8, Cat) > gh@o(~)
Nous savons aussi que la classe h des foncteurs fiddles est ouverte ~ gauche et
P-fib-ferm~e. En particulier, posant ~ub(~) = (h, ~), @ub(~), dont les objets sont
des foncteurs fiddles ~-valu~s, est une sous-cat4gorie pleine de (Cat, ~). De plus,
si T : ~ > ~ est une
fiddle ; ainsi F C (g) ~ h
diate du th~or&me 4.6.
Le foncteur compos4
= JoS~ : (Cat, S H
ou quelquefois sa restriction
8-th@orie, F C (T) C T C ~ C 8 un = : > est foncteur
et le th4or&me suivant est alors une consequence imm@-
(8, Cat) > ~hgo(8),
@ub(~) @, est appelg H-structure.
7.2. Th4or~me [8]. La restriction ~ de
passe par @ub(~) ~ et est adjoint ~ gauche de
(S, Cat)
gh~o (P.)
gh~o(8), mppel4e
H-structure.
> (Cat, G) ~
I > @ub(G) ~
Ainsi si T : 8 - ~ ~ est une ~-th4orie, le foncteur
obtenu en prenant le produit fibr~ suivant est fiddle.
p K . C~
H-s~mantique,
V = ~(T),
192
V peut donc ~tre consid~r~ comme un foncteur d'oubli vers G de is cat~gorie P
des "modUles" de la th~orie T. Ls "th~orie des modules" qui s'ensuit est particuli&-
rement simple et ~l~gante dans ce cadre g~n~ral. Par exemple, on montre facilement,
comme le fmit J. Lambeck dans [8], que V cr~e les limites projectives et qu'il est
triplable (au sens de J. Beck) d~s qu'il poss~de un adjoint ~ gauche ; en effet V
cr~e les co~galisateurs de psires V-absolues (R. Par~ [14]). Si le morphisme de
theories ~ : T > T' est un foncteur plein, le foncteur ~(~) : P' • > p
est mussi un foncteur plein. D'autre part si H est un plongement plein, il en est
de m~me de K : P > C ~, de sorte que P peut-~tre identifi~e g une sous-cat~gorie
pleine de C ~ et ~(@), g une restriction de C @ .Dans certains cas (e.g. Exemple
7.3.-B), le th~or~me d'extension de Ksn fournit alors un adjoint ~ gauche de ~(@).
7.3. Exemples
A- Structure et S~mantique de Yoneda (Linton [12])
Le th~or~me d'adjonction Structure et S~mantique de [12] est le cas psrti-
culier du th~or~me 7.2. obtenu en prenant pour
G ~ > ~ O
j~ Y
H le compos~
> gns ~
oO Y est le foncteur de Yon~da associ~ g la cat~gorie ~ et j : G > O
est un foncteur dense, i.e. pour lequel
G Y
est plein et fiddle. Pour j = I~ :
, qui est le compos@
) £ns > gns ~° gnsJ~
> G , Linton ~tablit un th~or~me de reprO-
sentation permettant d'interpr4ter lea constructions de Kleisli et d'Eilenberg-Moore
associ~es ~ un triple sur G corm~e des ~-th4ories et des cat4gories de modules
sur ces ~-th4ories respectivement. Ainsi une cst4gorie 4quationnelle (Linton [II])
eat simplement une cat4gorie triplable au-dessus de gns .
B- Structure et S~mantique alg~brique
Si I est un ensemble et si
directs finis librement engendr~e par
(Lawvere [18], B4nabou [i], et autres).
~(I) d4signe la cat~gorie avec produits
I, et si C est une cat~gorie avec produits
193
directs finis, on peut prendre pour H le foncteur ~vident
C C(C I) ~I = ~ : F(1) > ,
auquel correspond
: C I
qui ~tablit un isomorphisme entre C I
> C g(1)
et ia sous-cst~gorie pleine de C g(I)
d~termin~e par les foncteurs commutsnt aux produits directs. Ls cat~gorie des modules
d'une g(1)-th~orie T : g(1) ------> 4 est slors isomorphe ~ is sous-cat~gorie pleine
de C ~ constitute des foncteurs r : ~ ) C pour lesquels le compos~ ~ o T
commute aux produits directs. Ainsi pour I = I = [O} et C = gns on obtient
le th~or~me d'adjonction de Structure et S~mmntique mlg~brique de Lawvere.
Utilisant les r~sultats de ia § 6, en particulier le corollaire 6.4.,
on peut faire apparaltre des foncteurs de comparaison entre categories de modules
en faisant varier simultan~ment l'ensemble I, les g(1)-th~ories, et Is cat~gorie
C dans laquelle on prend les modules.
C- Structure et S~mantique op~rationnelles. (Lambeck [8])
Prenant H = I~ : C G > C ~ (H est slors le foncteur substitution
c ~
> C(C~)), on obtient Structure et S~mantique op~rationnelles de [8] .
D'ailleurs, g ~tant une sous-cat4gorie parfaite ~ droite de Cat , on
peut appliquer le corollaire 4.7. ~ H : 8 > C ~ pour se rendre compte que le
th4or~me 7.2. s'obtient de ce cas particulier en utilisant Is d4composition de la
proposition 3.9.
§ 8. Transferts de structures d'effscement
Soit C , une cat~gorie.
8.1. D4finition. Soient f : X > Y
Si pour tout u : X > A, il existe
est f-injectif et que f
et h : A - ) B, deux morphismes de C .
v : Y > B tel que vf = hu, on dit que
est un h-effacement de X.
194
X
u I A
>Y
>B
Si
morphismes f-injectifs pour tout f E ~ . Si
C , Y(~) est la classe des morphismes de C
h E ~ .
8.2. D~finition. (W. Zimmermann [15]). Soient
de C . On dit que le couple (~, ~) d~finit une structure d'effacement sur
si les conditions suivantes sont remplies.
i) ~ = ~(~)
ii) ~ = Y(~)
iii)
est une classe de morphismes de C , ~(~) d~note la classe des
est une classe de morphismes de
qui sont des h-effacements pour tout
et B , deux classes de morphismes
C
pour tout A E ICI, il existe f E ~ N ~ , de domaine A.
Si ~' est une classe de morphismes de C, I(~') = Co~'oC est l'id4al
bilat&re de C engendr4 par ~' ; on remsrque que Y(I(~')) = Y(~') .
8.3. D~finition. Soit (5, ~), une structure d'effacement sur C . On dit que ~'
est une base des morphismes injectifs de la structure si I(~') = ~ .
8.4. Proposition. Une classe ~' de morphismes de C est une base des morphismes
injectifs d'une structure d'effscement (5, ~) sur C si et seulement si pour tout
A E ICI, il existe f : A > C et h : C > D avec h E ~' et hf E Y(~').
8.5. D~finition. (J. Marauds [13]). Soit ~' une classe d'objets de C . Si
{IQI Q E ~'} est une base des morphismes injectifs d'une structure d'effacement
(5, ~) sur C , on dit que cette structure est (induite par) une structure injec-
tive, notre (5, ~), o~ ~ = {Q E ICI IIQ E ~] .
195
Supposons maintenant que U : C > ~ et F : ~ > C soient des
foncteurs pour lesquels (F --~ U ; ~, 0, ~). Le lemme suivmnt est elors une con-
s~quence innn~diate du Lemme Fondamental (§ 2).
8.6. Lemme. Soient ~ et 3 , des classes de morphismes de
vement. Alors
F-I(~(~)) = ~(U(~)) et U-I(~(3)) = ~(F(~))
8.7. Th~or~me.[15]. Si (3, ~)
C et ~ respecti-
est une structure d'effacement sur
est une base des morphismes injectifs d'une structure d'effacement
pour laquelle ~ = F-I(~).
C , u(~)
(3, h) sur
D~monstration. Utilisant la proposition 8.4., si D E I~I et si f : F(D)
est dans 3 N ~ , on a U(f) E U(~) et il est facile de voir que
S U (f) = U(f)pD E ~(U(~)) = F-I(~(~)) = F-I(~).
PD
PD U(f) D > UF(D) > U(C)
>C
Dans ce cas, on dit que (~, ~) est obtenue de (3, ~) par transfert
direct par la paire de foncteurs adjoints (F--~ U), et on ~crit (5, ~) = sU(3, ~).
On remarque que si ~' est une base des morphismes injectifs de la structure
d'effscement (5, ~), U(~') est aussi une base de morphismes injectifs de
sU(~, ~). En particulier on retrouve le processus de transfert des structures
injectives de [13] :
8.8. Corollaire (Maranda). Si (3, ~) est une structure injective sur C ,
sU(3, ~) est aussi une structure injective dont U(~) est une base d'injectifs.
Si 3 est une classe de morphismes de C , @ (3) d~note la classe des g
morphismes k de C pour lesquels il existe v tel que vk E 3 ; S(3) d~note
la classe des morphismes k de C pour lesquels il existe f E 3 et une somme
fibr~e du type suivant :
196
k C >D
f X >Y
On remarque que ~(~) = ~(@ (5)) = ~(S(~)) . g
8.9. Th~or~me. Si C admet des sommes fibr4es finies, ~ toute structure d'efface-
ment (5, ~) sur ~ correspond une structure d'effscement (3, ~) sur C pour
laquelle ~ = U-I(~) et ~ = @ (S(F(~))). g
D4monstration. Pour chaque A E ICI, choisissons un ~ : U(A) " > D dans ~ ~
f = M~(~) (c.f. Cor. 7). et posons
On obtient alors les deux diagrammes commutatifs suivants, celui de gauche
~tant une somme fibr4e, qui montrent que f E S(F(~)) et U(f) E ~ , i.e. f E U-I(~).
A
FU(A) F(~)
> F(D)
U(A) U(f)
> U(B)
D
Or U-I(~) = U-I(~ (5)) = ~ (F($)) = ~ (S(F(~))). Ainsi tout objet de C est le
domaine d'un morphisme dans S(F(~)) A ~ (S(F(~))) et il est alors facile de v~rifier
que ~(U-I(~)) = @ (S(F(~)), ce qui termine essentiellement la d~monstration. g
On dit alors que (3, ~) (not4e MU(~, ~)) est obtenue de (5, ~) par
trmnsfert inverse par la paire de foncteurs (F -~ U). On remarque cependant que
si (5, ~) est une structure injective sur ~ , il n'en est pas n4cessairement
de m~me de MU(~, ~) (voir [i0]).
Si (5, ~) et (5', ~') sont deux structures d'effacement sur C , on
dit que (5, ~) est plus fine que (5', ~') (et on 4crit (5, ~) ~ (5', ~') si
c 5' ou, ce qui est ~quivalent, si ~ 2 ~'. La classe £(C) des structures
197
d'effacement sur C est alors munie d'une relation d'ordre (~) faisant de g(C)
une cat~gorie.
8.10. Proposition. L'application transfert direct, S U : g(C) > g(~) preserve
la relation ~ et, consid~r~e comme un foncteur, admet transfert inverse,
M U : g(~) > g(C), co~e adjoint ~ gauche d~s que C est svec sommes fibr~es
finies.
Universit@ de Montreal
et Facult~ des Sciences d'Orsay
198
BIBLIOGRAPHIE
[I] BENABOU, J.,
[2] BENABOU, J.,
[3] EHRESMANN, C., Construction de structures fibres.
92 (1969), 74-104, Springer-Verlag.
Structure alg~briques dsns les categories. Th~se, Fscult~ des
Sciences, Universit~ de Paris, 1966.
Introduction to Bicategories. Lectures Notes in Mathematics,
47 (1967), 1-77. Springer-Verlag.
Lectures Notes in Mathematics,
[4] GRAY, J.W., The Calculus of Conmla Categories. Notices of the A.M.S., 14
(1967), p. 937.
[5] GROTHENDIECK, A., Categories fibr~es et deseente. S~minaire de g~om~trie
alg~brique, 1960-61. Fascicule II, Expos~ VI, I.H.E.S.
[6] KAN, D.M., Adjoint Functors. Trans. Amer. Math. Soc., 87 (1958), 294-329.
[7] KAPUT, J.J., Locally adjunctable functors. A parsitre dsns Ill, Jour. of Math.
[8] LAMBECK, J., Operational Categories and Grammars. Cours donn~ ~ l'Universit~
McGill ~ Montreal durant l'snn~e 1968-69.
Th~se, Columbia [9] LAWVERE, F.W., Functorial Semantics of Algebraic Theories.
University, New-York, 1963.
[iO] LEROUX, P., Sur les structures d'effacement, A paraltre.
[ii] LINTON, F.E.J., Some Aspects of Equational Categories. Proceedings of the
La Jolla Conference on Categories, 84-94, Springer, Berlin, 1966.
[12] LINTON, F.E.J°, An outline of Functorial Semantics. Lectures Notes in Mathe-
matics, 80 (1969), 7-52. Springer-Verlag.
[13] MARANDA, J.,
[14] PARE, R.C.,
[15] ZIMMERMANN, W., Injektive Strukturen und M-injektive Objekte.
Maximilians-Universit~t, Munchen, 1969.
Injective Structures. Trans. Amer. Math. Soc., IIO (1964), 98-135.
Absolute Coequalizers. Lectures Notes in Mathematics, 86 (1969),
132-145, Springer-Verlag.
Th~se, Ludwig-
LIMIT-COLIMIT COMMUTATION
IN ABELIAN CATEGORIES
Armin Frei and John L. MacDonald
Received Nov. Z4, 1970
The category ~ of relations in an abelian category A is iso-
morphic to its own dual. This entails that direct limits in ~ can be
computed as inverse limits and vice-versa. This, together with the
fact that limits of the same type commute, suggests the use of the
category ~ to obtain criteria for limit-colimit commutation in ~ .
The terminology used will generally be that of [l] and [2]. The
index categories I,J will be assumed to be qf' and qf respec-
tively, and we will write "Suppose (C,J',J)" for "Suppose that J'
and J are qf and that C : ~ ~ J is a cofinal functor". We use
both versions ~ and A of the category of relations in A . If
F : J ~ ~ is any functor, we denote by F,F the composition of F
with the embedding functors of A into A,A respectively.
A functor F : J ~ A is said to be d-conservative if lim F =
= l~m F , s i m i l a r l y F i s s a i d t o b e i - c o n s e r v a t i v e i f lim~ F = lim~ F .
Clearly F can be replaced by F : J ~ A .
Theorem i
If J is qf and A satisfies the Grothendieck axiom AB5, then
every functor F : J ~ ~ is d-conservative.
Let F : J ~ • be any functor and ~ a class of cones over F .
A cone ~ - F ~ L in ~ is said to be a direct V-limit of F (we
write {L,~} = ~-l~m F ) if, given any cone Q : F ~ X in ~ , there
200
exists a unique ~-morphism y : L ~ X such that ?~ = Q . Clearly
~-lim F is unique up to canonical isomorphism.
Let a\~ : ~--. --~X be a cone over F . The cone ~k~ is
called cocartesian if it has at least one cocartesian representation,
~j ~j that is, a representation {Fj ~ Gj ~ X} for which there exists
a functor G : J ~ A such that the diagram
(2)
X
Gj ) Gk
Fj F~ ) Fk
commutes in A for every ~ : j ~ k in J , and such that the square
in (2) is cocartesian.
Remark: A cone ~k~ is cocartesian if and only if its maximal repre-
sentation is cocartesian.
In a dual fashion we define cartesian cones under ~ : I ~ ~ .
Suppose (C,J',J) If F : J ~ A is a functor, let ~ be the
class of all cones a\~ over F for which ~\~IC is cocartesian.
Then ~[C is the class of all cocartesian cones over ~C .
Theorem 3
Suppose (C,JI,J) . Let F : J ~ A be a functor and
= l~m F . Then
{LF,~{C} = ~Ic-lim FC
{LF ,~) =
and equivalently
201
{T~,~T = ~-l!m F .
Corollary 4
Suppose
cone over FC
(C,J',J) If F : J ~ X is a functor for which every
is cocartesian, then F is d-conservative.
Corollary 5
Suppose (C,J',J) If F : J ~ A is a functor for which FC~
is epic for all morphisms ~ in J' , then F is d-conservative.
The anti-involution T : A ~ A yields an isomorphism between
and ~o As a consequence we have that when the direct limit of a
functor H : J ~ A exists it can be computed as the inverse limit of
the contravariant functor obtained by composing H with the anti-in-
volution of A , and vice-versa. More precisely, we have
Proposition 6
Let T : A -- A be the anti-involution and H : J ~ ~ a functor
for which i~ H exists. If {L,~) = i~ H , then {L,~} = lim TH . J ~o
Let F : IxJ ~ A be a functor and F the composition of F
with the embedding A ~ A . For every ~ : i ~ i I in I and
: Jl ~ j in J we obtain a commutative diagram
(7)
F(i,jl ) F(i,~) > F(i,j)
F(~,Jl) I IF (~, J )
F(il'Jl) F(ii,~) > F(il, j)
TO the functor F we make to correspond the two mappings:
202
F~ : I°×J ~ A defined by F{ (i,j) = F(i,j) ,
~{ (o,~) = F(iI,~)/F(~,j) = F(e,j) o F(il,¢) and similarly
Fr : I×J° ~ ~ defined by Fr(i,j) = F(i,j) ,
o Fr(~,~ ) = F(e,j)/F(ilJ~) = F(il,¢) o F(e,j)
Clearly F~ = T Fr
Proposition 8
FQ is a functor ~-~ (7) is exact ~ 5 r
is a functor.
Let F : I×J ~ A be a functor and Fi : J ~ A the corresponding
functor for fixed i ~ III . From now on we shall denote by {LFi,~..} 13
the direct limit of Fi and by {RFj,T..} the inverse limit of Fj . 13
Given a morphism e : i ~ i I in I the diagram
(9)
7[. ,
F(i,j) 13 • LFi
F(~,J) I i LFe
F(i1'J) x. ~ LFi I 11j
obviously commutes V j ~ IJl . Furthermore let {RLF,T i} = lim LF Y
and {LRF,~.) = l~m RF . 3 j
Theorem IO
Let F : I×J be a functor for which (7), (9) and
(11)
RF~ ) RFj RFJl
TiJll ITij
F(i,Jl) F(i,~) ~ F(i,j)
are exact for all (~,~) , (~,j) and (i,~) in I×J respectively.
Suppose in addition that
203
(i) For all i ~ IIl , Fi : J ~ A is d-conservative and LF is
i-conservative.
(ii) For all j ~ [J[ , Fj : I ~ & is i-conservative and RF is
d-conservative.
Then {RLF,~ij/T i} =io~jlim F~ and {LRF,Ti~ ~j} =ix~ Olim Fr . Furthermore ,
RLF ~ LRF , i.e. by abuse of language lim i~ F ~ i~ lim F . Y J J Y
As an application of Theorem IO we obtain
Theorem 12
Let I be qf' and J
which (7) is exact for every
all (~,j) in IxJ , and F(i,~) is epic for every
Then the conclusion of Theorem IO holds.
qf . Let F : IxJ ~ A be a functor for
(~,~) in IxJ , F(~,j) is monic for
(i,~) in IxJ .
In order to obtain a limit commutation theorem which is appli-
cable to a wider range of functors F : IxJ ~ & we make incisive use
of the notion of relative limit.
Theorem 13
Let I be qf' Let F : I×J ~ A be a functor for which
Fi : J ~ & is d-conservative for every i ~ III . Suppose in addition
that (9) is cartesian for every (~,j) in IxJ , and that
(14)
RFj
T ij~
F ( i , j )
3
13
> LRF
ILT i
LFi
is exact for every (i,j) in I×J I
204
Then LRF ~ RLF , i.e. by abuse of language
l~_m l i m F ~ l i r a l i m F .
By using cofinality we can slightly weaken the hypotheses of
Theorem 13. For this let Q : I/i i ~ I and P : jl/J ~ J be the pro-
jection functors. Then we have
Corollary 15
Let I be qfl and J qf . Let (il,Jl) ~ I I×Jl be fixed. Let
F : IxJ ~ A be a functor for which Fi is d-conservative for every
i = Q(i ~ i 1) . Suppose in addition that (9) is cartesian for every
: i - i I and every J = P(Jl ~ j) , and finally that (14) is exact
for i = i I and every J = P(Jl ~ j) . Then LRF ~ RLF
When ~ is an AB5 category , the hypotheses of Corollary 15 become
quite unrestrictive. Indeed, we have
Theorem 1 6
Let I be qf' , J qf and A AB5 . Let F : I×J ~ A be a
functor for which (7) is cartesian for every ~ : i ~ i i , with i i
fixed and ~ : j' ~ j with ~ = P~ . Then LRF ~ RLF , i.e.
l~m l~m F ~ l~m l~m F . J I I J
[1]
[2]
REFERENCES
B. Eckmann and P.J. Hilton, Commutinq Limits with Colimits. J. Alg. II (1969) , 116-144.
P.J. Hilton, Correspondences and Exact Squares. Conference on Categorical Algebra (La Jolla) 1965. Springer Verlag, 254-271.
[3] S. MacLane, Lectures in Cateqorical Alqebra. Bowdoin College,1969.
University of British Columbia and Forschungsinstitut f~r Mathematik, ETH
NON-ABELIAN FULL EMBEDDING;
ANNOUNCEMENT OF RESULTS
Michael Barr
The full embedding theorem of Mitchell states,
Theorem I. Let ~ be a small abelian category. Then there is a full,
faithful,exact functor ~ ~ Mod-R , the category of right R-modules.
It has long seemed to me that this theorem should be the additive
case of a theorem which will apply to a larger class of categories
which satisfy certain additional exactness conditions. The first step
in this direction was taken by Tierney, who proved the following.
Theorem 2. Let ~ be an additive category. Then ~ is abelian if and
only if it satisfies each of the following conditions.
EX O) ~ has kernel pairs and a terminal object as well as pullbacks
of any pair of maps, at least one of which is a regular epimorphism|
in addition the kernel pair of any map have a coequalizer.
EX i) In any pullback diagram
if f is a regular epimorphism, so is f'
EX 2) Equivalence relations are effective.
In these definitions, f is a regular epimorphism if it is the
coequalizer of some pair of maps! a subobject E L~XxX is an
206
equivalence relation if (-,E) represents an equivalence relation on
(-,X) It is effective if E ~ X is the kernel pair of some map.
Of course the limits assumed in EX O) could be replaced by the
assumption of all finite limits. However, there are some interesting
examples of categories which satisfy these conditions exactly as stat-
ed (e.g. non-empty sets, algebras of finite type over a noetherian
ring); in addition the proof of theorem 3 below is done by first prov-
ing theorem 4 and applying that to subcategories of ~ which, even
were ~ finitely complete, would generally only satisfy the version
of EX O) given above. Of course, by virtue of theorem i, additive cat-
egories satisfying EX O) - 2) are finitely complete.
Categories which satisfy EX O) and EX i) are called regular. If
in addition they satisfy EX 2), they are called exact. Exact catego-
ries bear the same relation to homotopy as abelian categories do to
homology. That is, it will be possible to define the property of a
simplicial ~ object being a Kan object and, for such a one, to de-
fine its homotopy objects. The objects so defined will be preserved by
functors which are exact according to the following definition.
Definition. Let U : ~ ~ ~ be a functor. We say that U is exact if
it preserves all finite limits (which ~ has) as well as regular epi-
morphisms.
Of course, as with all such definitions, the more finite limits
that ~ has, the more of a restriction this is.
The most obvious way to try to extend Mitchell's theorem is to
attempt to embed an exact category fully and exactly into a category
of M-sets for some monoid M . In fact, we will state a theorem of this
sort below, but there are simple examples to show that this is not, in
207
general, enough. For example ~in × ~in (~in = category of finite
sets) can have no such exact embedding. The essential reason is that
~×X = ~ for any set X , a fact which has no parallel in the abelian
case. Thus we have to replace M by a category with more than one ob-
ject. The theorem which results is
Theorem 3. Let ~ be a small category which satisfies EX O). Then the
following are equivalent:
i) There is a small category ~ and a full, faithful, exact embed-
ding ~ ~ (~,~) , the set valued functor category.
ii) There is a faithful, exact isomorphism reflecting functor
~ (~,~) where ~ is a discrete category.
iii) ~ is regular (i.e. satisfies EX i)).
Moreover ~ may be chosen so that its set of objects is the set of
non-empty subobjects of the terminal object.
In this statement, an object is called empty (= strict initial)
if it is initial and every map to it is an isomorphism. The following
is an immediate corollary, although, as mentioned above, the actual
proof of theorem 3 first proves this special case.
Theorem 4. Let ~ be a small category which satisfies EX O). Then the
following are equivalent:
i) There is a monoid M and a full, faithful, exact embedding
~ (M,~) the category of M-sets.
ii) There is a faithful, exact, isomorphism reflecting functor
~ .
iii) ~ is regular and its terminal object has no non-empty subobject.
208
Both of these results can be extended to large but cocomplete
categories with a set of generators, provided that for each of the ge-
nerators G there is a cardinal a such that G has (weak) rank ~.
This means that for any X and for any family {Xi} of subobjects of
X of which any a or fewer are contained in yet another subobject of
that family, the natural map
colim (G,Xi) ~ (G,colim Xi)
is an isomorphism.
These categories are called (weakly) locally presentable. It
seems that just about any small category embedding theorem can be ex-
tended to them. Included are all toposes. Since toposes are also exact,
it follows that every topos has a full exact embedding into a functor
category. Whether this can be proved directly from the definition of a
topos as a category of sheaves is not known.
THE MULTILINEAR YONEDA LEMMAS:
TOCCATA, FUGUE, AND FANTASIA ON THEMES BY EILENBERG-KELLY AND YONEDA
F. E. J. Linton
Wesleyan University, Middletown, Conn., U. S. A°
Received June i, 1970
Revised Jan. 15, 1971
=S____u~_~_ary_. Although the notion of a covariant ~-valued ~-functor, exemplified
by the ~-valued horn functors a(A, -) on a ~-category G, has been recognized by
Eilenberg and Kelly, in their comprehensive foundational treatise [EK, esp. pp. h54,
ff. ] on closed and monoidal categories, for general closed categories ~, those au-
thors pointedly renounce consideration of that notion's contravariant counterpart
until ~ is at least symmetric, and carefully refrain from even mentioning the two
analogous possibilities for general (not necessarily closed) monoidal categories ?f.
The purpose of the present note is to provide these definitions, to formulate,
somewhat after the fashion of Day and Kelly [DK, §§3, 4] or of Yoneda [Y, §§4.0,
4.1], the notions of the ~-object of ~-natural transformations between two such
~-functors of similar variance and the ~-obJect tensor product of a contravariant
~-valued ?;-functor with a covariant one, and to establish the pertinent Zoneda Lem-
mas (extending [DK, (5.1)], the ~I-valued case of [DK, (3.5)I, and [Y, (4.3.1), .2),
ol*), and .2*)]). These will facilitate the description (elsewhere), for not neces-
sarily symmetric ?f, of the algebras over a ~-triple [LI in terms of ?;-functors
on the associated Kleisli ~-category, generalizing Dubuc's work [D21 for closed
symnetric monoidal ~.
Preliminary speculations on these matters were aired in talks delivered at
McGill University, Oct° 18, 1968, and at a meeting of the Midwest Category Seminar
in San Antonio, Jan. 24, 1970.
During the preparation of the bulk of this paper, the author, on leave from
his home university, was a Killam Senior Research Fellow at Dalhousie University,
Halifax, Nova Scotia, and was supported in part by Canadian N.R.C. Grant @ A 7565.
210
~tion._____=== Whether we deal with a monoidal or a closed category ~,
we are in a position to describe as a multillnear ~¢-map <A1, ..., An> > B any
~/-morphism of the form ((...(AI®A2)®...)®An.I)®A n > B, if ?/ is monoldal,
or~ if ?/ is closed, but not monoidal closed, any T-morphism from A 1 to the
~-obJect T(A2, ~'( ..., ~;(An.1, ?/(An, B))...)) ; this terminology extends to
if we call each ?/-morphism I > B a O-linear ~-map <~> --> B and each
~-morphism A > B a l-linear ?/-map <A> ---> B.
> > <B1, ...~ Bk> of By a ~f-multimal~ f= <fl' "'" fk >: <AI' "'" An
distribution type ~ (~: [l...n} ~ {l...k} an order preserving function) we mean
n=O, 1
simply a sequence f= <fl' "'" fk > of multilinear ?/-maps
fi: <(Aj)j6~-1(i) > -->Bi (i <i <k).
... > > <B I ... Bk> and g: <B 1 ... The composition of ~-multimaps f: <A 1 A n
• .. Bk> ~ <C 1 ... Cm> , of distribution types ~ and B, respectively, can be
defined, in a manner more self evident, perhaps, when ?/ is monoldal than when ?F
> > (C 1 C > of distribu- is closed, as a certain multimap gof: <A1 "'° An "'" m
tion type $o~, with cc~ponents
(gof)~ g~o < < (Aj B_I --~ C~ = (fi)iES-i (~)>: ) jE~-i (~)> •
It is not our intent to beleaguer the reader with the fastidious details of this
construction, nor with the proof that there arises therefrom a strictly associative,
strictly unitary monoidal category ?~(?) with object class consisting of all the
multiobjects A = <A 1 ... An> belonging to the free monoid generated by the object
class obj(~) of ~, with morphisms all the multimaps between these multiobjects,
and with monoidal structure extending the free monoid structure of its object class,
nor with the demonstration that passage to the distribution type provides a monoidal
functor from 74(~) to the monoidal category A of finite ordinals and order pre-
serving maps (monoidal under ordinal sum), nor with this functor's further property,
reminiscent of the cleavage of a fibration, that for each multimap f: <A 1 ... An>
> <B 1 ... Bk> of type ~, each monic A-morphism 8: [1.o.k} ~ {1...%} , and
each multimap g: <C 1 ..o C~> --> <D 1 ... Din> for which Cs(i) -- B i, there is an
211
interleavin 6 composition gosf, a multimap to ~D I .. o Dm> from the multiobject
(X 1 ... X~> in which the entry Xj is either the single object C j, if j is not
a value of 8 , or the entire sequence <(Ah)h6~_ I (i)~ if j = 8(i) . Nor, finall~
do we intend to substantiate our claim that our axioms (MLC l) for multilinear cate-
gories have the twin virtues of honesty -- being true of the data arising, in the
manner here suggested, from closed or monoidal categories -- and prudence -- vouch-
safing, by their validity in a situation arising from the mere data [EK, pp. ~28 and
~71-2] required to specify a closed or a monoidal category, that those data will
themselves actually satisfy the axioms [EK, pp. cit. ] for such a category.
Rather# since it is our prime concern to establish the Yoneda I2mmas for contra-
variant and covariant ~-valued ~-functors defined on any ?f-category, where ~ is
closed or monoidal, and since we have found, as Alex Heller kindly predicted, that
the multilinear language serves not only to unify the parallel formulations both of
definitions and of arguments otherwise called for in the two separate but equal cases
closed and ~ monoidal, but also to generalize and simplify the ideas involved,
we present the Yoneda Lemz~s in the broader context of multilinear categories~ and
leave as a task for the reader to prove the allegations of the previous paragraph.
The idea of multilinear categories has quite an extended history. Multillnear
maps, of course, particularly billnear ones, were eon~nonplace long before categories
were born. As early as 1958, in first-year graduate algebra lectures at Columbia
University, Serge Lang formulated a tentative definition of multilinear categories
in terms of data resembling the interleaving cc~posltion~ Lang's formulation and our
axioms (MLC l) stand in much the same relation as do Hall's clones and Lawvere's al-
gebraic theories. More recently, as a result of his work with profunctors, B~nabou
has introduced promultiplicative categories (renamed "premonoidal over 8~" in [D1] )
in his public lectures. It appears that every multilinear category is promultiplica-
tive, and that every promultiplicative category can, in a variety of equivalent ways,
be enlarged to a multilinear one: the essential difference in the viewpoints of the
axioms is, in effect, that, of all the multilinear maps required as data for a multi-
linear category, only the billnear ones are explicitly required as additional data
for a promultipllcative category, but these are then subject to conditions that con-
212
tain the behavior of certain "derived" multilinear maps to within acceptable bounds.
The Yoneda Lemmas and their proofs will be found in §2, following the basic
definitions of §l. For the reading of §3, we recommend a grain of salt.
§i. The ten multillnear commandments
(MLC i) A multilinear category ~/ is an ordinary category ~o equipped with:
(MI~ la) data making the free monoid on obj(~o) into the object class of a category
~o(~) (a typical object <A 1 .oo An> of ~o(?f), with n > O and
A i E obj(~/o) , is called a multiobject of ~/ of length n ; there is just
one multiobject of length zero, which for convenience we denote <~> and
designate the empty multiobject; an ~o(?/)-morphism f: <A 1 ... An> >
<B 1 ... Bk> is called a multimap of ?f fr~n the first multiobject to the
second; when k = 1, f is called n-linear (or, imprecisely, multilinear);
the O-linear maps will play a special role);
(MLC lb) data making the inclusion A ~ <A>: obj(~ o) • > obj(~o(~)) "of the
generators" the object function of a faithfully full functor ~: ~o >
(MLC ic) data making the binary monoid operation on obj(,~o(~)) the object function
of a functor ®: > o(r) giving a strictly associ-
ative, strictly unitary (with unit the empty multiobject), monoidal cate-
gory structure ~(~f) ; and
(MID id) data making the passage <A 1 ... An>~ {l...n} the object function of a
monoidal functor ~ : ?~(~) > A ( A being the small, strictly associative,
strictly unitary, monoidal category (under ordinal sum) of finite ordinals
and order preserving functions) having the properties:
f E ~o(?f)(<A 1 ... An> , <B 1 ... Bk>), k_>l =
~' fi E ~o(~f)(<(Aj)jE~(f)_1(i)>, <Bi>) (i < i <k)
f -- fl ®''" ®fk (also denoted as = <fl' "'" fk > )'
and
I <° card (~)(<AI "'" Ak>' <¢>) = 1 (k = 0 "
213
[We forego the proofs that the procedures described in the introduction, pur-
porting to create multilinear data from closed (resp. monoidal) data on a category
~o' successfully give ~o a multilinear structure if and only if the original data
made ~ closed (resp. monoidal).] O
(MLC 2) A multilinear functor ~: ~ ~ ~' from one multilinear category ~ to
another ~' is a functor CPo: ~o ~ ~'o and a monoidal functor ~: ~(~) ~ ~(~')
extending ~o over ~F and ~, (and compatible with the conditions (MLC ld) for
and ~' if that is not automatically the case).
(MLC 3 ) A multilinear natural transformation k: ~ ~ ~ between multilinear
funct0rs ~, ~: ~ ~ ~' from one multilinear category ~ to another ~' is an
ordinary natural transformation k: ~o ~ ~o capable of being extended (in a unique
way, if at all) to a monoidal natural transformation between the monoidal functors
~ , ~ : ~(v) ~ ~ ( ~ , ) .
[It is left for the reader to verify that a closed (resp. monoidal) functor be-
tween closed (resp. monoidal) categories "is" a multilinear functor between the asso-
ciated multilinear categories, that a closed (resp. monoidal) natural transformation
between two such functors reduces to a multilinear transformation between them, and
conversely. The reader may also satisfy himself that ~(~)( <~>, ~(-)): ~o ~ J'
where J is any suitable (large enough) category of sets, is naturally endowed with
the structure of a multilinear functor
underlying set functor for ~ -- ,
tesian closed structure.]
V = V~: ~ ~ # -- the socalled canonical
being multilinear, say, by virtue of its car-
(MLC ~) A ~-category a, where ~ is multilinear, is an ~(~)-category a,
whose ~(~)-valued horn functo, a(-, =) is required to factor through ~ • (We
s h a l l t y p i c a l l y w r i t e ~AC: <a(B, C), G(A, B)> ~ a(A, C) and iA: <~> ~ a(A, A)
for the composition rules and ("names of") identity maps, respectively.)
[Note that a multilinear functor ~: ~ ~ ~' makes each ~-category a into a
~'-category ~a. In particular, V: ~ ~ ~ makes each ~-category a a "just
plain" category VG = a . ] O
214
(MLC 5) If G and 8 are two ~-categories (?f a multilinear category), a
~-functor F: G ~ ~ is just an ~(?f)-functor between the (admittedly rather special)
~(~)-categories G and ~.
[If F: G -
arises a ~'-functor ~F: ~G ~ $8. In particular, use of V = V~ converts
-S ] a "just plain" functor VF = Fo: G o o"
(MLC 6) If F, G: G - 8 are ?f-functors between ~-categories G and
being a multilinear category), a ?f-natural transformation
natural transformation k: F ~ G that happens to be an o o
tion between the ~(~)-functors F and G.
is a ?f-functor and ~: ?f ~ ~' is a multilinear functor z there
F into
(~
k: F ~ G is an ordinary
~(?f)-natural transforma-
[Again, if l: F - G is a ?f-natural transformation between ?f-funetors F, G:
G ~ ~, and if ~: ?f ~ ~r. is a multilinear functor, ~ remains a ~'-natural trans-
formation k = ~: ~F ~ ~G between the ?f'-functors ~F, ~G: ~G - ~. ]
Unless ?f is closed, the multilinear category ?f is not itself in any s e n s e a
~-category, so we cannot extract from (MLC 5) any notion of a ~-valued ~-functor on
a ?f-category a. This difficulty could be overcome by expanding (MLC 4) and 5) to
cover pro-~-categories -- i.e., categories whose horn functors take values not in
but in J ?f°°p, or perhaps rather # ~°(~)°p, and come endowed with appropriate compo-
sition data -- and the ~-functors between them. Although the desirability of a good
definition of pro-?f-categories will reappear in §3, in connection with the structure
of ?f-valued functor categories, we prefer here to follow a simpler, ad hoc approach.
(MLC 7) A covariant ~-valued ?f-functor F: G ~ ?f on a ~-category G is a
function F = obJ(F): objG - obj,, coupled with ?~o(?f)-morphisms F = FAC:
<G(A, C), FA>- FC making the diagrams
<G(A, A), FA>
i~A FA > FA , (unit condition)
and
215
<a(C, D),G(A, C), FA>
<Mc, FA> 1 <(~(A~ D)~ FA>
< id, FAC>
FAD
> <~(C, D)~ FC>
IFcD (composition condition)
> FD
commute, whatever A, C, D E obj G (both diagrams live in ~(9/)).
[For example, each ?f -valued horn functor (~(Q, -) becomes a covariant ~-val- o
ued ~ - f u n c t o r 1, Q = (I(Q~ - ) , when coupled wi th the obvious choice o f "composit ion"
~o (~)'~hi=s ~QAC = ~c: <a(A, C), a(Q, A) > ~ a(Q, c). ]
(MLC 8) A contravariant ~-valued Y-functor F: ~ - ? on a ~-category
a function F = obJ(F): obj (~ -obj ~o coupled with %(~)-morphisms F = FAC:
<FA, (I(C, A)> - FC making the diagrams
G is
<FA, i A ~ <FA, a(A, A)>
/ idFA FA > FA
and
<FA, a(C, A), (Z(D, C)>
< FA, LCA >
<FA~ (I(D~ A)>
> <FC, a(D, c)>
I FcD
> FD
commute in ~(~) ~ whatever A, C~ D E obj G.
[For examplej each ~-valued horn functor (~(-, Q) becomes a contravariant o
~-valued ~-functor R Q = a(-, Q) when coupled with the obvious choice of "composi-
tion" morphisms RQ C = LIQ ~ ~CQ: <G(A, Q), G(C, A) > - G(C, Q). ]
(MLC 9) If F and G are covariant ?/-valued ~-functors on a ?/-category (I,
a ~-natural transformation X: F - G is a family k = [kA]AEobjG of ~-morphismSo
kA: FA ~ GA ( A 6 obJ G ) rendering commutative each square of ~(~)-morphisms
<G(A, C), FA> ----=------> FC
<a(A, C), GA> GAC > GC
(A~ C 6 objG) .
216
(MLC i0) If F and G are contravariant ~-valued ~-functors on a ~-category
G, a ~-natural transformation k: F ~ G is a family k = [kA}A6objG of ~o-mOr-
phisms kA: FA ~ GA ( A 6 obj G ) rendering co~nutative each square of ~(~)-mor-
phisms
<FA, a(C, A)> > FC
<GA, a(C, A)> > GC
(A, C E obJG).
[It is no problem to see, with reference to (MLC 7)-(MLC i0), that each c~ntr~ -
variant ~-valued ~-functor on G is at least also an ordinary functor from G (°p) o
to ~o (of like variance), and that each ~-natural transformation between such
functors is also an ordinary natural transformation between the corresponding "just
plain" ~ -valued functors. Moreover, in the special case that ~ is closed, o
(MI~ 5) and (MIC 6) are compatible, respectively, with (MIC 7) and (MLC 9), as well
as with the Eilenberg-Kelly definitions; and in the case that ~ is symmetric closed,
(MID 8) and (MLC i0) are compatible with all the rest.]
§2. The Yoneda Lemmas
Preamble. Any attempt, such as that (see [DK] or [Y]) by means of ends and co-
ends, to define hom or tensor objects in ~ for pairs of ~-valued ~/-functors has a
natural tendency to fail -- even when ~ is a symmetric monoidal closed category --
in the absence of a suitable (completeness-of-~/, snm]luess-of-the-domain)-tradeoff
(except, of course, in those useful exceptional instances delineated by the Yoneda
Lemmas). An added complication in the present multilinear setting is that ends and
coends~ so useful and arising so naturally in the symmetric monoidal closed context,
are conspicuously unavailable notions here; indeed, we must dispense with them en-
tirely, being able, fortunately, to compensate for their absence by the use of cer-
tain universal problems they would pose (or rather, solve) were they present.
Because the resemblance to a familiar coend is the least farfetched, the sim-
plest illustration of this dilemma and its circumvention is the problem couniversally
solved by the tensor product of a contravariant with a covariant ~-valued ?/-functor.
217
This is, therefore, the aspect we consider first. Thereafter, we move on to the hom
object for a pair of contravariant ~-valued 9/-functors, and conclude this § with
the horn object for a pair of covariant 7/-valued ~-functors, establishing the appro-
priate Yoneda lemma for each notion before moving on to the next.
Definition i. Let (~ be a ~'-category, where ~ is a multilinear category,
and let F: G ~* ~, G: G ~ ~ be contravariant and covariant ~/-valued ~-functors,
respectively. A tensor product F®aG (in ~o ) o_~f F with G over a is an ob-
ject T of 7/o equipped with Z~o(~/)-morphisms ~A: <FA, GA>- T that couniver-
sally solve the problem of rendering the squares (in ~(~))
<FA, GBA > <FA, G(B, A), GB> > <FA, GA>
<FAB, GB> I IPA
<FB, GB> > X PB
(with X E obJ ~o) commutative, for ~11 A~ B E obj (~. That is to say, these squares
shall commute when X = T and PA ~ PB = TA ~ ~B' and, furthermore, whenever a
7/-object X 6 obj~ ° and ?~o(7/)-morphisms PA: <FA, GA> - X are given making the
above squares commute, there shall be a unique ~/-morphism p: T - X for which o
PA = p °TA "
Clearly, T = F®GG is uniquely determined (to within a unique compatible iso-
morphism) by these requirements. Further desirable attributes of F®GG, as sug-
gested by the considerations of Day and Kelly [DK, §2], may be formulated as side
conditions on an already existing tensor product F@GG = ( T, [TA]A6objG ) of F
with G over G . For example, we may call (T, [TA ]) a tensor product i_nn ~/ of
F with G over G if, whatever the multiobjects M and N in ~o(~) and what-
ever the ~o-object X and multilinear maps PA: <M, FA, GA, N> - X making each
square below commute
<M, FA, G(B, A), GB, N> I
<M, FAB , GB, N> I
<M, FB, GB, N>
there is a unique multilinear
<M, FA, GBA , N> > <M, FA, GA, N>
> X , PB
p: <M, T, N> ~ X satisfying PA = po<M, 7A' N> .
218
Or again, if the "test object" X in the above discussion is permitted to be any
multiobJect of ~ and the "test maps" PA are taken as general multimaps, we may
ca~1 (T, ~TA} ) a tensor product in %(~) or in ~(~) according to the exclusion
or inclusion of consideration of the auxiliary multiobJects M and N . The latter
two conditions boil down, of course, to the requirements that IT A} make T a co-
limit, in ~o(~) , preserved, in the second case, under the iterated tensoring of
%(~) , of the coend-like diagram vaguely hinted at by the picture
<FA, (~(B, A), GB> ~ ~
<FB, a(Q, B),
<FQ, G(Q, Q), GQ> / ~ - <FQ, GQ>
<FQ, G(B, Q), GB> /~/ ~ ' ~ <FA, GA>
(in ~B(~)) .
[When ~ is actually monoidal, a tensor product
diagram suggested by
F®~G in ~o will make the
/~FA ® GA ~
FA®G(B, A) ®GB _F®GG (in ~o )
. /
a colimit diagram, and conversely. ]
~_==_~or_e~= =i (Yoneda Density Immmas). Let G be a ~-category, let F: G * ~ be
a contravariant ~-valued ~-functor, let G: G- ~ be a covariant ~-valued
~-functor, and let Q E obJ G. Then:
i) RQ ®G G exists in ~(~) , being given by GQ via the multimaps
T A = GAQ: <RQ(A), GA> = <G(A, Q), GA> ~ GQ ; and
ii) F @G LQ exists in ?~(~) , being given by FQ via the multimaps
~A = FAQ: <FA, LQ(A)> = <FA, C(Q, A)> - FQ .
Moreover, both existence assertions result from absolute collmit information (cf.
[P1 ] or [P2]) in ~o(~) , as is made clear in the proof.
219
Proof. Consider the diagrams in ?~o(~/) :
~ < ( I ( A ~ Q), GA>
<a(B, Q), a(A, B), G A > ~ ,,
<a(Q, Q), a(B, Q), GB> ~-~.7. % m,
<a(Q, Q), a(Q, Q), GQ> ~ ' ~ , ~ , L ~ ' ~ - -/"
<FA, a(B, a), a(Q, B) ,,
<m, a(a, a), a(a,
Observe that, attention being restricted to the solid arrows, these diagrams
conmmte; that each dotted arrow, when followed by the solid arrow over it, composes
to give the identity on the right hand terminus; that the diamonds involving two
dotted arrows commute; and that the two endomorphisms of <G(Q, Q), GQ> (resp.
<FQ, G(Q, Q)> ) resulting from the remaining compositions of solid and dotted arrows
are equal. With these observations, we may conclude the proof by direct appeal to
Park's description ([Pl' (I.3.2)] or [P3' Th. ~.i~) of absolute colimits. Alterna-
tively, we may end by noting that, no matter what functor may be applied to these
diagrams, the two bottom rows remain, by what has Just been observed, contractible
coequalizer diagrams, and that compatible families [pA ] of maps PA from the en-
tries in the middle column are already uniquely determined by their components pQ ,
which then necessarily factor uniquely through the coequalizer of the bottom row.
[Since quite similar considerations are involved in concluding the proofs of the
remaining Yoneda Lemmas (Theorems 2 and 3), we may there phrase less carefully the
appropriate counterparts of the last two sentences of the above proof. ]
220
Definition 2. let F and G be contravariant ?/-valued ?/-functors G ~
defined on a ?/-category G , ?/ being, as usual, a multilinear category. By a
object ?/-nat(F, G) = IF, G] = Hom~(F, G) o_~f ?/-natural transformations from F
G is meant an object Z E ?/o equipped with %(?/)-morphisms ~A: <Z, FA> ~ GA
that universally solve the problem of rendering the squares
o
to
<X, FAB > <X, FA, a(B, A)> > <X, FB>
<gA' id> I IPB
<GA, (~(B, A)> > GB GAB
(in ~o(?/))
(with X 6 obj %'o ) commutative for all A, B E obj G . That is to say, these
squares shall commute when X = Z and PA' PB = ~A' ~B ' and furthermore, whenever
a ~o-object X and ~o(?/)-morphisms PA: <X, FA>-GA are given making the above
squares commute, there shall be a unique ~o-morphism 0: X - Z for which
PA = ~A ° <P' FA> •
We point out, for use in the proof of the next Yoneda Iemma, that the stated
conditions on the maps ~A are equivalent to the requirement that, whatever X E
obj %o ' the (immense) diagram of (perhaps "large") sets and functions faintly hinted
at by the picture below be an end, or inverse limit, diagram:
~ Z~o(?/)(<X, FA, G(B, A)>, GB)
Fdo(?/)( <X, FA>, GA) ~ Z - " F ( < x ~ii>, ~) ~>A)/ ((X,F~,>,3)~.I f.~(?/)( a(A, B)>, (~, ~ O <X, FB, GA)
/ A< ~o(?/)(<x, FB>, GB)
~o(X, Z) ~(X'%~ ~o(?/)( <X, m, C(B, Q)>, GB)
'~' ~V'-~ ~o(?/)( <x, ~, a(Q, Q)>, GQ) .
Equivalent again is the requirement that ~ = [~A ) be an ~(?/)-natural transforma-
tion <Z, F(-)> -G between the ~(?/)-valued ?~(?/)-functors <Z 3 F(-)> and G ,
and that composition with ~ 's components set up a biJection between ?/-morphisms
X ~ Z and ~(?/)-natural transformations < X, F(-) > ~ G.
221
As before, Z = [F, G] is uniquely determined (to within a unique compatible
isomorphism) by these requirements. Again, a desirable attribute of [F, G] , if it
exists, might be that the diagrams of the above mentioned type, with the left hand
vertex %(X, Z) replaced by ?~o(7/)(X, Z) , remain end (or inverse limit) diagrams
even when X be permitted to vary among all multiobjects in ~(7/) -- this is what
occurs, absolutely~ in the instance covered by the next Yoneda Lemma. If, in partic-
ular, the choice X = <~> results in an inverse limit diagram -- e.g., if the
functor V = V7/ = %(7/)(<~>, ~(-)) is (in a sense not defined here) 7/-represent-
able -- it will b~ seen that the elements of V(Z) correspond exactly to the indi-
vidual 7/-natural transformations from F to G.
[If 7/ is multilinear by virtue of being closed, the specifications on [F, G]
reduce to the requirement that [F, G] be the inverse limit of the end-like diagram
in 7/ suggested by the picture
Theorem g (Contravariant Yoneda Lemma). Let G be a 7/-category, let Q E obj G,
* R Q ' and let G: G ~ 7/ be a contravariant 7/-valued 7/-functor. Then ?/-nat( G) =
[R Q, G] = Hom~(R Q, G) exists, and is given, notably, by GQ via the muitimaps
~A = GQA: <GQ, RQ(A) > = <GQ, ~(A[ Q) > - GA . Indeed, these data solve the
extended universal problem absolutely.
Proof. That all the sqmares depicted in Definition 2 commute when F = R Q ,
X = GQ , and PA = ~A = GQA follows from (MIg 8). Consequently, letting X be any
multiobject in ~o(7/) , and attending~ for the moment, only to the solid arrows in
it, we see that the diagram of sets overleaf is commutative; moreover, each dotted
arrow composes with the solid arrow above it to give the identity function on the
222
left hand terminus~ the diamond involving two dotted arrows commutes~ and the two
remaining compositiEns of a dotted and a solid arrow, endomorphisms of the bottom
center set %(~r)( <X~ ~(Q~ Q)>, GQ) , are equal. So again, either by appeal to the
work of Psme or by the observations that the bottom row is a contractible equalizer
situation and that each compatible family of maps [pA } to the entries of the cen-
tral column has its components DB uniquely reconstructible~ via a dotted rising
arrow on the right~ from the component pQ , we obtain the desired inverse limit
information about %(~r)(X~ C~) .
[Note that when 7/ is multilinear by virtue of being closed, this absolute
inverse limit diagram in sets actua3_~ arises from an absolute inverse limit diagram
in ?f of the following form, where dotted arrows indieate splitting maps :
LO.(A,~)
t r ~
,~ ~ --~ ..... " ~ "~i~,~,O"q L'~" ~
~",,~ .~,~> ;. ,, ~ ~ , -~,,,,~- L ~ ~
"" "" ~'-- ' ' ' ~'I"" ",r~v ~'~" ~
223
Consequently, this limit will be preserved after application of any functor defined
on ~ , in particular, under application of any iterate of the ~-valued cov~riant
hom 9unctors on ~, and so is even better than what Day and Kelly [DK, §2] would
probably continue, in this context, to call a limit in ~. A similar procedure can
be used, when ~ is multilinear by virtue of being monoidal, to lift the absolute
colimit diagrams occuring in the proof ofTheorem 1 into ~ itself.]
Definition 3. Let F and G be covariant %'-valued ~/-f~nctors (~ ~ ~/ de-
fined on the ~-category G , where ~ is a multilinear category. By a %-object
~-nat(F, G) = [F, G] : Hom~(F, G) of ~-natural transformations from F t__oo G we
mean an object Z of % equipped with %(~')-morphisms CA: <FA, E> ~GA that
universally solve the problem of rendering all the squares
<FAB , X> <G(A, B), FA, X> > <FB, X>
<id' PA> I I~B
<(~(A, B), GA> > GB GAB
(in %(~))
(with X E obj ~o ) commutative, for all A, B E obj G . That is to say, the above
squares shall commute when X = Z and ~A' PB = CA' ~B ' and whenever an object X
of ~o and multimaps 0A: <FA, X > ~ GA are given making the above squares com-
mute, there sh~l] be a unique ~o-morphism 0: X ~ E satisfying PA = CA ° <FA' 0 > .
[As in the contravariant case, one might hope of Z, if it exists, that it also
solve the extension of the above problem in which X may be any multiobject of ~ .]
To facilitate the proof of Theorem 5, consider the diagram suggested by
224
The conditions set forth in Definition 3 on the maps ~A amount simply to the
requirement that this diagram of sets and functions be an inverse limit diagram.
Equivalently, composition with the ~A s should set up a bijective correspondence
between the maps X ~ Z and the ~(~)-natural transformations from the covariant
~(T)-valued ~(T)-functor <F(-), X > to the similar functor G.
[Unlike the situation for contravariant T-valued T-functors, there is nothing
more to be said here when T is closed, neither is there, by contrast with the
density lemma situation, when T is monoidal. When T is closed and symmetric,
however, so that Gop is a T-category, the present situation compatibly coincides,
after application of symmetry~ with that previously envisioned for contravariant
T-2unctors.]
Th_eor_e_m ~ (Covariant Yoneda Lemma). Let G be a T-category, let Q E obj G ,
and let G: G ~ T be a covariant T-valued T-functor. Then T-nat(L Q, G) = [L Q, G]
= Hom~(L Q, G) exists, being given~ notably~ by GQ via the multimaps C A = GQA:
<LQ(A)~ GQ> = <~(Q, A), GQ> ~ GA. Indeed, these data solve the extended
universal problem absolutely.
Proof. As in the proof of Theorem 2, consideration of the diagram of sets and
functions, of which the solid arrows in the picture
(~,A), 6n)
portray a representative portion~ shows that there are splitting maps, as indicated
by the dotted arrows in our picture, making ~o(T)(X, GQ) an absolute inverse limit
of the rest of the diagram, regardless what the multiobject X of ~ ; this ends
the proof.
225
§3. C0mmentar~
A. In defense of multilinear categories. When the author first discovered
Theorems 2 and 37 in the more limited context of closed ~r, it seemed improbable
that they could even be formulated, let alone proved, for ~ monoidal. By the same
token, it seemed folly to hope that Theorem 1, originally envisioned for monoidal
, could ever sustain itself in the closed environment. One of the cles~r virtues of
multilinear categories is their nutritive value to these results.
Another clear virtue of multilinear categories: while it is obviously nonsense
that any full subcategory of a closed (or monoidal) category should remain closed
(or monoidal) in a compatible manner, it is an equally obvious fact that each full
subcategory of a multilinear category inherits the multilinear structure of its
parent@ Incidentally, these remarks make it clear that there is a wide selection of
multilinear categories that are not closed or monoidal.
The most striking virtue of multilinear categories; however, is that they have
absorbed in the ordinary associativity of their composition rules all the coherence
problems surrounding the associativity of the multiplication bifunctor on a monoidal
category. Indeed, for a category ?f to have the structure of a monoidal category o
is the same as for it to have a multilinear structure in which each functor
~o ~> ~o (~) ~°(~')(X' -)' '> ~ ( X 6 obj ~o(~))
is representable in ~ . The fact; for example, that both ((A®B)®C)®D and
A®(B@(C@D)) represent the quadrilinear-maps-from-<A, B; C; D> functor assures
that there is a unique compatible isomorphism between these two objects; on the other
hand3 both reassociations are compatible isomorphisms: hence they are equal. (We
have not broached the related questions surrounding symmetry. )
_B. _Th_ee _a~o_g~ with modules over an __algebra over a ground ri_ng. Thinking; with
Yoneda [YS and Barry Mitchell, of a multilinear category ~ as a "ground ring", of
a ~f-category G as an "algebra" over ~f -- indeed~ as an algebra of obj G × obj G
matrices with entries from ~f -- and of a covariant (resp. contravariant) ~-valued
~-functor on (~ as a left (resp. right) G-module, we may write GG and G G for
226
the Yoneda left and right regular representations of ~ (meaning a~(Q) = L Q and
GG(Q) = RQ). The content of the Yoneda Iemmas then resembles, as the author is
grateful to Barry Mitchell for having pointed out, the familiar trivialities
concerning hom and tensor for modules over an algebra:
~oma(a a, Fa ) -~ - - -~a~aca , ~ ( a a , c~ ) -~ - - -Cc~aa~ .
(Theorems I, 2, and 3 provide such isomorphisms value-wise; only the question of
naturality would remain to be considered.)
Nor is there lacking a definition of a bimodule, i.e., of a ~-valued ~-bi-
functor of mixed varianee on a pair of ~-categories <a, 8> . This is a rule
F assigning to each pair (A, B) of objects A 6 a and B 6 ~ a value F(A, B)
in the object class of ~o and a quadruply indexed family of %(~)-morphisms
ABFA,B,: <@(B, B'), F(A, B), C(A', A)>
making the diagrams
> F(A', B')
= - ~" (A,~')
commute, whatever the objects A, A' A" , in a
example of such a bird is, needless to say, the
and B, B', B" in 8 . The prime
~/-valued horn ~-bifunctor on
<G, G> arising from a(-, =) ; the easiest, or most sensible, mY to verify this
claim is to establish the analogue of [EK, Prop. III.4.2] on the recognition of bi-
functors in terms of their partial functors. [Of course, in the absence of syn~netry,
one cannot readily deal with more variables or other variances, and the notation
<(%', ~> does not tend to designate even an 7R(?f)-category G °p ~ G .] Using
such bifunctors, one can reproduce many of the 'Using up" and "inheriting" operators
phenomena familiar from homming and tensoring modules over several algebras.
227
Finally, by ana/ogy with the notion of modules over the ground ring itself,
rather than over an algebra over the ground rlng~ one might be tempted to consider
categories a equipped with structure of the type
M a pairing ®: Z~o(~) x a ~ a and isomorphisms aAB: A® (B®M) ~ <A, B>®M
and raM: <~> @M ~ M satisfying the unit and associativity requirements
M M • A@a M = a M C~ aAi = A@~ M , a~ = ~A®M , aA<B,C> <A,B>C °aAB '
where I = <~> ~ i.e., a left ~-tensored category, or of the opposite type~ i.e.~
a right ~-tensored category (the latter can equally well be viewed as a monoidal
functor from ~(9/) to the monoidal (under composition, composing as in calculus)
functor category •a ) . The reader will find it easy to supply definitions for
covariant 9/-functors from a ~/-category to a left ~/-tensored category and for
contravariant 9/-functors from a ~-category to a right ~/-tensored category. With
a pairing [ ~ ]: (~(~/))°P x 6 4 G and isomorphisms JM: [<~>'M]~M and
bAB:M [A3[B,M] ] ~ [<A~B>,M] satisfying the unit and associativity laws
M = [A, JM] M M [A,b M M ob[C,M] bAI ' bIA = J[A,M] ' bA<B~C> ° C ] = b<A,B>C -AB '
we may define a ~-cotensored category -- this too can be done on the other side --
and the reader may discover for himself what sorts of ~-functors there are among
all the above. These ideas await exploitation.
C. Funetor categoric_s, comma c~tegori=es, l'~_ts, a_nd a_d_equac~. Were there any-
thing like a ~-functor ~-category ~G or ~/a°P, Theorems 2 and 3 would express
the faithful fullness of the Yoneda embeddings Y: G ~ ~ G°p and Y: a °p ~ ~/G.
unfortunately, short of requiring %~ to be as complete as a is large, there is no
reasonable way of ensuring that either ~ a°p ~a or is a ~/-category: the horn
objects [F, G] just need not exist. However, our very definition of [F, G] does
provide a # ~°°p J~°(~)°P-valued) -valued (indeed, even a hom functor on each of
these categories of ~-valued ~-funetors, provided only a suitable category # of
"large enough" sets is at hand. Disentanglement of the structure borne by these hom
functors will lead the reader to a suitable definition of a pro-~-category, as fore-
shadowed in the remarks preceding (MLC 7).
228
Just as Theorems 2 and 3 should be expressing the faithful fullness of the
Yoneda embeddings~ so Theorem 1 should be expressing the adequacy thereof.
Disregarding the problem that the ~-valued ~-functor '~-categorles" are, at best~
mere pro-~-categories, there remains the difficulty that adequacy of Y is usually
regarded as the sum of the assertions F = ~-l~[(Y3F ) ~ G ~ ~G ] , where (Y,F) is
the comma category and one of the G's has an op on it. Surely there must be
appropriate notions of a ~-comma-category and of a ~-colimit (John Gray has them
for ~ = C~Y) so that it is exactly the above displayed assertion that Theorem 1
establishes. Once such notions are available 3 it must surely also be possible to
make sense of and prove the statements
with which we close.
[D l]
[D 2]
[DK]
V,]
[Y]
229
REFERENCES
B. J. Day, On closed categories of functors, in Proc. M. W. C. S. IV,
Springer Lecture Notes in Math. 137 (1970)3 pp. 1-38.
E. J. Dubuc, Kan extensions in enriched category theorF, Springer
Lecture Notes in Math. 145 (1970), xvi + 173 pp.
B. J. Day and G. M. Kelly, Enriched functor categories, in Proc.
M. W. C. S. III, Springer Lecture Notes in Math. 106 (1969), PP. 178-191.
S. Eilenberg and G. M. Kelly~ Closed categories, in Proc. C. O. C. A.
(La Jolla, 1965), Springer, Berlin, 1966, pp. 421-562.
F. E. J. Linton, Relative functorial semantics: adjointness results,
in Category Theory, Homology Theory, and thelr Applications, III~
Springer Lecture Notes in Math. 99 (1969), pp. 384-418.
R. Par~, Absoluteness Properties in Category Theory (thesis), MCGill
Univ., MontrSal~ 1969.
R. Par~, On absolute colimits (preprint)~ Dalhousie Univ., Halifax~ 1970.
N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo 8
(1960), pp. 507-576.
LOCALLY ~-PRESENTABLE AND LOCALLY a-GENERATED CATEGORIES
.) Friedrich Ulmer
The notions of a locally a-presentable and locally a-generated categories are intro-
duced, where a is a regular cardinal. The properties of these categories are studied
extensively, in particular their close relationship with other types of categories. Also
the subclasses of topos, algebraic categories and locally ~-noetherian categories are in-
vestigated in detail. A "classification" of locally a-presentable, locally a-generated
categories, locally ~-noetherian categories and algebraic categories is given.
We begin with some examples motivating the terminology and then introduce the basic
definitions. This will make up the first third of this summary. In the remaining part we
give an outline of the main results. At the end the reader can find a list of the no-
tions and symbols used in this summary. The page number indicates where the definition
can be found.
Recall that an object in an abelian category is called noetherian if every properly as-
cending chain of subobjects is finite. An abelian category is called noetherian if every
object is nostherian. Let 3 be a locally noetherian category [7], i.e. a Grothendieck
[lO] ABS) category with a set of noetherian generators. Let I : U -*A denote the inclu-
sion of the full subcategory consisting of all naetherian objects in ~ . Then ~ is a
small exact subcategory and the functor
o A.--,[_u ,A~.G~.] , A ,,~[I-,A]
o induces an equivalence of A onto the full s~bcategory of [_U ,A__b.Gr.] consisting of
ell contravariant left exact functors on ~ with values in the category Ab.Gr. of abe-
lien groups [7]. Conversely, the contravariant left exact functors on a small noetherian
category ~ with values in Ab.Gr. form a locally noetherian category, the noetherian
*) This note is a summary of a joint paper with P. Gabriel. It is an outgrowth of mostly
unpublished papers and manuscripts of both authors from 1965-69, cf. [8], [20], [21],
[22]. Details will appear elsewhere.
231
objects of which are precisely the representable functors, cf. [T]. Hence there is a bi-
jection between equivalence classes of small noetherian categories and equivalence class-
es of locally noatherian categories.
In order to generalize this reault to "arbitrary categories" we first need a modifi-
cation of the notion noetherian. Recall that an object A in a locally noetharian cate-
gory ~ is noatharian iff the hom-functor [A,-] : A -~Ab.Gr. associated with A E
preserves filtered colimits (= direct limits), cf. [7]. The latter makes sense in more
general categories. For instance, let Mod(A) be a category of left A-modules over a not
necasserely noetherian ring A . Then the set valued hom-functor [A,-] : Mod(A) -~Sets
associated with a module A ~ Mod(A) preserves filtered colimits iff A is finitel~
presentable. Moreover a module A i__~_s finitel W qanerated iff the hom-functor
[A,-] : Mod(A) -~Sets preserves monomorphic filtered colimits, i.e. the transition mor_-
phisms are all monomorphic. The same holds for the categor~ of qroups, rinqs7., universal
alqebras.
More generally, let
tared if
a) for every family (Dv)~ N of objects in ~ with
D ~ ~ and a family of morphisms (D v -~D)~ ~ .
b) for every family (~k : Do -+DI)XCL of morphisms in
morphism ~ : Dl-~D 2 such that ~k =~H for all
be a regular cardinal. A small category is called ~-fil-
card(N)< ~ there is an object
D with card(L)< ~ there is a m
X,U c L . *)
A functor is said to preserve a-filtered colimits if it preserves colimits over a-filter-
ed categories. For ~ =% this specializes to the usual definition of filtered colimita.
Recall that a group G admits a presentation by less than ~ generators and less
than ~ relations iff there is a cokarn~1 di~r~
_LL .LLz- *--~ We would like to thank to H. Reichel who pointed out to us that our previous defini-
tion of amfiltered was too weak for our purposes. He suggested to us the modified con-
dition b). Note a regular cardinal is --~-~o-- " that
232
in the category G_.~r of groups such that both coproducts (= free products) have less than
factors ~ (= integers). One can now easily show that ~ qroup G e Gr admits such a
presentation iff the hom-functor [G,-] : Gr-~Sets preserves a-filtered colimits. Like-
wise G has less than a generators iff the functor [G,-] : Gr-~Sets preserves mono-
morphic a-filtered colimits.
This suggests calling an object A in an arbitrary cocomplete category ~ G-present-
able (a-generated) if the hom-functor [A,-] : A -~Sets preserves a-filtered colimits
(monomorphic a-filtered colimits). An object A E ~ is said to be presentable (generated)
if it is a-presentable (a-generated) for some regular cardinal ~ . In general, an object
is neither presentable nor generated. For instance, in the dual category Sets ° of sets
only sets with a single element are generated. In the category Comp of compact spaces
only the empty space is generated. However in the dual cateqory Comp ° (~ commutative
C*-algebras with unit) A space i&s ~-presentable iff it is finite, and ~l-presentabl8 iff
it is metrizable. In a cateqory of universal alqebras in the sense of Birkhoff [2] o__r_r
Lawyers [13] a_n.nalqebra i__ss a-presentable (~-qenerated) iff it admits a presentation with
less than ~ generators and less than ~ relations (loss than a qenerators). For
a =~ this specializes to "finitely presentable" and "finitely generated" in the usual
sense. In the category Cat of small categories an object ~ is a-generated iff there is
a set M of morphisms in ~ such that card(M) ~ a and every morphism in ~ is a
finite composition of morphism:belonging to N . An object X e Cat is a-presentable iff
there is a set M of morphisms in X with card(M)~-~ a such that X can be obtained
as a quotient category of the free category ~(M) on M viewed as a graph by identify-
ing less than e morphisms in ~(M) .
The notions a-generated and a-presentablehave the usual properties. Recall that in a
category ~ an epimorphism A-~A" is called reqular if it is the cokernel of a pair
A' ~A , and proper if it does not factor through a "proper" subobject of A" . A set M
of objects in a cocomplete category ~ is called a reqular (proper) set of qenerators if
every object in ~ is a regular (proper) quotient of a coproduct of objects of M . I__~n
cocomplete cateqor.y A with a reqular set M of a-presentable generators an object
233
A E 3 i8 a-presentable iff there is a cokernel diagram
~U. ~ ~Ui-~A
with Ui,U j E M and card(I)~ a ~ card(J) . The proviso is that the composite of two
regular epimorphisms is regular. Likewise, in ~ cocomplete category A with a proper set
M of a-generated generators an o bimct A E A is ~-generatad iff it admits a proper epi-
morphism __ ~U i -~A with U. E M and card(1)~ a e
~I i
A category 3 is called locally a-presentable if it is cocomplete and has a proper
set of a-presentable generators. *) A category 3 is called locally presentable if it is
locally a-presentable for some regular cardinal ~ . The least such cardinal is called the
rank TF(3) of 3- For instance, the equations ~(Gr)=~, T~(Cat)=~o presentation
-n~ Comp° ) =~i hold. If k is locally a-presentable, then the full subcatoqory 3(a) and
of the a-presentable obiects is small and the inclusion A(~) -~A is dense (= left
adequate [12], [21] 1.~). Moreover A has limits (= inverse limits) and every object
X E A is ~-presentable for some cardinal ~ .
Likewise a category ~ is called locall~ a-qenerated if ~ is cocomplete and has a
proper set M of a-generated generators such that the proper quotients of every ~-copro-
duct of generators form a set. A category ~ is called locall# qenerated if it is locally
~-generated for some regular cardinal ~ . The least such cardinal is called the qenera-
rank ~(3) of 3 - For instance the equations ~(G~r) =~o ' &(Cat) =~o and tion
~(C°mp°) =~l hold. If A is locally ~-generated, then the full subcateqory _~(a) of
the a-generated obiects is small and the inclusion ~(~) -~A is dense. Moreover A has
limits and every object X E A is ~-qenerated for some cardinal ~ . It is not hard to
see that a locally a-presentable category 3 is locally a-generated and thus ~(~) ~(~)
holds. The converse however is not true.
The class of locally presentable categories is quite large. It includes the categories
*) We first called such categories algebraic. A talk of S. Breitsprecher at Oberwolfach
in the spring of 1970 led us to change the definition, cf. also [3].
234
of sets, monoids, groups, rings and more generally universal algebras in the sense of
Hirkhoff [2] (reap. Lawvere [13]), 51ominski [19] (resp. Linton [15], with rank) and
Benabou [1]; the category Cat of small categories, the category of ordered sets, the
dual category ComR ° of compact spaces, the category of set valued sheaves on a small
category with respect to a Grothendieck topology, the category of set valued functors on
e small category ~ which preserve a given set of limits in ~ , etc.
However the categories Camp and Top of (compact) topological spaces are not locally
presentable. Neither is the dual of Sets . More qenerall~, a non small cateqory B is
not locally presentable, if B ° is locall N presentable.
Recall that in a category [~,Sets] of set valued functors on a small category
the hom-functor IF,-] : [~,Sets] -~Sets associated with an object F ~ [~,Se_~ pre-
serves colimits iff F is a retract of a representable functor. This suggests calling an
object A in a cocomplete category 3 D-presentable if the functor [A,-] : A ~Sets
preserves colimits. A category ~ is called locally O-presentable if it is cocomplete
and has a proper set of O-presentable generators. One can show that ~ cateqory A is
locall~ O-presentable (i.e. 1¢(~) = O ) iff it is equivalent to a functor cateqor.y
[~,Sets] with X small, cf. also Roos [18], Bunge [4].
Let ~ be a regular cardinal. A category is called a-cocomplete (~-complete) if it
has coproducts (products) with less than ~ ~ummands (factors) and cokernels (kernels). A
functor is called a-cocontinuous (a-continuous) if it preserves these colimits (limits)
which we refer to also as a-colimits (s-limits). For instance, the ful_~l subcateqory ~(a)
of the a-presentable ob.iec~of ~ locally s-presentable cateqory A is ~-cocomplete cate-
qory and the inclusion A(a) -~A is ~-cocontinuous. Note that an~-cocomplete category
is a category with finite colimits. A category ~ is called O-cocomplete (O-complete)
if it has contractible cokernelS(contractible kernel), i.e. every diagram A-~ A'
with ~ = idA, and a~ = a~ has a cokernel. It is easy to see that a category is
O-complete iff it is O-cocomplete iff every pair A J~-~ A with ~ = ~¢ has a cokernel
(d. Beck).
235
An ~-cocomplete category ~ is called properly ~-cocomplete if every system
CX-~Xc)c~ of proper quotients of X ¢ ~ has a colimit. Note that card(1)~ ~ is
allowed. An ~-cocontinuous functor is called properly ~-cocontinuous if it preserves
these colimits For instance, the full subcateqor W ~(~) of the ~-qenereted objects of~
locally ~-qenersted cateqor W ~ i ss properly ~-cocomplete and the inclusion ~(~) -~
i sproperly ~-cocontinuous. The dual notions of properly a-complete and properly ~-conti-
nuous are left to the reader.
The main body of the paper consists of a study of the following classes of categories:
KZ Locally presentable categories,
K2 Locally generated categories.
K3 Categories equivalent to a category Cant [~°,Sets] of contravariant a-continuous
set valued functors on a small ~-cocomplets category ~ ~ and ~ are veriable~.
Examples: a) Let I : U-~Gr be the inclusion of the full subcstegor Y of all finitely
presentable groups in Gr . Then U is small and ~o-cocomplate and the functor
G r-~[U_.°,Sets] , G ~LI-,G] induces an equiveleqce of Gr onto the full subcateqory
of all ~o-continuous functcrs.
b) Let I : Met-~Comp be the inclusion of the full subcategory of metric compact
spaces. Then Met is ~l-complete and the functor Camp -~[Me_.tt,Sets] , X ~[X,I-]
induces an equivalence of Camp onto the full subcateqary o~f ~l-contlnuous functors.
K4 o
Categories equivalent to a category Cant [~ ,Sets] of contravariant properly a-con-
tinuous set valued functors on a small properly ~-cocomplete category ~ ~ and
are variable)
Example: Let
generated
I : U-~Gr be the inclusion of the full subcategory of all finitely
groups in Gr . Then U is small and ~roperly ~o-COcomplets an~
K5
K6
236
the functdr Gr-~[~°,Sets] , G "~[l-,G] induces an equivalence of Gr onto the
full subcatsqor~ of all properl~%~ontinuous functors.
Categories equivalent to a category of contravariant set valued functors on a small
category ~ which take a given set of colimits in ~ into inverse limits (~ and
the set of colimits are variable)
Examples: Algebraic categories in the sense of Lawyers [13], Linton [15] (~ith rank),
Categories eq-ivalent to a cat~gory £ont~[_U°,Sets] of oontravariaot Z-continuous
set valued functors on a small category U , where ~ is a given set of morphisms in
U ° [_ ,Sets] , ~ and ~ are variable). A functor t : U2-~ Sets is called ~-continu-
(~: d0~-~r~) E~ the induced map [r~,t]-~[d~,t] , @ ~m-~ is ous if for every
a bijection.
Example: Let be a Grothendieck topology on a small category ~ and let ~_ be
the inclusions of the crlbles associated with ~ in representable lunchers, cf.
Verdisr [23]i. Then ~ne E-continuous func~ors U°-~Sets are precissl~ the sheaves
on U with respect to ~.
K7 Categories equivalent to a category
where Sets is an arbitrary product of copies of Sets and a triple with rank
I
in Sets and an idsmpotent triple with rank in Sets 1 , cf. d. Becks introduc-
tion to Lecture Notes vol. 80. The number of factors in Sets and and are
variable).
A triple ~T = (T,~,U) in a category ~ is said to have a rank if the functer
T : A -~A preserves ~-filtared colimits for some cardinal ~ .
237
Example: Any coreflexive full subcategory ~ of a functor category [~,8ets I such
~hat ~ is small and the inclusion ~-~[~,Sets] preserves ~-filtered colimits for
some regular cardinal ~ .
K8 Categories equivalent to a categor W ~
where Sets is an arbitrary product of copies of Sets and , ,... is a
finite sequence of triples with rank in Sets , -- , .... (The number of factors
~F~ in Sets , the number n and ~l ' ...~ are variable) n
Example: The categor W of contramodules over an associative coalgebra, cf. Eilenberg-
Moore [5] .
The main result of the paper is that these eight classes coincidE, in particular that
an~ cateqor Y belonqinq t__o_o K~-K8 is locall N ~-presentable for some regular cardinal
which is Easily computable in each case, and that the locall N a-presentable (~-qsnerated)
cateqories can be classified bw means of their u-presentable (~-qsnerated) ob.~ects. I__n_n
more detail, for every reqular cardinal
I~ IA E K1 and ?~(A)~ ~ I
o__~r ~ = 0 the map
>I: small and / ~-cocomplete
• [ o Sets] is defined variant e-continuous set valued functors on U (For ~ = Ol Cant U ,
to be [Ug,Sets ] ~. LikEwise for aver.y reqular cardinal ~ the map
I A~ A ( K2 and E(A)~ ~ I > I ~ ] U small and I / properl W ~-cocomplete
which assiqns to ~ locally S-presentable cataqor~ A the full subcateqor W ~(~) of its
a-presentable objectsjinduces a biiection between equivalence classes of locally a-pre-
sentable categories and equivalence classes of small ~-cocompleta categories. The inverse
assigns to a small ~-cocomplate cateqor~ U the categor~ CEnt [~°,Sets] of contra- map
238
which ass±~ns to a locally a-qenerated cateqorE ~ the full subcateqor W of its a-qanerat-
ed objects/induces ~ bijection between equivalence classes of locally 0~-qenerated cateqo-
mias add equivalence cla~ses o~fsmall properly ~-cocomplete cate@ories. The invarse map
to u the category C ,Setsl . assigns
Before we investigate the above mentioned classes K1-K8 and some of their subclasses
we list some properties of locally presentable categories. Every locally presentable cate~
gory is cowell-powered. If both A and A ° are locally presentable, then A is equiva.
lent to a partially ordered set which is inf-complete. This is a generalization of the
well known result that a 6rothendieck ABS) and ABS)* category is zero. In a locall~ a-pre-
sentable category ~ ~-filtered colimits commute with a-limits. Likewise in a locally
a-generated category ~ monomorphic a-filtered colimits commute with a-limits. Moreover
there is for every cardinal ~ a cardinal 7 > ~ such that ~(7) is ~-complete and that
~(y) = ~(y) , i.e. every y-generated object is y-presentable. Also every object in ~ has
an "underlying set" the cardinalit¥ of which can be estimated by means of certain "invari-
ants" of ~ . These "invariants" are easily describable if ~ is given in the form
Contz[~°,Sets] , cf. K6.
A continuous functor between locally presentable categories has a left adjoint iff it pre-
serves monomorphic a-filtered colimits for some cardinal a .
Let ~ be a locally a-presentable (a-generated) category and d : U -~A the inclu-
sion of the full subcategory of its a-presentable (a-generated) objects. Let ~ be a co-
complete category. Then a functor T : A-~Z is the Kan extension of its restriction on
U iff T preserves a-filtered colimits (monomorphic 0~-filtered colimits), cf. Hilton
Ill] ~7.
Now let ~ : ~(~) and let ~ be a diagram type with card(~)~ ~ . If a functor
t : U-~Z preserves colimits of type ~ , then so does its Kan extension Ed(t) : A ~Z .
If ~ has limits of type ~ and if in ~ a-filtered colimits commute with limits of
type ~ , then Ed(t) : A-~Z preserves limits of type ~ provided t : U-~Z does.
Now let d : U -~A be an arbitrary functor between categories with finite limits~
small. (J need not preserve finite limits). If Z is a topos and t : U-~Z pre-
serves finite limits, then so does the Kan extension Ed(t) : A-~Z . If instead ~ is
239
]0caJly presentable and if U and A_ are a-complete and for every A (~ A_ the category
N lJ
J/A of objects in U over A is ~(Z)-filtered, then Ed(t) : A-~Z preserves &-limits
~rovided t : U ~Z does. (Note that a and ~(~) are independent of each other).
The full subcatagories of functor categories [~°,Sets] as described in KJ-K8 are
coreflexive. An explicit construction of the coreflection can be given in the case of K3.
U ° It is based on the fact that a functor F : -- -~Sets is a-continuous (~ a-cocomplete)
iff the cateqor¥ ~/F of')reprasentable functors over F 11
, -- i_~s a-cocomplete and the under-
lyinq functor
Yr : U / F - - , U , ( U , [ - , U ] ~ r ) ~ U
is a-coeontinuous. First soma preparation.
For a small category ~ let ~a(~) be the full subcategory of [X_°,Sets] consisting of
ell a-presentable objects. The category K_a(~) is called the ~-cocompletion of ~ in
[~°,Sets] because every functor in ~ (~) is an a-colimit of representable functors and
because K (X) is ~-cocomplete. Moreover every functor t : X --)U can be extended to an
a-cocontinuous functor Ed(t) : K_ (~) -~ which is unique up to equivalence/namely the
Ken extension of t with respect to the Yoneda embedding d : X-~K (X) , X ~)[-,X] .
O The value of the coraflection L : [U_ ,Sets] -~Cont~°,Sets] at a functor F ~ [U°,Sets]
can now be obtained in the followinq way. Let X = U/F and t = YF then LF is the co-
limit of the composite
where Y is the Yoneda embeddinq U ~[-,U] . This construction shows A duality between
the problem of cocompletinq small cateqories and the problem of makinq set valued functors
continuous. This can be made precise by means of the full embedding
and its left adjoint @,v .--,lira [ - ,HV] . - - -~v
240
In the description of the above classes K3-K6 the category Sets of sets apparently
plays a distinguished role. This however is not so. One can replace it by an arbitrary
locally presentable cateqory without chanqinfl (i.e. enlarqinq) the classes K3-KS. In par-
ticular, th___ee category Sh[U°,A]_ -- __°f sheaves on small category U with respect to a Gro-
thendieck topology and with values in a locally presentable cateqory A is aqain locally
presentable. Moreover the functor "associated sheaf" [~°,A] -~Sh[uO,A]_ _ preserves finite
x inverse limits if A is locally ~o-presentable (e.g. A = Cat , A = Br etc.)/ cf. Verdict
[23] II no. 5, Gray [9]- Likewise the cateqory o_/f qroup objects or cateqory ob.iect ~ etc.
in a locally a-presentable cateqory is aqsin locally a-presentable (rssp.~o-presentable
i# ~ = 0 ). More generally, if ~ is a locally ~-presentaB}ecategory and ~ a set of
a U ° merphisms in category [_ ,Sets] , ~ small, then the category Cont~[U2,A ~ of~-conti-
U ° nuous functors on _ with values in ~ is locally ~-presentable where ~ ~ ~ is the
least cardinal such that for every ~ E ~ the domain d~ and the range r~ are ~-pre-
sentable in [~e,sets] ~Nots that a functor F : ~o ~ is said to be ~-continuaus if
for every A E ~ the functor [A,F-] : U_ ° ~Ssts is~-continuous as defined in K6 .)
Also, if ~ is a set of morphisms in a locally presentable cateqory A and ~ denotes
th~ full subcateqory of ~ consistinfl all objects X E A such that [~,X] : ~X]--~5~ ]
is a bi.iection for every ~ E~ , then ~ is aRain locall~ presentable and the
inclusion ~-~A has a left adjoint and preserve~ a-filtered colimits for some cardi-
nal @ . This illustrates that the class of locally presentable categories has good
closure properties.
A subclass of K6 consists of categories equivalent to a category Cont~[~°,Sets] ,
where ~_- is a set of monomorphisms in [U2,Sets ] which is stable under change of base
in the following sense. If in a pullback diagram
R >d~
[-,u] - - ~ , r ~ -
241
belongs to ~ , then so does ~ . One can show that the coreflection
[U2,Sets ] -~Con~[~°,Sets] preserves finite inverse limits. Thus this subclass consists
of all topos in the sense of Grothendieck-Giraud-Verdier [23]. This shows that the local
property T2 in the definition of a Grothendieck topology (cf. Verdict [23] I p. 13) is
redundant. As above this subclass is closed in the sense that the category Sets can be re-
placed b_2 L any topos A . In particular Cont~[U2,~] i sAtopos if A is and ~ is as
above. Moreover the functor "associated sheaf" [uo,A]_ -- _~Contz[ ~o,~] preserves finite
inverse limits. (The latter is also true for any locallY~o-presentable category ~ , but
Con~[~°,~] is of course in general not a topos). As before in the case of locally pre-
sentable categories, if ~ is a set of monomorphisms in a topos A with a reqular set
M of qenerators, then the above described full subcateqory ~ of~&sheaves in A is
aqain A topos and the left ad.ioint A-~A2 of the inclusion A~ f~ preserves finite
inverse limits.jprovided in every pullback diaqram in A with U ~ M
R > d~-
U - >r~
@ balanqs to 2 i j_f %- does. (This was also proved by Lawvere-Tierney [14] within the
framework of a more general set theory but under the additional assumption that ~ satis-
fies the above mentioned local property of a Grothendieck topology .)
A subclass of K5 consists of categories which are equivalent to a category
~[ o Sets] of a-product preserving functors U ° Cent U , --*Sets , where U is a small cate-
gory with ~-coproducts. Following Lawvere ILinton~Benabou [13] [15] [1] we call such cat@ ~
gorias algebraic. As above in the case of locally presentable categories and topos the
role of Sets is not distinguished. If A is an alqebraic cateqory, then so is
0 Cont~[~ ,3] - In particular, the category of group objects (or algebra objects, etc.)in an
algebraic category ~ is again an algebraic category. ~ cateqory A is alqebraic iff it
242
satisfies the following conditions :
s) ~ is cocomplete.
b) Ever~ equivalence relation is effective
c) A has a proper set M of a-presentable ~enerators s.uch that for every V ¢ M the
functor [V,-] : A -~5ets preserves reqular epimorphisms (i.e. the generators are pro-
jective with respect to regular epimorphisms).
In the special case card(M) = 1 these conditions are essentially those of Lawvere,Linton
[13] [15] although a) - c) are somewhat weaker. CThe existence of kernel pairs is redund-
ant, and the proper generator V does not have to be regular; also [V,-] : A -~Sets
doesn't have to reflect regular epimorphisms). Thus the onl~ difference between an "alge-
braic" categor~ in the sense of Lawvere [13] and Benabou [1] is that the former has one
proper qenerator while the latter has a proper set of qenerators.
The projective presentation rank l~p(~) of an algebraic category ~ is the smallest
regular cardinal a such that ~ has a proper set of a-presentable projective genera-
tots. Of course ~(A)~(~) holds. The full subcateqorw A (a) of the ~-presentable p -- --p
projectives of an algebraic category ~ is small and it is closed in ~ under a-copro-
ducts and contractible cokernels.
Th_~e map
I~ I A-- algebraic and l.~p~(A) ~ a ) I~ I U small with c~cOPr°ductsand contractible coke enel~ fj /
which assigns to an algebraic categor ~ the full subcataqor~ of its a-presentable project=
ives linduces a bijection between equivalence classes of alqebraic categories with
~p(-) ~ a and equivalence classes of small categories with a-coproducts and contractible
cok@r~e]~. The inverse map assiqns two a_ small cate.qor.y U the cateqor.y
Con~U°,Sets] . It should be noted that an algebraic theory T (cf. Lawvere [13], Bena-
bou [1]) is not e categorical invariant of the algebraic category A it gives rise to,
but only the closure of T in A under contractible co~@rmels <in other words not
the free algebras but the projectives, Morita [16]~.
243
Likewise one can define the projective generation rank of an algebraic category,
etc ....
Let N be a set of objects in a small category ~ with ~-coproducts such that every
object in ~ is e retract of a coproduct of objects O~ N . ~Choose N as small as "pos-
sible". Note that if ~o is an algebraic theory [13] [15], then one can choose ~ s~ ~at
card (N) = l>. If A is a locall~ presentable cateqor~ in which cokernels of effective
equivalence relations commute with ~-products (e.g. ~ algebraic and ~ arbitrary, or
a topos and ~ =% )' then this also holds in Con~°,A~ and the functor
is tripleable and preserves reqular epimorphisms, where T~ denotes the N-fold product N
of A . If in addition [U,U'] ~ ~ for everN U,U' ~ N , then also : Con ~ ,A~ -~)
F ~ ~TFu is tripleable. If A ~B T for some triple ~ in e categor X B with pull-
backs, then the composite of the underl~inqs
Cont,,~U° A]~A =~ BT__~ B_
is tripleable, provided equivalence relations are effective in A . (The latter then
also holds in Cont~U°,A~_ ). This shows that the "algebraic" cateqories of Benabou [i]
are those categories which are tripleable with rank~o over-a product --°f copies of Sets
(the number of factors Sets is arbitrary). The above also implies the existence of the
tensor product of theories, cf. Freyd [6], and the tripleability of algebra valued
sheaves on a site over set valued sheaves with rank ~ , cf. H. Wolff [24]. y. @
An object A in a category ~ is called ~-noetherian if for every well ordered
chain
A -~A 1 -~A 2 -~ ... ~A~ --~
of proper quotients of A with u E I and car~(1) = ~ there is a largest element
G~ I , i.e. the morphisms Ac~--~A~I ~A~+2 ~ ... are isomorphisms. In an abelian
244
category an object i_~S~o-noetherian iff it is noetherian in the usual sense. In the dual
cateqory Comp ° o~_f compact spaces ~ space is~l-noetherian if it is metrizable, and
~ o-noetherian iff it is finite. A proper quotient of an ~-noetherian object is again
~-noetherian, but an ~-coproduct of ~-noetherian objects is in general not ~-noetherian-
For instance, in the category G_~r of groups the group ~ of the integers iS~o-neethe-
rian, but a finite coproduct (= free product) of copies of ~ is obviously not~o-noe-
thsrian. Thus a category is called ~-noetherian if it is properly ~-cocomplete and every
object is ~-noetherian. For ~ small properly a-cocomplete category U the followinq are
equivalent:
(i) U is ~-noetherian
( i i ) Con-'Jt [U_°,Setsl = Cont~U°,Sets]
(iii) a) Every strict epimorphism is regular. (strict in the sense of Grothendieck, SGA
1959/60, Def. 2.2) b) (PL~ : U -~U ~ M be an.y system of strict quotients of an object U ¢
\
Le__~_t
end let p : U -~lim U be the canonical morphism. Then for a_anO Z pair of mor-
phisms f,g : U' _~U in U with pf = pg there is a v ¢ M such that already
pv f = pvg holds.
c) The least ordinal number ~ such that every proper epimorphism in U can be
decomposed i_~n ~ +l strict epimorphisms has cardinality less than ~ . (Note
that ~ is zero if every proper epimorphism is strFcfi . In Cat ~ equals 1 .)
In a locally a-presentable category ~ every object is y-noetherian for a sufficiently
large cardinal y . This suggests calling a category ~ locally a-noetherian if ~ is
cocomplete and has a set of a-generated ~-noetherian generators such that every ~-copro-
duct of generators is again ~-noetherian and has only a set of proper quotients and if
~(~) = ~ . From this ~(~) =~(~) follows. <In many cases ~-noetherian implies ~-gene-
rated .) A locally presentable category ~ is locally ~-noetherian iff E(~) = ~ and
has a proper set M of a-generated ~-noetherian generators such that every ~-coproduct
of objects of M is again ~-noetherian. For instance, the "locally noetherian" catego-
ries mentioned at the beginning [7] are .locallY~o-noetherian , the category Comp ° i~s
245
Jl ,k iocall~-noetherian with the unit intervall beinq ~ proper~l-noetherian qenerator, the
boolean algebras i~s~o-noetherian and so is the cataqor~ of commutative alqe- cate~0r ~
bras pver a commutative noetherian rinq A , the polynomial alqebra over A be_~
proper~o-noetherian qenerator. In a locally ~-noetherian category the full subcate-
gory of its a-generated objects is small, ~-noatherian, dense and properly ~-cocomplete.
It coincides with ~(~) and it is reduced in the sense that for any regular cardinal
~' ~ ~ the full subcategory of its ~'-generatad objects is not dense. For instance, the
dual category Mat ° of compact metric spaces ist~-noetherian and reduced. The category
of finitely generated commutative algebras over a commutative noetherian ring A is
~o-noetherian and reduced. The category of countable groups is~l-noetherian but not re-
duced because the finitely generated groups form a dense subcategory. Note however that
in a locally c~-noatherian category an ~-noetherian object need not be ~-generated. For
instance, in the above mentioned category of commutative A-algebras, every A-field is
~o-noetherian.
The map
I ~ ~ locally ~-noetherian ) L~ ~ small, reduced -- and ~-noetherian i
which asaiqns to a locally a-noetherian cateqor W the full subcateqory of its ~-qenerated
objects, induces a bi.iection between equivalence classes of locall~ ~-noetherian cateqo-
ries and equivalence classes of small reduced 0~noetherian categories. Th_~B inverse map
aasiqns to such a category U the category Cont [U ,Sets] = Cont~U_°,Sets] .
If a reader prefers the language of closed categories to Sets, the transcription of
this paper should not involve difficulties, with the exception of sheaves, where apparent-
ly some problems are still open.
Notions Symbols
algebraic category p. 241 ~'(~) p. 233 0-cocomplete p. 234 ~(~) p. 233 ~-cocomplete p. 234 A (~) p. 242 ~-cocompletion p. 239 --A--P~ po 240 ~-cocontinuous p. 234 Cat p. 232 ~-colimits p. 234 Comp p. 232
246
N o t i o n s C-comple%e p. 234 ~-compiete p. Zd4 ~-continuous p. 234 -continuous p.236/2~0
contractible cokernel p.23Y contractible kernel P- Z3Y ~-filtered P- 131 ~-generated object p. 2 3 2 ~-limits p. 134 locally ~-generetad category p. Z 3 3 locally ~-noetherian category p. Z ~ locally O-presentable category p.23~ locally ~-presentable category P- 2 3 monomorphic colimit p. l ~/ ~-noetherian category p . Z ~ ~-noetherian object p. 2~ 3 O-presentable object p. 23~ ~-presentable object P- 2 3 l
projective P - l ~ proper epimorphism p-Z3Z proper set of generators P. 23 properly ~-cocomplete p. Z3~ properly ~-cocontinuous P- Z~- properly ~-complete P-2 35- properl~ ~-continuous P-23~ rank of a triple p . 2 ~
reduced P-~5- regular epimorphism P-2 32 regular set of generators p.~32
~ymb~is r o C~nt U ,Sets]
o Cont U ,sets] C ~ o ont,~[ u , Sets]
-
~(A) G._£_r Met ~(_x)
p. 235
p . 235
p. 2.. "~ / p z ~ /2~0 p 233 p 232_
p 23 .5-
p 23~ p Z33 p Z~I
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[24] H. Wolff. Fractions and closed categories, Dissertation, University of Illinois, 1970.
THE NEETING OF THE ~IDWEST CATEGORY SEMINAR IN ZURICH
AUGUST 24-30, 1970
John W. Gray
This was the first meeting of the Midwest Category Seminar in
Europe and was attended by 52 participants from 8 countries. The For-
schungsinstitut fur Mathematik under the direction of Prof. B. Eckmann
was host to the conference. The following lectures were delivered:
M. Barr: Lubkin's construction
J. Beck: Infinite loop spaces
M. Bunge: Bifibration induced adjoint pairs
A. Burroni: Esquisses des cat6gories ~ limites et des quasi-topologies
E. Burroni: Cat4gories discr~tement structur4es - Triples
C. Lair: Construction d'Esquisses - Transformations Naturelles G6n6ralis6es
B. Eckmann: Categories of fractions and simple homotopy type
H. Egli: Picard categories
J. Gray: Representable 2-categories
D. Gildenhuys: Equational Completions of categories
R. Guitart: Relations - fermetures - continuites
F.W. Lawvere: (2 lectures) Applications of Elementary Topos to set
G.M. Kelly: Closed category structures
A. Kock: Symmetric monoidal closed categories
P. Leroux: A general structure-semantic-theorem
R. Pari: ~ and colimits o
H. Schubert: Cocompleteness of Eilenberg-Mac Lane categories
Th. Thode: A characterization of Birkhoff subcategories
M. Tierney: (2 lectures) Elementary topos
F. Ulmer: Algebraic categories
V. Z~berlein: 2-categories
theory
Several of these lectures are presented either in final or in
announcement form in this volume. Of those that are not, some infor-
mation is available. The paper of Beck will appear in the Proceedings
249
of the Neuchatel Conference on H-spaces, and those of the Burronis,
Guitart and Lair are available as "Esquisses Math4matiques", numbers
1, 2, 4, 5, from Dept. of Math., Tours 45-55, 9 Quai St. Bernard,
Paris 5 eme. Eckmann's paper will appear in a Rome Proceedings volume
and Egli's is available as a preprint from the ETH.
I would like to report briefly on some of the remaining talks,
which will be published elsewhere.
I. D. Guildenhuys reported on joint work with J. Kennison on equa-
tional completions. Here if I : ~ ~ Z is a set valued functor then
the model induced triple is given by
T n = l~m [(n,I) ~ 2]
and the category of algebras Z q?
of ~ (= models). A triple '7['o
if there is a comparison functor
is called the equational completion
is called a separating triple for I
I : ~ ~ Z II'° over Z which is o
fully faithful, 1-1 on objects and ~ is closed under the formation
of qro-subobjects.
Suppose a) ~o exists, and b) i II0 ~ Z factors through groups. Suppose
further that ~ has and I preserves finite products.
Theorem (Guildenhuys and Kennison)
i) If I(M) is finite for all M , then $ is the smallest
Birkhoff subcategory of the compact topological qr-algebras containing o
the models. (This does not use b).)
Furthermore, this category is the same as the category of pro-objects.
ii) If the ~-kernels of models satisfy d.c.c., then Z is the o
category of linearly compact topological qCo-algebras.
250
Examples
1. finite sets ~ compact T 2 spaces
2. finite groups ~ profinite groups
3. fields ..... > topological products of fields and continuous homomorphisms.
2. G. M. Kelly reported on the work of his student B. Day concerning
biclosed structures on functor categories [A,~] where ~ is a
closed (= symmetric, monoidal, closed) category and ~ is a ~-catego-
ry (Biclosed means monoidal plus right adjoints for A® - and - ®B.)
These turn out to be equivalent to promonoidal structures on X ! i.e.,
a pair of functors
p : ~op × ~op × ~ ~ ~ , j : ~ ~
and natural transformations
k : SAjA ® P(ABC) % A(BC)
Q : ~CP(ABC) ~ JC A(AB)
fXp(ABX) ® P(XCD) ~ ]YP(BCY) ~ P(AYD)
plus coherence conditions. Here ]A denotes "coend". Morphisms of
promonoidal categories are defined and more generally one has the fol-
lowing result.
Theorem (B. Day) If A °p ~ • is dense, with A promonoidal and
suitably completed, cocomplete, tensored, etc., then there is a bi-
closed structure on • such that e admits enrichment to a morphism
of promonoidal categories iff suitable ends and coends exist.
251
3. The work described by Par4 is a technical and far reaching elabo-
ration of the following ideas. Let G : A ~ ~ be a functor. Then the
operation which assigns to each B ~ S the comma category (B,G) de-
termines a functor (-,G) : S °p ~ Cat (the category of small catego-
ries). Let ~ : Cat ~ Sets be the "set of path components" functor. o
Then ~o(-,G) = l~m Y o G where Y is the Yoneda embedding for S .
Hence for all F : ~ ~ ? , ~ (-,G) ~ ~ (-,F) iff i~ FG = i~ FF . o o
Taking G = Id gives ~ (-,G) = I and this yields the usual descrip- o
tion of a final functor. Taking G :'~ ~ S gives the description of
absolute colimits. This leads to other useful things via a couple of
observations. First, let G, --~ Sets G . Then ~o(-,G) = G, (±) . This
is the basis of the generalization mentioned above. Second, Tierney
pointed out that there is another interpretation; regarding
~o(-,G) : S °p ~ Sets as a fibration with discrete fibres, the opera-
tion G, in terms of fibrations is left adjoint to pulling back along
G . This point of view has been extensively developed by Tierney, and
Bunge's paper in this volume is a generalization of one aspect of this.
4. The work which has probably aroused the most interest was that of
Lawvere and Tierney on sheaf theory and set theory. A category ~ is
called an elementary topos if i) it has finite limits and colimits,
ii) it is cartesian closed, iii) there is a universal object
I ~ w ~ w (this is the axiom of infinity plus induction), iv) there
is a subobject classifier, ~ . This last means that there is a map
"true" : i ~ ~ such that any subobject Y ~ X is the pullback of
~op "true" by a unique map X ~ ~ . Note that in case ~ = Sets -- , then
is the functor whose value on C ~ ~ is all cribles in C (i.e.,
all subfunctors of Hom(-,C).) These axioms imply the following
252
I. monomorphisms are equalizers
2. Epimorphism plus monomorphism implies isomorphism
3. Equivalence relations are kernel pairs
4. Partial maps are representable
5. QY is a covariant functor of Y in two ways giving rise to trip-
les expressing inf and sup.
6. There are canonical functorial injective resolutions
7. The pushout of a monomorphism is a monomorphism and the pullback of an epimorphism is an epimorphism.
8. Pulling back along a fixed map has left and right adjoints. In par- ticular, colimits are universal, the initial object is strict, co- products are disjoint, there are epi-mono factorizations, and epi- morphisms are coequalizers.
9. The category of abelian group objects in • is abelian.
IO. ~ is a Heyting algebra object in ~ ~ in particular, it has, O, l, ^, v , and ~ .
If @ is an elementary topos then a topology in • is a map
j : fl -~ t~ such that i) j o "true" = "true" , ii) ^ o (j×j) = j o A ,
iii) jj = j . The pullback of "true" along j is a subobject J c
~op For instance, if • = Sets , then J ¢ ~ gives, for each C ~ C ,
a family J(C) of cribles satisfying the usual axioms for a Grothen-
dieck topology. Examples in general are the discrete topology
"true" (id : ~ ~ ~) , the indiscrete topology (~ ~ I ..... 9 ~) , and the
double negation topology (-I7 : ~ ~ ~ , where -I = ~ <id,false~ ~×~ ~
"false" : I ~ ~ being the map whose pullback along "true" gives
~ i .)
For each X , j induces a closure operator on the subobjects X'
of X , denoted by X' . X' ~ X is called dense (resp. closed) if
X' = X (resp., X' = X' ). The dense objects are classified by J and
the closed ones by the equalizer ~ of j and id . X is called 3
f j-separated if for all y i y _~ X with i dense, fi = gi implies
g
f = g . It is called a j-sheaf if it is separated and for all dense
i : Y' ~ Y and f' : Y' ~ X , there exists f : Y ~ X with fi = f'
(Equivalently, xY ~ X Y' is an isomorphism.) The associated sheaf
253
functor "a" is constructed as follows: For each X , X) )X×X is
classified by X×X ~ ~. , which corresponds by adjointness to X i ~X 3 3
whose epi-mono factorization yields the separated reflection
S(X) >--~ ~2X . Then a(X) = S(X) . One shows that "a" is left exact via 3
fractions. The category ~j of j-sheaves is then itself an elementary
topos (with ~ as subobject classifier.) Every exact reflective sub- 3
category of ~ is the category of sheaves for a unique topology on ~.
A morphism of topos is a pair of adjoint functors U* ~ U, : ~' ~
with U* left exact. If ~ = U - sets for a universe U , then there
is at most one morphism from ~' to • and ~' is a U - topos in
the sense of Grothendieck-Verdier-Giraud iff it has generators indexed
by a U - set.
An elementary topos is called O-dimensional if epimorphisms split
(axiom of choice) and Boolean if ~ is a Boolean algebra (Equivalent-
ly i+i ~ ~ is an isomorphism, or every X' ~-+ X has a compliment,
or the double negation topology is the same as the discrete topology.)
A O-dimensional Boolean topos in which I has no proper subobjects
(equivalently, there are exactly two maps of i into D) is called a mo-
del of set theory.
Theorem: There exists a model of set theory in which the negation of
the continuum hypothesis holdsl i.e., in which there is an X with
proper containments ~ ¢ X c ~
The proof depends on three things:
i) For any • , the category ~7 of sheaves for the double negation
topology is Boolean.
ii) If • is a model of set theory and i~ is a poset in • then o o r
the internal functor category ~;Po satisfies (~)]7 is Boolean and
O-dimens ional.
254
iii) There is a generalized ultraproduct construction (via fractions)
which allows one to collapse such a category in order to obtain a mod-
el of set theory.
5. J.W. Gray. I reported on representable 2-cateqorie 9. These are
2-categories ~ in which for each object A there is an object ~A
and a 2-cell A A : ~A ~ A such that given any 2-cell ~ : B ~ A ,
there is a unique i-cell ~ : B ~ ~A with A A ~ = ~ . A is called
stronqly representable if
~(-) = cot (z,-) : ~ ~
If A is representable then # is a category object in Trip A ° (A o
is the underlying category of A .) and the total category of A is
isomorphic to the Kleisli category of ~ .
Representability is equivalent to the existence of comma objects (with
playing the role of (-)~) with the usual algebraic properties, and
in the presence of finite completeness, to the existence of finite
cartesian 2 -limits. o
Some examples of strongly representable 2-categories are Cat ,
H-Cat (where ~ has pullbacks), Cat(x) (the category of category
objects in % , where % has pullbacks) , and Cat-valued sheaves on a
Grothendieck topology. A is strongly representable iff it has a cano-
nical strict embedding into Cat(A o) . Certain aspects of category
theory can be carried out in a strongly representable 2-category A
• ~ B , i = O,I by as follows. Denote the comma object of fi : Al
(fo,fl) (fo,fo) ~ Ao is a category object and it operates on
(fo,fl) on the left. Similarly for (fl,fl) on the right. In parti-
cular, (B,B) = @(B) , so ~(B) operates on (B,f)
a) If T : (B,f) ~ (B,f') is over B×A , then T is a ~(B)-homomor-
phism.
255
b) given A f B ~ A and T : (B,f) ~ (g,A) over BxA . Then
g : 9B ~ (g,g) is functorial and <g,T> is a bimorphism.
Theorem (Yoneda) The functor from A(A,B) to objects over BxA
taking f : A ~ B to (B,f) ~ BxA is full and faithful.
Theorem Given A ~ B ~ A , then adjunctions between f and u (i.e.
pairs of 2-cells satisfying the usual identities) are in one-to-one
correspondence with morphisms (f,B) ~ (A,u) over AxB .
Theorem Given f : A ~ B , then in
PA A ( (f,B)
-9 q f
B B < B
i) there exists qf---4 PA with pAqf = id and pfqf = f
ii) pf is the universal split normal fibration over B associated
to f
iii) (Adjoint functor theorem) f has a right adjoint iff pf has a
v v right adjoint pf with pfpf = id