directed graphs or quivers f j 2 g f 2 j
TRANSCRIPT
1. Directed graphs or quivers
What is category theory?
• Graph theory on steroids• Comic book mathematics• Abstract nonsense• The secret dictionary
Sets and classes: For S = X | X /∈ X, have S ∈ S ⇔ S /∈ S
Directed graph or quiver: C = (C0, C1, ∂0 : C1 → C0, ∂1 : C1 → C0)Class C0 of objects, vertices, points, . . .Class C1 of morphisms, (directed) edges, arrows, . . .For x, y ∈ C0, write C(x, y) := f ∈ C1 | ∂0f = x, ∂1f = y
tail, domain // ∂0ff ∈C1 // ∂1f head, codomainoo
Opposite or dual graph of C = (C0, C1, ∂0, ∂1) is Cop = (C0, C1, ∂1, ∂0)
Graph homomorphism F : D → Chas object part F0 : D0 → C0
and morphism part F1 : D1 → C1
with ∂i F1(f) = F0 ∂i(f) for i = 0, 1.
Graph isomorphism has bijective object and morphism parts.
Poset (X,≤): set X with reflexive, antisymmetric, transitive order ≤
Hasse diagram of poset (X,≤): x→ y if y covers x, i.e.,x 6= y and [x, y] = x, y, so x ≤ z ≤ y ⇒ z = x or z = y.
Hasse diagram of (N,≤) is 0 // 1 // 2 // 3 // . . .
Hasse diagram of (1, 2, 3, 6, | ) is 3 // 6
1 //
OO
2
OO
1
2
2. Categories
Category: Quiver C = (C0, C1, ∂0 : C1 → C0, ∂1 : C1 → C0) with:
• composition: ∀ x, y, z ∈ C0 ,C(x, y)× C(y, z)→ C(x, z); (f, g) 7→ g f• satisfying associativity: ∀ x, y, z, t ∈ C0 ,∀ (f, g, h) ∈ C(x, y)× C(y, z)× C(z, t) , h (g f) = (h g) f
yg
xxqqqqqqqqqqqqqhg
<<<<<<<<<<<<<<<<<
z
h++VVVVVVVVVVVVVVVVVVVVVVVVVV x
f
iiSSSSSSSSSSSSSSSSSSSS
h(gf)=(hg)f
gfoo
t
• identities: ∀ x, y, z ∈ C0 , ∃ 1y ∈ C(y, y) .∀ f ∈ C(x, y) , 1y f = f and ∀ g ∈ C(y, z) , g 1y = g
y
g
1y
&&MMMMMMMMMMMMM x
f
foo
z ygoo
Example: N0 = x , N1 = N , 1x = 0 , ∀m,n ∈ N , nm = m+n ; —
one object, lots of arrows [monoid of natural numbers under addition]
Equation: 3 + 5 = 4 + 4 Commuting diagram:x
4 //
3
x
4
x5// x
Example: N1 = N , ∀ m,n ∈ N , |N(m,n)| =
1 if m ≤ n;
0 otherwise
— lots of objects, lots of arrows [poset (N,≤) as a category]
These two examples are small categories: have a set of morphisms.
Example: The category Set has the class of all sets as its object class,with Set(X, Y ) as the set of all functions from X to Y , composition offunctions: g f(x) = g
(f(x)
), usual identities 1X : X → X;x 7→ x.
This example is large (not small), but locally small:just a set of arrows between each pair of objects.
3
3. Special morphisms and objects
Consider morphism f : x→ y.
• Isom. or invertible: ∃ f ′ : y → x . f f ′ = 1y and f ′ f = 1x.Ex: Bijective function in Set.• Monomorphism: ∀ gi : z → x , f g1 = f g2 ⇒ g1 = g2.
Ex: Injective function in Set.• Epimorphism: ∀ gi : y → z , g1 f = g2 f ⇒ g1 = g2.
Ex: Surjective function in Set.• Retract or split epimorphism: r : y → x with r f = 1x.
Ex: r : n 7→ max0, n− 1 retracts successor function on N.• Section or split monomorphism: s : y → x with f s = 1y.
Ex: Successor function on N is a section of r : N→ N.• Idempotent: x = ∂0f = ∂1f and f f = f .
Ex: R2 → R2; (x1, x2) 7→ (x1, 0) in Set.
Lemma. For xr((y
shh with r s = 1y:
• r is an epimorphism.• s is a monomorphism.• s r is an idempotent (said to split).
Isomorphic objects x ∼= y: Have isomorphism f : x→ y.
Terminal object > for C has ∀ x ∈ C0 , |C(x,>)| = 1.
Examples: 0 in Set, upper bound in a poset, . . .
Initial object ⊥ for C has ∀ x ∈ C0 , |C(⊥, x)| = 1.
Examples: Ø in Set, lower bound in a poset, Z in Ring, . . .
Zero object 0 for C is both initial and terminal.
Examples: 0 in categories of groups, vector spaces, . . .
Groupoid: Category where all morphisms are invertible.
Examples: For a set X:
• Discrete category (X, 1x | x ∈ X, ∂0 : 1x 7→ x, ∂1 : 1x 7→ x).• Symmetric group X! of all bijections X → X.• The collection InvX of all bijections between subsets of X.
4
4. Functors
Functor: A graph homomorphism F : D → C, thus with restrictions
(4.1) ∀ x, y ∈ D0 , F1 : D(x, y)→ C(F0x, F0y) : f 7→ F1f ,
respecting identities, compositions: F11x = 1F0x, F1(g f) = F1g F1f .
Global conditions:
Isomorphism: F0 and F1 are isomorphisms.
Essentially surjective: ∀ c ∈ C0 , ∃ d ∈ D0 . c ∼= F0d
Local conditions:
Full: Each restriction (4.1) is surjective.
Faithful: Each restriction (4.1) is injective.
Example: While the forgetful or underlying set functor
U : Grp→ Set; [f : (G1, ·,−1 , 1)→ (G2, ·,−1 , 1)] 7→ [f : G1 → G2]
is faithful, U0 : Grp0 → Set0 is not injective (same set, different groups).Also U not full: Some functions between groups are not homomorphic.
Example: For a monoid (M, ·, 1M), write M∗
for the group of invertible elements or units. Then Mon→ Grp;
[f : (M1, ·, 1)→ (M2, ·, 1)] 7→ [f |M∗1 : (M∗1 , ·,−1 , 1)→ (M∗
2 , ·,−1 , 1)]
is a functor between large categories, the group of units functor.
Moral: Mathematical constructions are functors!
Example: A monoid homomorphism f : M1 → M2 yields a functorbetween the corresponding small one-object categories.
Note (R, ·, 1)→ ([0,∞[, ·, 1);n 7→ n2 is full, but not faithful.
Example: A functor F : (P1,≤) → (P2,≤) between poset catgeoriescorresponds to an order-preserving function:
x ≤ y in P1 ⇒ F0x ≤ F0y in P2 .
Trivially faithful.
Example: Inclusion of a subcategory always gives a faithful functor.
Full subcategory: The inclusion functor is full.
Example: Category FinSet of finite sets is full in Set.
Example: Functor
(N,≤)→ FinSet; [n < n+1] 7→ [0, 1, . . . , n−1 → 0, 1, . . . , n−1, n]is essentially surjective.
5
5. Natural transformations
Given graph maps F,G : D → C from a graph D to a category C, anatural transformation τ : F → G is a “vector” (τx | x ∈ D0) ofcomponents τx : Fx→ Gx in C1 such that,
for all f : x→ y in D1, the rectangle of the naturality diagram
x
f
Fxτx //
Ff
Gx
Gf
y Fy τy// Gy
. . . in D . . . in C
commutes in the category C.
Natural isomorphism: Each component τx is an isomorphism in C.
Example: For a set A, have a functorLA : Set → Set; [f : X → Y ] 7→ [A × X → A × Y ; (a, x) 7→ (a, fx)].Then a function α : A→ B gives a natural transformationLα : LA → LB with components LαX : A×X → B×X; (a, x) 7→ (α a, x)and naturality diagram
X
f
(a, x) LαX //_
LAf
(α a, x)_
LBf
Y (a, fx) LαY
// (α a, fx)
Example:Category L of (linear transformations between) real vector spaces.Dual space V ∗ = L(V,R) of linear functionals on vector space V .Double dual V ∗∗ = L(V ∗,R) = L(L(V,R),R).Identity functor I : L → L.Double dual functor DD : L → L;V 7→ V ∗∗.Natural transformation τ : I → DD with “evaluation” components
τV : V → V ∗∗; v 7→ [θ 7→ θ(v)].
Gives a natural isomorphism in finite dimensions.
Contrast: Given basis e1, . . . en of V , define ei : V → R; ej 7→ δij.
Then V → V ∗; ei 7→ ei does not set up a natural isomorphism.
6
6. Duality and contravariant functors
Dual or opposite Cop of a category C is built on the dual graph Cop:Same identity morphisms, but composition as shown:
y
g
xfoo
gf
||xxxxxxxxxxxxxxxxxxy
f // x
z In C z
g
OO
gf or gf
;;wwwwwwwwwwwwwwwwwwIn Cop
For Eulerian notation in C, algebraic notation would be natural in Cop.
Example: The dual of a one-object monoid category (M, ·, 1M) is theone-object monoid category of the monoid (M, , 1M) with xy = y ·x.
Example: For a set X, the dual of the poset category of(P(X),⊆
)is the poset category of
(P(X),⊇
).
Contravariant functor F : D → Cis a (“covariant” or usual) functor F : D → Cop or F : Dop → C.
Thus F (1x) = 1Fx as usual, but F (g f) = Ff Fg.
Generic examples: Locally small C, e.g., Set or lin. trans. cat. L.Fix a dualizing object T ∈ C0, e.g., 2 = 0, 1 ∈ Set0 or R ∈ L0.Functor ∗ : C → Setop;Z 7→ Z∗ := C(Z, T ) with (g f)∗ = f ∗ g∗:
Y
g
Xf
oo
θ g θ g f∗ // θ g f
T
θ_
g∗
OO
θ_
g∗
OO
:
(gf)∗
==zzzzzzzzzzzzzzzzzzzzz
Z
EE
From Set, set Z∗ is the power set 2Z of characteristic functions θ.
From L, vector space Z∗ is the dual space of linear functionals θ.
7
7. Diagram categories and functor categories
Diagram category CD for diagram D and category Chas graph maps F,G : D → C as objectsand natural transformations σ : F → G as morphisms. Composition:
x
f
Fx(τ•σ)x //
Ff
σx((QQQQQQQQQQQQQQQ Hx
Hf
Gx
Gf
τx
66mmmmmmmmmmmmmmm
Gy
τy((PPPPPPPPPPPPPPP
y Fy(τ•σ)y
//
σy
66nnnnnnnnnnnnnnnHy
. . . in D . . . in C
Constant objects and morphisms: x
f
cθ //
1c
c′
1c
y cθ
// c′
Functor category: Category D, functors F,G, . . .
Example: Linear representations of a group G are objects R : G→ Lof the functor category LG for the one-object group category G,so group homomorphisms R : G→ L(V, V )∗ = AutV = GL(V ).
The morphisms are intertwiners or equivariant maps τ : R1 → R2,
so ∀ g ∈ G , V1
R1(g)
τx // V2
R2(g)
V1 τx// V2
E.g., G = S3, V1 = R3 = Spane1, e2, e3, R1(π) : ei 7→ eπ(i).
V2 = R2 = Spane2−e1, e3−e2, R2(π) : (ei+1−ei) 7→ (eπ(i+1)−eπ(i))
τx︷ ︸︸ ︷1
3
[−2 1 1−1 −1 2
]·
R1(1 2 3)︷ ︸︸ ︷0 0 11 0 00 1 0
=
R2(1 2 3)︷ ︸︸ ︷[0 −11 −1
]·
τx︷ ︸︸ ︷1
3
[−2 1 1−1 −1 2
]
8
8. Products and coproducts
Product X × Y = (x, y) | x ∈ X , y ∈ Y of sets X, Y :
X X × YπXoo πY // Y
Zf
[[
g
CC
fug
OO
Universality property: ∀ Z ∈ Set0, “solid”↓ implies ↓“dashed”bijection Set(Z,X)× Set(Z, Y )→ Set(Z,X × Y ); (f, g) 7→ f u gwith f = πX (f u g) and g = πY (f u g). Thus f u g : z 7→ (fz, gz).
Picture in Set2 for discrete “two spot” diagram 2 = • • :
Zf //
fug ''PPPPPPPPPPPPP X
X × YπX
77nnnnnnnnnnnnn
X × YπY
((PPPPPPPPPPPPPP
Z g//
fug77nnnnnnnnnnnnnn
Y
Examples: Product in Set carries products in Grp, Ring, Mon, etc.
Example: Product in a poset category is a greatest lower bound.
a b a× b = c exists,
c
OO AAd
OO
but c× d does not.
Coproduct in C is the product in Cop: XιX //
f ))
X + Y
ftg Y
ιyoo
guuZ
Example: Coproduct in Set is the disjont union.
Example: Coproduct in a poset category is a least upper bound.
Biproduct UιU --
U ⊕ VπU
jjπV
44 VιUqq
in L is product and coproduct.
9
9. More limits and colimits
Pullback: Z
k
&&
h --
r//___ X ×B Y πY
//
πX
Y
g
X
f// B
or with poset diagram
• → • ← • category picture: Zh //
r((PPPPPPP X
f
X ×B YπX
66nnnnnnnnnnnnnn
Zr //______
fh=gk
66X ×B YfπX=gπY // B
X ×B YπY
((PPPPPPPPPPPPPP
Zk
//
r
66nnnnnnnY
g
OO
Ex: Domain of category composition is pullback of C1∂0−→ C0
∂1←− C1.
Pushout is the dual of a pullback.
Equalizer: Z
k
$$
r//___ E e
// Xf //g// Y so fk = gk ⇒ ∃! r . er = k
In L, E = Ker(f − g)e→ X. In Set, E = x ∈ X | fx = gx e
→ X.
Coequalizer: Xf //g// Y u
//
k
##C r
//___ Z ; kf = kg ⇒ ∃! r. ru = k
In Set, C is quotient of Y by equiv. rel’n. gen. by (fx, gx) | x ∈ X.In L, u projects from Y to C = Coker(f − g) := Y/Im(f − g).
Extended First Isomorphism Theorem in L is the exact sequence
0 // Kerf // Xf // Y // Cokerf // 0
where exact means Im g1 = Ker g2 for eachg1 // • g2 // .
Similar in Ab, RMod, ModR, ModK (commutative unital ring K),or any abelian category A where each A(X, Y ) is an abelian group.
10
10. General limits and colimits
Diagram D, category C, constant or diagonal for θ ∈ C(c, c′) is nat.
tr. ∆θ : ∆c→ ∆c′ with ∆: D → C; [f : x→ y] 7→ cθ
// c′
cθ // c′
.
Limit of graph map F : D → C is projection π : ∆ lim←−F → F suchthat ∀ κ : ∆Z → F , ∃! r = lim←−κ ∈ C(Z, lim←−F ) . π ∆ lim←−κ = κ.
x
f
Zκx //
r=lim←−κ ''NNNNNNNNNNNNN Fx
Ff
lim←−Fπx
77ooooooooooooo
lim←−Fπy
&&NNNNNNNNNNNNN
y in D in C Z κy//
r=lim←−κ88qqqqqqqqqqqqq
Fy
A.k.a “projective limit” or “inverse limit”, written as lim.
Colimit of graph map F : D → C is insertion ι : F → ∆ lim−→F suchthat ∀ κ : F → ∆Z , ∃! r = lim−→κ ∈ C(lim−→F,Z) . ∆ lim−→κ ι = κ.
A.k.a “inductive limit” or “direct limit”, written as colim.
Example: Functor (order-preserving) between poset categoriesx : (N,≤)→ (R ∪ ∞,≤) : n 7→ xn. Then lim−→x = limn→∞ xn.
Example: F : Nr 0, 1 → Ring;n 7→ Z/nZ. Thenr = ∆ lim←−κ : Z → lim←−F =
∏∞n=2 Z/nZ for κn : Z→ Z/nZ;x 7→ x+nZ.
Directed diagram D: ∀ x, y ∈ D0 , ∃ z ∈ D0 . x→ z ← y.
Then have directed limits and directed colimits.
Example: (Real) vector space V ,directed poset
(Pfin(V ),⊆
)of finite subsets.
Functor F : Pfin(V )→ L;X 7→ Span(X). Then lim−→F = V .
Theorem: Each algebra is the (directed) colimitof its finitely generated subalgebras.
11
11. Product categories and bifunctors
Product B × C of quivers B,C has (B × C)0 = B0 × C0,(B × C)1 = B1 × C1, pointwise ∂i(f, g) = (∂if, ∂ig) for i = 0, 1.
Product B×C of categories B,C : pointwise identities, composition:(B × C)
((x, x′), (y, y′)
)× (B × C)
((y, y′), (z, z′)
)→ (B×C)
((x, x′), (z, z′)
):((f, f ′), (g, g′)
)7→ (f ′f, g′g).
Universality: B B × CπBoo πC // C
DF
[[
G
CC
FuG
OO
— graph maps or functors.
Example: B′ B′ × C ′π′Boo
π′C // C ′
B
F
OO
B × CπBoo πC //
F×G
OO
C
G
OO
Bifunctor S to D on B and C is a functor S : B × C → D
— graph, diagram or quiver bimap if B,C are just quivers.
Proposition: Given bifunctor S : B × C → D: For (b, c) ∈ (B × C)0,define Rb := S(b, ) : C → D and Lc := S( , c) : B → D.
Then ∀ f : b→ b′, g : c→ c′: S(b, c)Rb(g)=S(b,g)
//
Lc(f)=S(f,c)
S(f,g)
((QQQQQQQS(b, c′)
Lc′ (f)=S(f,c′)
S(b′, c)Rb′ (g)=S(b′,g)
// S(b′, c′)
Conversely, given Rb : C → D and Lc : B → Dwith ∀b ∈ B0, c ∈ C0 , Lc(b) = Rb(c)and commuting solid square,the diagonal defines a bifunctor S : B × C → D.
Example: Locally small C, B = Cop, Rb : C → Set; b 7→ C(b, c),Lc : Cop → Set; b 7→ C(b, c) (like dualizing), Rb(c) = C(b, c) = Lc(b).
For h ∈ C(b′, c), so bf // b′
h // cg // c′ , have
h f Rb(g) // g h f C(b, c)Rb(g) // C(b, c′)
h_
Lc(f)
OO
Rb′ (g)
// g h_Lc(f)
OO
C(b′, c)
Lc(f)
OO
Rb′ (g)// C(b′, c′)
Lc′ (f)
OO
12
12. Cartesian monoidal categories
Cartesian monoidal category: category C with all finite products.
Idea: Think of (C,×,>) as like a monoid, say (N,+, 0) or (R, ·, 1).
Problem of non-associativity: e.g. in Set, (x, (y, z)) 6= ((x, y), z).
Fix: Bifunctor C × C → C; (X, Y ) 7→ X × Y ,trifunctors C ×C ×C → C; (X, Y, Z) 7→ X × (Y ×Z) or (X ×Y )×Z,
nat. isom. α with components αX,Y,Z : X × (Y × Z)→ (X × Y )× Z
which commute with projections; both sides give a product of X, Y, Z.
[Typical two-stage projection πY : X × (Y × Z)πY×Z−−−→ Y × Z πY−→ Y .]
Problem of non-unitality: e.g. in Set with> = ∗, have (∗, x) 6= x.
Fix: Functors C → C;X 7→ > ×X or X × > nat. isom. to identity,
so components λX : >×X → X and ρX : X ×> → X .
Potentially large “monoid” (C,×,>) “up to natural isomorphisms”.
Pentagon:
(W ×X)× (Y × Z)αW×X,Y,Z
ttiiiiiiiiiiiiiiii
((W ×X)× Y )× Z W × (X × (Y × Z))
αW,X,Y×ZjjUUUUUUUUUUUUUUUU
1W×αX,Y,Z
(W × (X × Y ))× Z
αW,X,Y ×1Z
OO
W × ((X × Y )× Z)αW,X×Y,Zoo
Triangle: X × (>× Y )αX,>,Y //
1X×λY ''OOOOOOOOOOO(X ×>)× Y
ρX×1Ywwooooooooooo
X × Y
Digon: >×>λ>
((
ρ>
66 >
Coherence [CWM §VII.2]: If these three diagrams commute, thenw.l.o.g. have a strict monoidal category: the nat. isoms. are identities.
• Coherence holds for Cartesian monoidal categories.
13
13. Groups in categories
Group (G,∇ : G×G→ G,S : G→ G, η : > → G) in Set, satisfying:
G×G×G 1G×∇//
∇×1G
G×G∇
G×G∇
// G
and G×G∇
%%KKKKKKKKKKK G×>1G×ηoo
ρG
>×GλG
//
η×1G
OO
G
(so a monoid)
and G×G 1G×S // G×G∇
##GGGGGGGGG
G
∆;;wwwwwwwww
∆ ##GGGGGGGGG// > η // G
G×GS×1G
// G×G∇
;;wwwwwwwww
with G×G πG //
πG
G
G G
1G
OO
1Goo
∆ffM M M M M M
(so a group).
Group in a Cartesian monoidal category: interprets the diagrams.
Example: A topological group is a group in the categoryTop of continuous maps between topological spaces.
Example: The additive group functorGa : K 7→ (K,+,−, 0) is a group in GrpCRing.
Example: The multiplicative group or group-of-units functorGm : K 7→ (K∗, ·,−1 , 1) is a group in GrpCRing.
Example: The p-th roots of unity functorµp : K 7→ (k ∈ K | kp = 1, ·,p−1 , 1) is a group in GrpCRing.
Example: (SL2,∇, S, η) as a group in GrpCRing:
SL2 : CRing→ Grp;K 7→[
a bc d
] ∣∣∣∣ a, b, c, d ∈ K , ad−bc = 1
.
∇K :
([a bc d
],
[a′ b′
c′ d′
])7→[aa′ + bc′ ab′ + bd′
ca′ + dc′ cb′ + dd′
],
SK :
[a bc d
]7→[d −b−c a
], and ηK : 1 7→
[1 00 1
].
14
14. Spaces, bases, adjunctions
Category L of linear transformations of vector spaces over a field K.
Forgetful or underlying set functor U : L → Set.
For set X, v. sp. with basis X is FX = r∑i=1
λi ·xi | λi ∈ K , xi ∈ X
,
the space of formal linear combinationsr∑i=1
λi · xi of elements of X.
At X, have unit ηX : X → UFX;x 7→ 1 · x which inserts the basis.
At V , have counit εV : FUV → V ;r∑i=1
λi · vi 7→ v, where
v = λ1v1 + . . .+ λrvr, the formal combination worked out in V .
For f : X → Y , lin. transf. Ff : FX → FY ;r∑i=1
λi · xi 7→r∑i=1
λi · f(xi).
So have functors F : Set→ L and U : L → Set.
Nat. isom. with components ϕX,V : L(FX, V ) ∼= Set(X,UV ) (*)
Mutually inverse ϕX,V : [FXθ−→ V ] 7→ [X
ηX−→ UFXUθ−→ UV ]
(informally, restricting θ to X); note unit ηX = ϕX,FX(1FX);
and dually, ϕ−1X,V : [X
f−→ UV ] 7→ [FXFf−→ FUV
εV−→ V ]
(informally, extending f to FX); note counit εV = ϕ−1UV,V (1UV ).
Adjunction (F,U, η, ε) with left adjoint F and right adjoint U .
Thus ϕX,V : θ 7→ Uθ ηX and ϕ−1X,V : f 7→ εV Ff .
Triangular identities: ∀X ∈ Set0 , 1FX = εFXFηX [= ϕ−1X,FX(ηX)]
and ∀ V ∈ L0 , 1UV = UεV ηUV [= ϕUV,V (εV )].
The triangular identities are necessary and sufficient for an adjumction.
Other notations: F a U or LU
55⊥ Set
F
vv
Mnemonic: In the box (*), put the functors at the extreme edges.The left adjoint (F ) is on the left; the right adjoint (U) is on the right.
15
15. Three adjunctions with monoids
• Free module functor F : Set→Mon is left adjointto the forgetful functor U : Mon→ Set.
“Tensor” notation x1 ⊗ . . .⊗ xn for n-tuple (x1, . . . , xn).
For set or alphabet X, coproduct FX :=∑n∈N
Xn, with X0 = 1 and
word concatenation associative product(x1⊗. . .⊗xm, y1⊗. . .⊗yn) 7→ x1⊗. . .⊗xm⊗y1 . . .⊗yn.
Note λX = 1X gives 1⊗ x = x and similarly x⊗ 1 = x by ρX = 1X .
Unit ηX : x 7→ x (“alphabet letter makes a one-letter word”) and
counit εM : FUM →M ;m1⊗m2 7→ m1 ·m2 (“multiplication table”).
• Group of Units functor U : Mon→ Grp; (M, ·, 1) 7→ (M∗, ·,−1 , 1)is right adjoint to Forgetful F : Grp→Mon; (G, ·,−1 , 1) 7→ (G, ·, 1).
Natural isomorphism ϕG,M : Mon(FG,M) ∼= Grp(G,UM); θ 7→ θ ,
since g · g−1 = 1 ⇒ θ(g) · θ(g)−1 = 1 and dually, so θ(g) ∈M∗.
Unit ηG : G→ G∗; g 7→ g (note G∗ = G) and
counit εM : M∗ →M ;u 7→ u (embedding group of units into monoid).
• M-sets for a monoid M — categorification of a monoid M —e.g., permutation representations for M a group.
Functor L : M → Set; ∗ 7→ X, m 7→ [Lm : X → X;x 7→ mx],can also be written as (X,M), a set X with “scalars” from M ,or as the monoid homomorphism L : M → Set(X,X);m 7→ Lm.
Category SetM of M -sets.
Forgetful functor U : SetM → Set;L 7→ L(∗) or (X,M) 7→ X.
Free M-set functor F : Set→ SetM ;X 7→ (M ×X,M)with m(n, x) = (mn, x).
Free algebra functor is left adjoint to the underlying set functor:
Unit ηX : X →M ×X;x 7→ (1, x)(embedding generators into the free algebra) and
counit ε(X,M) : (M ×X,M)→ (X,M); (m,x) 7→ mx(action in the M -set).
16
16. Poset adjunctions and Galois correspondences
Poset categories (A,≤), (B,≤), functors R : A→ B, S : B→ A.
Galois connection: adjunction A(Sb, a) ∼= B(b, Ra),
so Sb ≤ a ⇔ b ≤ Ra.
Unit: ∀ b ∈ B , b ≤ RSb. Counit: ∀ a ∈ A , SRa ≤ a.
Thus ∀ a ∈ A , Ra ≤ RSRa and ∀ b ∈ B , SRSb ≤ Sb (plug in).
Also ∀ b ∈ B , Sb ≤ SRSb and ∀ a ∈ A , RSRa ≤ Ra (use S,R).
Closed elements: In S(B) ⊆ A or R(A) ⊆ B.
Closure of a ∈ A is SRa = dom εa, and of b ∈ B is RSb = cod ηb.
Galois correspondence: Mut. inverse(S(B),≤
) R // (R(A),≤
)Soo .
Polarity is a relation α ⊆ I × J . Gives Galois connection
S : (2I ,⊆)→ (2J ,⊇);X 7→ y ∈ J | ∀x ∈ X , xα y
R : (2J ,⊇)→ (2I ,⊆);Y 7→ x ∈ I | ∀y ∈ Y , x α y
Note ∀ X ⊆ I , ∀ Y ⊆ J ,
SX ⊇ Y ⇔ ∀ x ∈ X , ∀ y ∈ Y , xα y ⇔ X ⊆ RY .
Galois theory: Group permutation representation or G-set (X,G).Fixed point relation (x, g) ∈ X ×G | gx = x.Right adjoint R : 2G → 2X is the fixed point functor.Left adjoint S : 2X → 2G is the (pointwise) stabilizer functor.
Polar geometry: Vector space V with quadratic form 〈u,v〉.Polarity (u,v) | 〈u,v〉 = 0 ⊆ V × V .Closure of a subset is its orthogonal complement.
Alg. geometry: On Cn×C[X1, . . . , Xn], polarity (x, f) | f(x) = 0.Closed subsets of Cn are algebraic sets or varieties.Closed subsets of C[X1, . . . , Xn] are radical ideals.Hilbert’s Nullstellensatz: The closure of an ideal IC[X1, . . . , Xn]
is its radical√I = f | ∃ 0 < n ∈ N . fn ∈ I.
Example: Radical of 〈X21 〉 in C[X1, . . . , Xn] is 〈X1〉.
17
17. Slice categories and comma categories
For b ∈ C0, slice category (C ↓ b) or (1C ↓ b) of C-objects over b
has object class ∂−11 (b), morphisms c
p<<<<<<<<f // c′
p′
b
(commuting),
composition c
p======== f//
f ′f''
c′
p′
f ′// c′′
p′′
b
, terminal object 1b : b→ b.
Dually, slice category (b ↓ C) or (b ↓ 1C) of C-objects under b.
Examples: Down-sets, and up-sets (or principal filters), in posets.
Example: For a group G and G-module A in AbG,the split extension p : A×G→ G; (a, g) 7→ g in (Grp ↓ G).Here (a, g)(a′, g′) = (a+ ga′, gg′).
For b ∈ C0 and T : E → C,comma category (T ↓ b) of objects T -over b
has morphisms Te
p@@@@@@@@Tf // Te′
p′~~
b
.
Dually, for b ∈ C0 and S : D → C,comma category (b ↓ S) of objects S-under b.
Proposition: For adjunction (F : X→ A, U : A→ X, η, ε),unit ηX : X → UFX is an initial object of (X ↓ U) and
counit εA : FUA→ A is a terminal object of (F ↓ A).
Proof. XηX
xxxxxxxxp
!!CCCCCCCC
UFXUϕ−1
X,Ap
//_______ UA
and FX
p!!CCCCCCCCFϕX,Ap //_______ FUA
εAvvvvvvvvvv
A
Cor: Given A
U
66⊥ X
F
vv, unit and counit uniquely determined.
18
18. The Yoneda Lemma
Yoneda Lemma: Let A be locally small.For object A1 of A, and f : A2 → A3 in A1, rememberA(A1, f) : A(A1, A2)→ A(A1, A3);h 7→ f h post-composes with f .
Then forK : A→ Set, have SetA(A(A1, ), K
) ∼= KA1; τ 7→ τA1(1A1)
Proof. • Injectivity:
A1
h
A(A1, A1)τA1 //
A(A1,h)
=L(h)
KA1
Kh
1A1_
// τA1(1A1)_
A2 A(A1, A2) τA2
// KA2 h // τA2(h) = Kh(τA1(1A1)
)
In A In Set
• Surjectivity, ρ : A(A1, A2)→ K, ρA2 : h 7→ Kh(x) for x ∈ KA1 nat:
A2
f
A(A1, A2)ρA2 //
A(A1,f)
=L(f)
KA2
Kf
h_
// Kh(x)_
A3 A(A1, A3) ρA3
// KA3 f h //
Kf(Kh(x)
)=
K(f h)(x)
In A In Set
Corollary: Full, faithful (covariant) Yoneda embedding
∃ : D → D = SetDop
; [f : x→ y] 7→[D( , f) : D( , x)→ D( , y)
]Category D of (set-valued) pre-sheaves over D.
Note: “∃” is Katakana for “Yo”.
Example: Poset category (P,≤).For element x, slice category D( , x) is (ess.) the down-set ↓ x of x.Then for f : x ≤ y,natural transformation D( , f) is the inclusion ↓ x →↓ y.
19
19. Reflective subcategories and counit properties
Reflective subcategory A of B meansthe inclusion K : A → B is full (not required in CWM), and has a
left adjoint L : B→ A, called the localization or reflector.
Example: K : Ab → GrpThen L : G 7→ G/[G,G], the largest abelian quotient of G.
Reflective adjunction: A(LB,A) ∼= B(B,A)
Unit: ηB : B → LB; counit: εA : LA→ A is an isomorphism.
So, when are counits of adjunctions isomorphisms? Need lemmata:
Lemma 1: τ : S → T is
epi
mono
in SetA iff each τA′′
epi
mono
in A.
Proof. Sτ //
τ
p-o
T
1T
T1T
// T
⇔ ∀ A′′ ∈ A0, SA′′τA′′ //
τA′′
p-o
TA′′
TA′′ TA′′
Lemma 2: For f : A′ → A, natural transformation R(f) or
A(A, f) : A(A, )→ A(A′, ) is
monoepi
iff f is
epi
split mono
.
[Note R(f) 7→ R(f)(1A) = f under the Yoneda Lemma.]
Proof. A(A,A′)R(f)−−−→ A(A′, A′) epi ⇒ ∃ r ∈ A(A,A′) . r f = 1A′ .
Conv., f oprop = 1A′ ⇒ ∀ A′′ , ∃A′′(f op) ∃A′′(rop) = ∃A′′(1A′)⇒ R(f)A′′ R(r)A′′ = 1A(A′,A′′) ⇒ R(f)A′′ surj., epi; so R(f) epi.
∀ A′′ , A(A,A′′)R(f)−−−→ A(A′, A′′) mono⇔ h1f = h2f ⇒ h1 = h2 .
Theorem: In AU
44⊥ XF
tt,
U is . . . iff εA . . .full has retractfaithful is epifull, faithful is iso
Proof. Natural transformation α : A(A, )→ A(FUA, ) with
component αA′ : A(A,A′)UA,A′−−−→ X(UA,UA′)
ϕ−1UA,A′−−−−→ A(FUA,A′).
Under Yoneda Lemma, α 7→ αA(1A) = εA. Then by Lemma 2:εA split mono ⇔ α epi ⇔ ∀A′ , αA′ surj. ⇔ ∀A′ , UA,A′ surj;εA epi ⇔ α mono ⇔ ∀A′ , αA′ mono ⇔ ∀A′ , UA,A′ inj;.
20
20. Category equivalence
Equivalence: Full, faithful, essentially surjective functor F : X→ A.
Recall essentially surjective: ∀ A ∈ A0 , ∃ X ∈ X . εA : FX ∼= A .
Preorder: Set (Q,≤) with reflexive transitive relation ≤ on set Q,or a small category with ∀ x, y ∈ Q , |Q(x, y)| ≤ 1.
Define α on Q by x α y ⇔ x ≤ y and y ≤ x , an equivalence relation.Set P of equivalence class representatives: ∀ q ∈ Q , ∃ p ∈ P . p ∼= q .
Inclusion functor F : (P,≤) → (Q,≤) is an equivalence.“Election” functor U : (Q,≤)→ (P,≤) chooses representatives.Then ∀ q ∈ Q , εq : FUq ∼= q, isomorphic counit of an adjunction.Note (P,≤) is a poset — antireflexive!
Adjoint equivalence: AU
44⊥ XF
ttwith unit, counit iso.
Equivalence: AU
55 XF
uuwith 1X → UF , FU → 1A iso.
Theorem: Functor F : X→ A. TFAE: (a) F is an equivalence;(b) F is part of an adjoint equivalence of categories;(c) F is part of an equivalence of categories.
(a)⇒(b): ∀ A ∈ A0 , ∃ UA ∈ X . εA : FUA ∼= A . Full, faithful F ⇒∀ f ∈ A(FX,A) , ∃! ϕX,Af ∈ X(X,UA) . FϕX,Af = ε−1
A f , . . .[Complete the adjunction, dual to the construction for linear algebra.]
(c)⇒(a): Need F full and faithful.
F faithful: X1
g
ηX1 // UFX1
UFg
X2 ηX2
// UFX2
and U faithful: FUA1
k
εA1 // A1
FUk
FUA2
εA2 // A2
F full: For h ∈ A(FX1, FX2), want h = Ff for f ∈ X(X1, X2).
Have X1
f
ηX1 // UFX1
Uh
X2 ηX2
// UFX2
for f = η−1X2 Uh ηX1 and X1
f
ηX1 // UFX1
UFf
X2 ηX2
// UFX2
so Uh = UFf . Then U faithful gives h = Ff.
Corollary: Essentially surjective K : A → B gives a reflection.
21
21. Typical equivalences
• Skeleton S of C: unique representative for each isomorphism class.Like poset (P,≤) induced in preorder (Q,≤),essentially surjective K : S → C has reflection L : C → S.
Ex:i ∈ N | i < n
∣∣n ∈ N
as object set of skeleton of FinSet.
• Morita equivalence: Ring R, ring Rnn of n× n-matrices over R.
ModRU
22⊥ ModRnn
Frr
with U : M →n︷ ︸︸ ︷
M ⊕ · · · ⊕M ,
F : [Rnn → End(N)] 7→ [R→ Rn
n → End(N)].
Concrete category:Category of sets with structure (algebraic, topological,. . . )and structure-preserving functions (homomorphisms, continuous,. . . ).
• Duality: Equivalence AU
33⊥ Xop
Ftt
of concrete categories.
Dualizing object: Set T with structure T ∈ A or T ∈ X,where: ∀ A ∈ A0 , A(A, T ) ≤ Set(A, T ) = TA ∈ X
and: ∀ X ∈ X0 , X(X,T ) ≤ Set(X,T ) = TX ∈ A.
Then U = A( , T ) and F = X( , T ).
Example: Category Lfin of fin.-dim. vector spaces over a field K.Then A = X = Lfin, T = K and ε−1
V : V → V ∗∗; v 7→ [f 7→ f(v)].
Example: Fourier transforms, Pontryagin duality.
Then A = Ab, X = CAb (compact abelian groups),and T = (R/Z,+, 0) “1-dimensional torus” or (S1, ·, 1) “circle group”.
A := UA = Ab(A, T ), the group of characters χ : A→ T .ε−1A : A→ FUA; a 7→ [χ 7→ χ(a)].
Example: Category A of finite Boolean algebras, X = FinSet,dualizing object T = 2 := 0, 1, so power set FX (char. fns.).
ηX : X → UFX;x 7→ [χ 7→ χ(x)].
Note: Can extend from a category Af.g. of finitely generated algebrasto a category A of all algebras: treat as colimits of f.g. algebras,which will dualize to limits of Xf.g.-objects.
22
22. Preservation, reflection and creation
Diagram D : J → A, functor G : A→ B. J
D
AG// B
G preserves J-limits if it “pushes limits forward”:
Diagram D : J → A has a limit[
lim←−Dπj−→ Dj
]implies GD : J → B has a limit
[G(lim←−D)
Gπj−−→ GDj
].
G reflects J-limits if it “pulls limits back”:
Diagram GD : J → B has a limit of the image form[GL
Gπj−−→ GDj
]implies D : J → A already had a limit
[lim←−D = L
πj−→ Dj
].
G creates J-limits if it both preserves and reflects,and if lim←−GD exists, then it exists in the image form.
Corresponding definitions for colimits.
Example: U : Grp→ Set preserves, reflects limits, directed colimits.[Consider “pointwise” structure on the underlying sets.]Doesn’t preserve or reflect general colimits.
Example: U : Top→ Set preserves, but doesn’t reflect, limits:
EπY //
πX
p-b
Y
g
Xf// B
in Top
means E = (x, y) ∈ X × Y | f(x) = g(y) has the subspace topology.
Example:Full and faithful K : Ab → Grp preserves limits, but not colimits.
Theorem: Full and faithful G : A→ B reflects limits and colimits.
Example: In Ab, coproduct C2 + C3 or C2 ⊕ C3 is C6.In Grp, coproduct C2 +C3 or C2 ∗C3 is the modular group PSL2(Z).Doesn’t violate K : Ab → Grp reflecting colimits: PSL2(Z) 6= KC6.
23
23. Preservation and adjunction
Diagram D : J → A, adjoint functors F,U : J
D
A
U
66⊥ X
F
vv
Suppose limit[
lim←−Dπj−→ Dj
]exists: A
κj //
r=lim←−κ &&NNNNNNNNNNNNN Dj
lim←−Dπj
77ppppppppppppp
Thus AJ(∆A,D) ∼= A(A, lim←−D).κj = πj r 7→ r
Theorem: Right adjoints preserve limits.
Proof. UD h a s l i m i t U lim←−D:
XJ(∆X,UD) ∼= AJ(∆(FX), D
) ∼= A(FX, lim←−D) ∼= X(X,U lim←−D) .
Corollary: Left adjoints preserve colimits.
Example: In L(FX, V ) ∼= Set(X,UV ),have U(V1 ⊕ V2) = V1 × V2 and F (X1 +X2) = FX1 ⊕ FX2.
Example: Multiplicity of the Euler ϕ-function or totient functionϕ(n) = |r | 1 ≤ r ≤ n and gcd(r, n) = 1| =
∣∣(Z/n,×, 1)∗∣∣.
Recall group of Units functor U : Mon→ Grp; (M, ·, 1) 7→ (M∗, ·,−1 , 1)is right adjoint to Forgetful F : Grp→Mon; (G, ·,−1 , 1) 7→ (G, ·, 1).
For coprime m,n, have (Z/mn,×, 1) ∼= (Z/m,×, 1)× (Z/n,×, 1).
Then ϕ(mn) =∣∣(Z/mn,×, 1)∗
∣∣ =∣∣∣[(Z/m,×, 1)× (Z/n,×, 1)
]∗∣∣∣=∣∣(Z/m,×, 1)∗×(Z/n,×, 1)∗
∣∣ =∣∣(Z/m,×, 1)∗
∣∣×∣∣(Z/n,×, 1)∗∣∣ = ϕ(m)ϕ(n).
Example: Equivalence AU
55 XF
uu
implies A
U
66⊥ X
F
vvand A
U
66> X
F
vv,
so F and U preserve limits and colimits.
24
24. Heyting algebras and topologies
Preorder (P,≤) with all finite products, sufficiently including:
The empty product (terminal object) > with ∀ x ∈ P , x ≤ >;
The (comm., assoc.) meet or g.l.b with ∀ x, y ∈ P , x← x · y → y.
For each fixed a in P , functor S(a) : (P,≤)→ (P,≤);x 7→ (x · a).
Suppose each S(a) has a right adjoint R(a) : z 7→ (a( z):
∀ x, y, z ∈ P , x · y ≤ z ⇔ x ≤ y( z (∗)
Example: Propositions, “and” is product; “deduce q from p” is p→ q.Then p( q would be proposition “p implies q”.
Bounded lattice: poset, finite products, coproducts, 0 = ⊥, 1 = >.
Complete lattice: poset with all products and coproducts.
Heyting algebra is a bounded lattice with the adjunctions (∗).
Prop: Heyting algebras are distributive: S(a) preserves coproducts.
Prop: Complete Heyting algebras are completely distributive.By (∗), have y( z =
∑x | x · y ≤ z.
Example: Boolean algebra with implication p( q = p→ q = (¬p)∨qNegation (pseudocomplement) ¬x := x( 0 in any Heyting algebra.
Example: 0 ≤ 12≤ 1, where 1
2( 0 = maxx | x · 1
2≤ 0 = 0.
Then ¬¬12
= ¬0 = 1 6= 12; “Law of the excluded middle” does not hold.
Regular elements x = ¬¬x in Heyting algebra form Boolean algebra.
Topology: In any topological space (X,O), the subset O of 2X
comprising the open sets forms a complete Heyting algebra.Unions in 2X , but infinite intersections differ, take interior.Here P ( Q = [(X r P ) ∪Q]
• Indiscrete topology O = Ø, X• Discrete topology O = 2X
• Alexandrov topology of poset (P,≤) is the set of all downsets.
• Cofinite topology of set X has O = Ø∪S ⊆ X | XrS finite• For monoid M and an M -set X, take O as the set of M -subsets.
If M is a group, get a Boolean algebra.
25
25. Currying
Heyting algebra: ∀ x, y, z ∈ P0 , P (x · y, z) ∼= P (x, y( z)In particular, ∀ y, z ∈ P0 , P (y, z) ∼= P (1, y( z).
Currying: ∀X, Y, Z ∈ Set0 , Set(X×Y, Z) ∼= Set(X,Set(Y, Z)
)In particular, ∀ Y, Z ∈ Set0 , Set(Y, Z) ∼= Set
(>,Set(Y, Z)
).
Tensor product: ∀ X, Y, Z ∈ L0 , L(X ⊗ Y, Z) ∼= L(X,L(Y, Z)
)In particular, ∀ Y, Z ∈ L0 , L(Y, Z) ∼= L
(K,L(Y, Z)
),
“linear spaces” as modules over commutative ring K., e.g., Z for Ab.
Note L(X,L(Y, Z)
)⊆ Set
(X,Set(Y, Z)
) ∼= Set(X × Y, Z), so
L(X,L(Y, Z)
)tracks the bilinear maps X × Y → Z.
In all three cases, ∼= is a natural isomorphism of sets, so on theleft hand side of the lower ∼= is a hom-set of the locally small category.
Strict symmetric monoidal category (C,⊗, I): X ⊗ Y = Y ⊗X,X ⊗ (Y ⊗ Z) = (X ⊗ Y )⊗ Z, and I ⊗X = X = X ⊗ I.
E.g: Heyting algebra (P, ·, 1), Cartesian (Set,×,>), linear (L,⊗, K).
Closed monoidal category: Adjunction C(X⊗Y, Z) ∼= C(X, [Y, Z])with internal hom-object [Y, Z], set isom. C(Y, Z) ∼= C(I, [Y, Z]).
Bifunctors: monoidal product ⊗ : C×C→ Cand internal hom [ , ] : Cop ×C→ C.
Heyting algebras: monoid product → adjunction → internal hom.Linear spaces: internal hom → adjunction → monoid product.
Note ⊗ Y a left adjoint ⇒ preserves coproducts ⇒ distributivity:
(X+X ′)⊗Y = (X⊗Y )+(X ′⊗Y ) or(∑
Xi
)⊗Y =
∑(Xi⊗Y )
Also [Y, ] a right adjoint ⇒ preserves products ⇒ “exponentiation”:
[Y, Z1 × Z2] = [Y, Z1]× [Y, Z2] or [Y,∏Zi] =
∏[Y, Zi]
Compare C(Y,∏Zi) ∼=
∏C(Y, Zi)
Arithmetic: (l + l′) ·m = l ·m+ l′ ·m and (n1 · n2)m = nm1 · nm2in the skeleton (N, ·, 1) of (FinSet,×,>).
26
26. Enriched categories
Bicomplete category: All limits and colimits.
Base category: bicomplete symmetric monoidal category (B,⊗, I),
e.g., (Set,×,>), (L,⊗, K), poset((
[0,∞],≥),+, 0
)with x+∞ =∞.
B-enriched category: quiver C with ∀ x, y ∈ C0 , C(x, y) ∈ B0 and:
• composition: ∀ x, y, z ∈ C0 , ∈ B(C(x, y)⊗ C(y, z), C(x, z)
)• identities: ∀ x ∈ C0 , jx ∈ B(I, C(x, x)) with commuting:
C(w, x)⊗ C(x, y)⊗ C(y, z)⊗1 //
1⊗
C(w, y)⊗ C(y, z)
C(w, x)⊗ C(x, z) // C(w, z) and
C(x, y)
SSSSSSSSSSSSSSSSS
SSSSSSSSSSSSSSSSS
jx⊗1 //
1⊗jy
C(x, x)⊗ C(x, y)
[recall B = I ⊗B, etc.]
C(x, y)⊗ C(y, y) // C(x, y) . . . for w, x, y, z ∈ C0 .
Locally small category is enriched over (Set,×,>).
Pre-additive category is enriched over (Ab,⊗,Z).
Linear category L is enriched over (L,⊗, K).
Closed monoidal category is enriched over itself.
Preorder is enriched over the Boolean algebra 2 =(⊥ < >,∧,>
).
Directed metric spaces are enriched over ([0,∞],+, 0).Thus d(x, y) ∈ [0,∞] for x, y ∈ C0,
and composition means d(x.y) + d(y, z) ≥ d(x, z).
If the symmetric monoidal (B,⊗, I) is closed, can “impoverish” theenriched category C to Co with Co(x, y) = B(I, C(x, y)) for x, y ∈ C0.
27
27. Copowers and free enriched categories
For a set S and an object b of a cocomplete category B, the colimit ofthe constant diagram S → b is the copower or multiple
S · b =∑
s∈S b, with insertions ιs : b→ S · b for s ∈ S.
Example: For X ∈ Set, have ιs : X → S ×X = S ·X;x 7→ (s, x).
Example: For V ∈ L, have S · V =
|S| copies︷ ︸︸ ︷V ⊕ . . . ⊕ V for S finite.
For arbitrary S, have power V S = Set(S, UV ) ∼= L(FS, V ) ∈ L0,and copower S · V =
f : S → V
∣∣ ∞ > |s ∈ S | f(s) 6= 0|
,
a subobject of V S, proper if S is infinite.
Category C, bicomplete closed symmetric monoidal base category (B,⊗, I).
Free B-enriched category BC on C: left adjoint to impoverishment.
Object class BC0 = C0.
For x, y ∈ BC0 := C0, define BC(x, y) := C(x, y) · I =∑
f∈C(x,y)
I.
For x ∈ C0, define jx = ι1x : I → C(x, x) · I.
For x, y, z ∈ C0, distributivity and unitality give BC(x, y)⊗BC(y, z) =∑f∈C(x,y) I⊗
∑g∈C(y,z) I =
∑f∈C(x,y)
∑g∈C(y,z) I⊗I =
∑(f,g)∈C(x,y)×C(y,z) I.
Then have composition∑
C(x,y)×C(y,z) I //___∑
C(x,z) I
I
ι(f,g)
OO
ιgf
66nnnnnnnnnnnnnn
.
Example: For a category C, and Boolean algebra 2,the free 2-enriched 2C is the preorderobtained by “forgetting arrow labels” of C.
Group rings: For linear (L,⊗, K),and a one-object group G = G1 on G0 = ∗,the group ring over K is the one-object free L-category LG,
with morphism set G ·K.
Standard Hopf algebra notation: ηG = j∗ = ι1∗ : K → G ·K.
28
28. Pointed sets, kernels, and cokernels
Pointed set Xe has chosen element e, so e : > → X with image e.
Category of pointed sets is the slice category (> ↓ Set).
Internal hom [Xe, Yd] = [X, Y ]d with constant d : X → Y ;x 7→ d.
Currying: (> ↓ Set)(Xe ∧ Yd, Zc) ∼= (> ↓ Set)(Xe, [Yd, Zc]
)with the
smash product Xe∧Yd =((
(Xre)× (Y rd))∪(e, d)
)(e,d)
.
Suppose category C has a zero object 0, e.g., (> → >) in (> ↓ Set).
Zero morphism in C(x, y) is the composite (x0−→ y) = (x→ 0→ y).
Kernel: Ker fker f−−→ x is the equalizer of x
f**
0
44 y .
Cokernel: ycoker f−−−−→ Coker f is the coequalizer of x
f**
0
44 y .
Lemma: Ker fker f−−→ x is mono; and dually y
coker f−−−−→ Coker f is epi.
Proof. ∀ z r,r′−−→ Ker f , (ker f) r = (ker f) r′ =: κx
⇒ Ker fker f // x
f**
0
44 y
z
r,r′
OO κx
<<zzzzzzzzzz⇒ r = r′ .
Example: For f ∈ (> ↓ Set)(Xd, Ye), have Ker f =(f−1e
)d→ Xd.
Object c of C with zero, (co)kernels, preorders(∂−1
1 c,∣∣ ) and
(∂−1
0 c,∣∣op )
.
Ker gker g // c adjunction c
coker f //
g
Coker f
zzttttttttttttt
(ker g)∣∣ f ⇔ d
f
OOddHHHHHHHHHHHH⇔ g f = 0⇔ b ⇔ (coker f)
∣∣ gSo: ker g = ker coker ker g and coker f = coker ker coker f
29
29. Factorization of morphisms, abelian categories
First Isom. Thm. for sets: Xf //
e !!DDDDDDDDD Y so f = m e ,
f(X)
m
OO
m mono, e epi
Abelian category [Freyd]: (A0) A has a zero object;(A1) For A,B ∈ A0, product A×B and coproduct A+B exist;(A2) For A,B ∈ A0 and f ∈ A(A,B),
have kernel Ker fker f−−→ A and cokernel B
coker f−−−−→ Coker f
(A3) Every monomorphism is a kernel; every epimorphism is a cokernel.
Image: [Im fim f−−→ B] := [Ker(coker f)
ker coker f−−−−−→ B] ,a monomorphism, smallest subobject of B that divides f .
Coimage: [Acoim f−−−→ Coim f ] := [A
coker ker f−−−−−→ Coker(ker f)] ,an epimorphism, smallest quotient of A that divides f .
Factorization [Af−→ B] = [A
q−→ Im fim f−−→ B] with q an epimorphism.
Indeed, coker q 6= 0 would mean f divided by a smaller subobject of B.
Theorem: f : A→ B mono and epi ⇒ f is an isomorphism.
Proof. Have Bcoker f−−−−→ 0 since f epi, and 1B = ker coker f .
Since f is mono, Af−→ B is also a kernel of coker f [and so A ∼= B].
Thus f has a section: f s = 1B. Dually, it has a retraction: rf = 1A.Since r = r 1B = r f s = 1A s = s, have f invertible.
Corollary: Im f ∼= Coim f , and f = im f coim f
See https://math.stackexchange.com/questions/3268091/
coimage-and-image-in-abelian-categories
30
30. Enriching abelian categories
Abelian category A (Freyd’s definition).
Matrices: X X × Yoo // Y X //
f
--
X + Y[f
g
]
Yoo
gqqA
[f g]
OO
f
VV
g
HH
A
Exact: 0→ XιX−→ X + Y
[0
1
]−−→ Y → 0, 0→ X
[1 0]−−→ X × Y πY−→ Y → 0
Theorem: 0→ X + Y
[1 00 1
]−−−−→ X × Y → 0 exact,
so X+Y ∼= X×Y =: X⊕Y , biproduct.
Diagonal: ∆: X[1 1]−−→ X ⊕X. Summation: Σ: X ⊕X
[1
1
]−−→ X.
For f, g ∈ A(A,B), define Af+Lg //
∆
B
A⊕ A[f
g
];;wwwwwwwww
and Af+Rg //
[f g] ##GGGGGGGGG B
B ⊕BΣ
OO .
Proposition: 0 +L f = f = f +L 0, 0 +R f = f = f +R 0.
Proposition: (f +L g) +R (h+L k) = (f +R h) +L (g +R k).
Proof. Both sides are A∆−→ A⊕ A
[f h
g k
]−−−−→ B ⊕B Σ−→ B.
Theorem: +L = +R, commutative and associative.
Proof. Setting g = h = 0, have f +R h = f +L h =: f + h.Setting h = 0, have (f + g) + k = f + (g + k).Setting f = k = 0, have g + h = h+ g.
Theorem: A(A,B) is an abelian group.
Proof. For f : A→ B, have A⊕A
[1 f
0 1
]−−−−→ B⊕B monic and epic, so an
isomorphism with inverse B ⊕B
[1 g
0 1
]−−−−→ A⊕ A, then f + g = 0.
31
31. Categories and 2-categories
Category Cat of (small) categories;with Cat(D,C) = CD =: [D,C] (functor category).
Cartesian closed monoidal (Cat,×,1), terminal category 1 = •yy
as the monoidal unit, and Currying Cat(A×B,C) ∼= Cat(A, [B,C]).
(Strict) 2-category: (1-)category C enriched over Cat.0-cells: objects A,B, . . . of C.1-cells: morphisms of C, i.e., objects in the categories C(A,B)
or elements of the sets Cat(1,C(A,B)
).
2-cells: morphisms in the categories C(A,B).
Example: 2-category Cat. Categories as 0-cells. Functors as 1-cells.
Natural transformations τ : F → G as 2-cells AF %%
G99⇓ τ B
Horizontal 2-cell composition: Cat(A,B)×Cat(B,C)−→ Cat(A,C);
AF %%
G99⇓ τ B
F ′ %%
G′99⇓ τ ′ C 7→ A
F ′F ''
G′G77⇓ τ ′ τ C
with F ′ FaF ′τa
τ ′Fa //
(τ ′τ)a
&&MMMMMMMMMM G′ FaG′τa
F ′ Gaτ ′Ga
// G′ Ga
Identity: Cat(1,Cat(C,C)) 3 jC : •yy7→ C
1C %%
1C
99⇓ id C
Vertical 2-cell composition on Cat(A,B): Fa
σa !!DDDDDDDD(τ•σ)a // Ha
Ga
τa
<<zzzzzzzz
Entropic or interchange law: (τ ′ • σ′) (τ • σ) = (τ ′ τ) • (σ′ σ),as bifunctorial horizontal composition respects vertical composition.
(n+ 1)-category: an n-category enriched over Cat.
32
32. The braid category
Monoid (M,∇ : M ×M →M, η : > →M) in C with products.
Monoidal category (C,⊗, I) is a monoid in (Cat,×,1).
Sequence (Gn | n ∈ N) of groupswith trivial G0, and homomorphisms ρm,n : Gm ×Gn → Gm+n
satisfying Gl ×Gm ×Gn
ρl,m×1//
1×ρm,n
Gl+m ×Gn
ρl+m,n
Gl ×Gm+n ρl,m+n
// Gl+m+n
. Category G with G0 = N
and G(m,n) = Gm for m = n and Ø for m 6= n.
Monoidal category (G,+, 0) is (N,+, 0) at the object level,with + =
⋃(ρm,n : Gm×Gn → Gm+n
)on morphisms.
Symmetric groups Sn =〈τ1, . . . , τn−1 | τ 2
i = 1, τiτj = τjτi for |i− j| > 1, τiτi+1τi = τi+1τiτi+1〉.
General linear groups GL(K) with GLn(K).
Braid groupsBn = 〈σ1, . . . , σn−1 | σiσj = σjσi for |i− j| > 1, σiσi+1σi = σi+1σiσi+1〉
give the braid category B.
Braid relation:
i i+ 1 i+ 2t t tZZZZ
σi t t t
ZZZZ
σi+1 t t t
ZZZZ
σi t t t
i i+ 1 i+ 2
=
i i+ 1 i+ 2t t tZZZZ
σi+1t t t
ZZZZ
σit t t
ZZZZ
σi+1t t t
i i+ 1 i+ 2
First string on top layer, second in middle, third on bottom layer.
• “Third Reidemeister move” in knot theory terms.
• “Yang-Baxter equation” in physics.
33
33. Endofunctors, (co)algebras, monads
Endofunctor category XX = [X,X] of category X.
Algebra (X,α) for an endofunctor T : X→ X is givenby a structure map α : TX → X in X(TX,X) for some X ∈ X0.
Algebra (homo)morphism θ : (X,α)→ (Y, β)
is given by commuting diagram TX
α
Tθ // TY
β
Xθ// Y
in X.
Example: Finite subset endofunctor Pfin : Set→ Set,bounded semilattice (commutative, idempotent monoid) (X, ·, 1),structure map α : PfinX → X; x1, . . . , xr 7→ x1 · . . . · xr · 1.
Coalgebra (X,α) for an endofunctor T : X→ X is givenby a structure map α : X → TX in X(X,TX) for some X ∈ X0.
Coalgebra (homo)morphism θ : (X,α)→ (Y, β)
is given by commuting diagram TXTθ // TY
X
α
OO
θ// Y
β
OO in X.
Example: Coalgebra with structure map α : X → PfinXrepresents a non-deterministic dynamical system.
Endofunctors form a monoidal category(XX, , 1X
).
Monad on X is a monoid (T, µ : T 2 → T, η : 1X → T ) in(XX, , 1X
):
T 3 Tµ //
µT
T 2
µ
T 2
µ
>>>>>>>> TTηoo Here, µT is:
T 2µ// T T
ηT
OO
T XT // X
T 2%%
T99⇓ µ X
or
XT &&
T
88⇓ idT XT 2
%%
T99⇓ µ X = µ idT , whiskering.
Will get various kinds of algebras from monads.
34
34. Adjunctions yield monads
Adjunction (F : X→ A, U : A→ X, η : 1X → UF, ε : FU → 1A).
Triangular identities 1F = εF • Fη and 1U = Uε • ηU .
Trace in X: UF -coalgs. ηX : X → UFX, µ := UεF : UFUF → UF .
Proposition: Unital law UFUFUεF
$$JJJJJJJJJ UFUFηoo
UF
ηUF
OO
UF
Proof. Triangular FUF
εF ""FFFFFFFFF FFηoo
F
and UFUUε
""FFFFFFFFF
U
ηU
OO
U
Proposition: µ : UFUF → UF associative.
Proof. Need UFUFUFUFUεF//
UεFUF
UFUF
UεF
UFUFUεF
// UF
or FUFUFUε //
εFU
FU
ε
FU ε
// 1A
.
Naturality: FUA
εA
FUFUAεFUA //
FUεA
FUA
εA
A FUA εA// A
. . . in A . . . in A
Theorem: Adjunction (F,U, η, ε) gives monad (UF,UεF, η) on X.
Example: Free monoid adjunction, for set or alphabet X.
Then UFX or X∗ is the set of words or lists 〈x1 . . . xr〉 in the alphabet.
Coalgebra ηX : X → UFX; letter x 7→ one-letter “word” or list 〈x〉.Multiplication µX = UεFX : UFUFX → UFX;
list of words or lists 7→ concatenation of list:〈〈x11 . . . x1r1〉 . . . 〈xs1 . . . xsrs〉〉 7→ 〈x11 . . . x1r1 . . . xs1 . . . xsrs〉
— removes inner brackets.
35
35. Eilenberg-Moore algebras
Does a monad (T, µ, η) on X give adjunction A
U
66⊥ X
F
vv?
• Monoid (M,m : M2 →M, e : > →M) is a monoid in (Set,×,>).
M-sets X have action a : M ×X → X with
associativity: M2 ×X 1M×a //
m×1X
M ×Xa
M ×X a
// X
, unitality: Xe×1X //
HHHHHHHHHH
HHHHHHHHHH M ×XaX
morphisms: M ×X1
a1
1M×f // M ×X2
a2
X1f
// X2
, category SetM of M -sets.
Free FX = (M2 ×X m×1X−−−→M ×X), adjoint U(M ×X a−→ X) = X.
Unit ηX : X →M ×X;x 7→ (e, x), counit εa = a.
Gives a model endofunctor T : X 7→M ×X on Set.
• Monad (T, µ : T 2 → T, η : 1X → T ) is a monoid in (XX, , 1X).
Eilenberg-Moore algebra a : TX → X in X(TX,X) for X ∈ X0:
with associativity: T 2XTa //
µX
TX
a
TX a// X
and unitality: XηX //
DDDDDDDD
DDDDDDDD TX
aX
,
morphisms: TX1
a1
Tf // TX2
a2
X1f// X2
, cat. XT of Eilenberg-Moore algebras.
Forgetful UT : XT → X; (TXa−→ X) 7→ X.
Free F T : X→ XT ;X 7→ (T 2XµX−−→ TX). Note UTF TX = TX
Unit ηTX : XηX−→ TX, counit εTa : (T 2X
Ta−→ TX)a−→ (TX
a−→ X)
Eilenberg-Moore adjunction(F T , UT , ηT , εT
), yields monad (T, µ, η).
36
36. The Kleisli category of a monad
Alternative adjunction A
U
66⊥ X
F
vvfrom monad (T, µ, η) on X.
Kleisli category XT of monad (T, µ, η) on X:
(XT )0 = X0, XT (X, Y ) = X(X,TY ), identities 1X = ηX : X → TX,
composition (Yg−→ TZ) (X
f−→ TY ) = (Xf−→ TY
Tg−→ T 2ZµZ−→ TZ) .
Adjunction XT (FTX, Y ) = XT (X, Y ) = X(X,TY ) = X(X,UTY )
with left adjoint FT (Xf−→ Y ) = (X
f−→ YηY−→ TY ),
right adjoint UT (Yg−→ TZ) = (TY
Tg−→ T 2ZµZ−→ TZ), unit X
ηX−→ TX,
counit εY = TY1TY−−→ TY , and monad (UTFT , UT εFT , η) = (T, µ, η).
Power set monad (P , µ, η) with ηX : x 7→ x and set family unionµX : x, . . . , y, . . . , . . . 7→ x, . . . , y, . . . , . . . — like with lists.
Category Rel of relations XR−→ Y = (x, y) | x R y on sets:
Have Rel0 = Set0,
with 1X as the identity function or equality relation on a set X.
Relation product (XR−→ Y
S−→ Z) := (x, z) | ∃ t ∈ Y . x R t S z.
Theorem: Category Rel is the Kleisli category for (P , µ, η).
Proof. XR−→ Y gives SetP-morphism X
R−→ PX;x 7→ y | x R y.Kleisli identity is ηX : X → PX;x 7→ x = x′ | x = x′,and Kleisli composition gives the relation product:
XR
// PYPS
// P2Z µZ// PZ
x // t | x R t // z | t S z | x R t // z | ∃ t . x R t S z
E // z | e S z | e ∈ E
37
37. Compact closed categories
Compact closed category: Symmetric, monoidal (C,⊗,1) with:
• Contravariant duality functor ∗ : C→ C;• Evaluation natural transformation evX : X∗ ⊗X → 1; and• Coevaluation natural transformation coevX : 1→ X ⊗X∗,
yanking conditions(X
coevX⊗1X−−−−−−→ X ⊗X∗ ⊗X 1X⊗evX−−−−−→ X)
= 1X
and (X∗
1X∗⊗coevX−−−−−−−→ X∗ ⊗X ⊗X∗ evX⊗1X∗−−−−−→ X∗)
= 1X∗
Lemma: Internal hom [X, Y ] = X∗ ⊗ Y .
Example: (Lfin,⊗, K) with duality X∗ = L(X,K).If X has basis e1, . . . , en,and X∗ has dual basis e1, . . . , en with ei(ej) = δij,then coev : K → X ⊗X∗; 1 7→
∑nj=1 ej ⊗ ej.
Yanking: ei 7→∑n
j=1 ej ⊗ ej ⊗ ei 7→∑n
j=1 ej ⊗ ej(ei) =∑n
j=1 ejδji = ei
and ei 7→ ei ⊗∑n
j=1 ej ⊗ ej 7→∑n
j=1 ei(ej)⊗ ej =∑n
j=1 δij ej = ei.
Lemma: For Y with basis d1, . . . , dm,morphism ei 7→ dj corresponds to tensor ei⊗dj.
Example: Relation category Rel, biproduct is the disjoint union with
X 3 x&
ιX--x ∈ X + Y 3 y$
πX
kk
πY
33 y ∈ Y
ιYqq
Monoidal category(Rel,×, 0
), compact closed with X∗ = X.
Evaluation ((x, x), 0) | x ∈ X, coevaluation (0, (x, x)) | x ∈ X.
First yanking condition:relation product of (x, (x′, x′, x)) | x, x′ ∈ X
with ((x′, x, x), x′) | x, x′ ∈ X is (x, x) | x ∈ X.
Second is similar.
38
38. Monoidal functors
Monoidal categories (C,⊗,1), (C′,⊗,1′).
Monoidal functor F : (C,⊗,1)→ (C′,⊗,1)with natural transformations µX,Y : F (X)⊗ F (Y )→ F (X ⊗ Y )and C′-morphism ε : 1′ → F (1) such that:
F (X)⊗(F (Y )⊗ F (Z)
)1F (X)⊗µY,Z
α′F (X),F (Y ),F (Z)//
(F (X)⊗ F (Y )
)⊗ F (Z)
µX,Y ⊗1F (Z)
F (X)⊗ F (Y ⊗ Z)
µX,Y⊗Z
F (X ⊗ Y )⊗ F (Z)
µX⊗Y,Z
F(X ⊗ (Y ⊗ Z)
)F (αX,Y,Z)
// F((X ⊗ Y )⊗ Z
),
— associativity, and unitality:
F (X)⊗ 1′ρ′F (X) //
1F (X)⊗ε
F (X)
F (X)⊗ F (1) µX,1// F (X ⊗ 1)
FρX
OO, F (X) 1′ ⊗ F (X)
λ′F (X)oo
ε⊗1F (X)
F (1⊗X)
FλX
OO
F (1)⊗ F (X).µ1,Xoo
Example: Underlying set functor U : (L,⊗, K)→ (Set,×, 1).Here ε : 1 → K; 1 7→ 1, and µX,Y : UX × UY → U(X ⊗ Y ) is the
usual quotient by relations (k1x1 + k2x2, y)!
= k1(x1, y) + k2(x2, y), etc.StrongStrict
monoidal functor: ε and the µX,Y are
isomorphisms.
identities.
Example: Free vector space functor F : (Set,×, 1)→ (L,⊗, K).
Here F (X × Y ) = F (X)⊗ F (Y ) and F1 = K, so strong, strict.
Braided monoidal functor: F (X)⊗ F (Y )
µX,Y
σ′F (X),F (Y )// F (Y )⊗ F (X)
µY,X
F (X ⊗ Y )
FσX,Y
// F (Y ⊗X),
Symmetric monoidal functor if (C,⊗,1), (C′,⊗,1′) symmetric.
Example: ∗ : C→ Cop in a compact closed category (C,⊗,1).
39
39. Dagger categories
Dagger category C has a contravariant functor † : C→ C, with:X† = X for X ∈ C0, adjoint f † : Y → X of f ∈ C(X, Y ), f †† = f .
Example: One-object linear categories C,H with x† = x.
Morphism f in dagger category C is:• Hermitian or self-adjoint if f † = f ;• unitary if invertible and f † = f−1.
Dagger monoidal category: † is a strict monoidal functor.
Dagger compact closed category: ∀ X ∈ C0 , coevX = (evX)†.
Lemma: ∀ f ∈ C1 , f†∗ = f ∗†. (Necessary, not sufficient, for DCCC.)
Biproduct dagger compact closed category: ∀X ∈ C0 , π†X = ιX .
Example: Category FDHilb of finite-dimensional Hilbert spaces,with ∀ x ∈ X , ∀ y ∈ Y , 〈f(x) | y〉 = 〈x | f †(y)〉 for f : X → Y .
Example: Rel with R∗ = R† as the converse relation.
Information theory: A bit in a BDCCC is 2 := 1⊕ 1.
Examples: 0, 1 in Rel, or qubit C⊕ C = C2 in FDHilb.
Extract information from [1,1],e.g., false = Ø and true = 11 in Rel, or scalar 1 7→ c in FDHilb.
Trace of f ∈ [X,X] = X∗ ⊗X is
1coevX−−−→ X⊗X∗ τ−→ X∗⊗X 1X∗⊗f−−−−→ X∗⊗X evX−−→ 1.
Example in FDHilb:∑j ej⊗ej 7→
∑j ej⊗ej 7→
∑j ej⊗f(ej) 7→
∑k
∑j ej⊗fjkek 7→
∑j fjj
Positive endomorphism f : X → X if ∃ g : X → Y . f = g† g.
Examples: In Rel, x R y ⇒ y R x and x R x.In FDHilb, ∀ x ∈ X , 〈f(x) | x〉 ≥ 0.
Complete positivity of f : [X,X]→ [Y, Y ] or f : X∗⊗X → Y ∗⊗Y :
∀ Z ∈ C0 , ∀ positive g : 1→ Z∗ ⊗X∗ ⊗X ⊗ Z ,
1g // Z∗ ⊗X∗ ⊗X ⊗ Z
1Z∗⊗f⊗1Z// Z∗ ⊗ Y ∗ ⊗ Y ⊗ Z is positive.
40
40. Subobjects and subobject classifiers
For object X of category C, define Presub(X) as the full subcategoryof (C ↓ X) whose defining morphisms s : S → X are monomorphisms.
Lemma: Presub(X) is a preorder: X
S
j1**
j2
44
s
??T
t
__????????
⇒ j1 = j2.
Skeleton poset SubC(X) or just Sub(X) consists of [2nd-order concept]
subobjects of X: Equivalence classes of monomorphisms s : S → X.
Note X
Sj //
1S
77
s
??S ′
s′
OO
j′ // S
s
__????????
⇒ j′ j = 1S, similarly j j′ = 1S′ .
Well-powered category C: Each object has a set of subobjects.
Now assume C has finite limits,so terminal > giving elements > → X of objects X.
Subobject classifier > true−−→ Ω in C [makes subobjects first-order!]:
∀ X ∈ C0 , ∀ (Ss−→ X) ∈ SubC(X) ,∃! χ . S //
s
p-b
>true
X χ//___ Ω
Example: > true−−→ false, true in Set with S = χ−1true ⊆ X.
Also works in category SetG of G-sets, for any group G,with trivial action of G on Ω.
Proposition: If C is well-powered and locally small,
SubC∼= C( ,Ω) = ∃Ω ∈ C0.
Proof. S ′ //
p-b
S //
p-b
>true
Yf// X χ
// Ω
Remark: S ′ is the inverse image of S under f : Y → X.
41
41. Dinatural transformations and power objects
Given graph maps F,G : Dop×D → C for graph D and category C, adinatural transformation δ : F ⇒ G is a “vector” (δx | x ∈ D0) ofcomponents δx : F (x, x)→ G(x, x) in C1 such that,
for all f : x→ y in D1, the hexagon of the dinaturality diagram
x x
f
F (x, x)δx // G(x, x)
G(1x,f)
%%KKKKKKKKK
F (y, x)
F (f,1x)99sssssssss
F (1y ,f) %%KKKKKKKKKG(x, y)
y
f
OO
y F (y, y)δy
// G(y, y)G(f,1y)
99sssssssss
in Dop in D in C
commutes in the category C.
Example: δnx : C(x, x)→ C(x, x); k 7→ kn — Church numeral n.
h0
11
--
h f 7→ (h f)n
f h 7→ (f h)n
%**
44
f (h f)n = (f h)n f
Example: For C with (finite limits and) a subobject classifier Ω,
SubC(X × Y ) ∼= C(X × Y,Ω) ∼= C(X,PY )
defines the power object PY of an object Y .
In C(PY × Y,Ω) ∼= C(PY,PY ), suppose 3Y 7→ 1PY .
In Set, have 3Y as characteristic function of (S, y) ∈ PY ×Y | S 3 y.For f : X → Y , define Pf : PY → PX as the unique morphism making
PX ×X 3X // Ω
>>>>>>>>
>>>>>>>>
PY ×X
Pf×1X77ppppppppppp
P1Y ×f ''NNNNNNNNNNN Ω
PY × Y 3Y// Ω
commute.
So dinatural 3 : P × 1C ⇒ ∆Ω for P : Cop → C.
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42. Elementary and Grothendieck topoi
Elementary topos: Category C with finite limits and a power object.
Properties: Finite colimits, subobject classifier, Cartesian closed.
Examples: Presheaf categories D = SetDop
for small D.
Grothendieck topos: E with reflective full K : E → D, for some D,
where the left adjoint L : D → E preserves finite limits.
Example: Sheaves — the presheaves F ∈ E0 ⊆ D,where D = (O,⊆) for a topological space (X,O), satisfying:
For each open cover U =⋃i∈I Ui of each U ∈ O, require equalizer
FUiF (Ui∩Uj⊆Ui) // F (Ui ∩ Uj)
FUe //___∏
k∈I FUk
p //___
q//___
πi
OO
πj
∏k,l∈I F (Uk ∩ Ul) .
πi,j
OO
πi,j
FUj
F (Ui∩Uj⊆Uj)// F (Ui ∩ Uj)
Typically, F (V ⊆ U) : FU → FV ; f 7→ f |V (restriction of functions).
Equalizer condition means match of FUi and FUj on F (Ui ∩ Uj).
Elementary definition: A topos is a category C with the following.
(a) A terminal object >.
(b) Pullback of each X → B ← Y .
(c) Monic > true−−→ Ω, and ∀ monic Ss−→ X , ∃! χ . S //
s
p-b
>true
X χ//___ Ω
(d) ∀ Y , ∃ (3Y : PY × Y → Ω) . ∀ (ρ : X × Y → Ω) ,
∃ unique X
r
such that X × Yr×1Y
ρ // Ω
PY PY × Y 3Y// Ω
Lawvere: First-order theory of topoi as a foundation for mathematics.