# Morphisms of Colimits: from Paths to Profunctors

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Appl Categor StructDOI 10.1007/s10485-013-9357-0

Morphisms of Colimits: from Paths to Profunctors

Robert Pare

Received: 29 October 2012 / Accepted: 18 June 2013 Springer Science+Business Media Dordrecht 2013

Abstract A general kind of morphism of diagrams in a category is introduced. It is themost general notion of morphism which induces a morphism between the colimits of thediagrams. The sense in which it is the most general is made precise. It is expressed in termsof total profunctors which generalize everywhere defined relations. Their functorial proper-ties are developed leading to the notion of cohesive family of diagrams. A complementarynotion of deterministic profunctor is also introduced generalizing single valuedness.

Keywords Profunctor Diagram Colimit Total profunctor Deterministic profunctor Cohesive family of diagrams

Mathematics Subject Classifications (2010) 18A30 18A35

Introduction

The following problem was motivated by our work on the composition of modules in doublecategory theory [17]. This involved taking some rather complicated colimits and the func-toriality of the construction was far from obvious. In fact it was not functorial in generaland finding the conditions under which it was, covering all known examples, is perhaps themain contribution of that paper.

The problem we will consider is this: given two diagrams in A,

I

A

I JJ

A

R. Pare ()Department of Mathematics and Statistics, Dalhousie University, Halifax, NS B3H 4R2, Canadae-mail: pare@mathstat.dal.ca

R. Pare

what is the most general kind of morphism

which will induce a morphismlim lim ?

The trivial answer is: a morphism lim lim. But of course this is useless. We have

in mind something more syntactic like a functor over A

I

A

I J

F J

A

or more generally an oplax triangle, i.e. a triangle with a natural transformation in it:

I

A

I J

F J

A

This certainly works but is not general enough for our purposes.Here is an example. Suppose we are given two diagrams, a coequalizer diagram (the As)

and a pushout diagram (the Bs) and morphisms f1, f2, f3 as in

A0 A1a0

A0 A1a1

A1 Aa

f2

f3

f1

f

B0

B2

b1

B1

B0

b0

B1

B2

B

B2

b3

B1

B

b2

B1

B2

satisfying the equations

f1a0 = b0f2f1a1 = b0f3b1f2 = b1f3

Then

b2f1a0 = b2b0f2= b3b1f2= b3b1f3= b2b0f3= b2f1a1

so there exists a unique f : A B such that f a = b2f1.

Morphisms of Colimits: from Paths to Profunctors

Because the above is equational, we get more. For any functor F : A B for whichthe coequalizer Q and pushout P below exist

FA0 FA1Fa0

FA0 FA1Fa1

FA1 Qq

Ff2

Ff3

Ff1

g

FB0

FB2

Fb1

FB1

FB0

Fb0

FB1

FB2

P

FB2

p3

FB1

P

p2

FB1

FB2

there exists a unique g : Q P such that gq = p2Ff1.So we refine the question. What is the most general kind of morphism which

will produce a morphismlimF limF

natural in F ? (We assume here, and for the rest of the paper, that we only consider those Ffor which the colimits exist).

In what follows, we give two versions of a solution to this problem. The first, a nuts andbolts solution in the same vein as our coequalizer pushout example above. The second, amore functorial approach similar to the lax triangle idea but involving profunctors.

This leads us into the subject matter proper of the paper, namely the use of profunctorsin basic category theory. Profunctors are to functors as relations are to functions, and ourcolimit problem naturally leads to a notion of total profunctor extending the notion of every-where defined relation. To emphasize the analogy, we introduce what we call deterministicprofunctors which correspond to single-valued relations. A profunctor is representable ifand only if it is total and deterministic.

In the 1970s Benabou [1] remarked that Grothendiecks correspondence between pseudofunctors into Cat and fibrations could be extended to lax normal functors intoProf and arbi-trary functors, and in [2] suggests that interesting subbicategories of Prof would produceinteresting conditions on functors. The total profunctors are such a subbicategory and thecorresponding functors are what are called homotopy fibrations according to [2]. These weuse to define cohesive families of diagrams which allows us to express in the most generalterms the functoriality of colimits.

The point of the paper is the interplay between explicit calculations involving equa-tions and equivalence classes of paths on the one hand and profunctors on the other. Theprofunctors give a more conceptual perspective. The calculations suggest new ideas aboutprofunctors.

We thank the anonymous referee for a thoughtful report, going well beyond what mightnormally be expected. He/she gave a detailed account of how the paper might be reorga-nized in a more conceptual way using Kan extensions, thereby eliminating most of thecalculations involving the paths and equations characteristic of Section 1 and, to a lesserextent, Section 2. Unfortunately, we were not able to incorporate those suggestions herewithout disrupting the overall structure of the paper. We merely give, at the end of Section 2,an indication of how readers, well-versed in the lore of Kan extensions, might do this forthemselves.

R. Pare

1 Arrows and Equations Solution

In this section we determine diagrammatic conditions which produce natural morphismsbetween colimits. More precisely, consider the following conditions on two diagrams in A,

I

A

I JJ

A

(MC) For every F : A B for which limF and limF exist, we are given a morphismtF : limF limF

which is natural in F in the sense that if we have another functor G : A C for whichlimG and limG also exist, then for any H : B C and isomorphism : G

= HFwe have that

H limF H limFHtF

limG

H limFh

limG limGtG limG

H limFh

commutes, where h and h are the canonical comparison morphisms induced by (H, ).(I.e., h is the unique morphism such that

HFI H limFH injI

GI

HFI

I

GI limGinjI limG

H limFh

and similarly for h with replacing ).

Remark 1.1 There is a stronger naturality we could require, where is not necessarilyinvertible, which also holds for the families tF we get, but we dont need this. In the otherdirection, we could also get by with being an equality, but this requires some unnaturalset theoretical fiddling. To be clear about it, our naturality condition is as we gave it abovewith an isomorphism .

We will show that (MC) is equivalent to the following equational conditions.(AE) For every I in I we are given a JI in J and a morphism aI : I JI with theproperty that for every morphism i : I I there is a zigzag path joining aI i to aI

JI J1j1

I

JI

aI

I

J1

I

I

I

i

I I I

JI J1j1

I

JI

I I I

J1

a1

JI J2

j2

I

JI

I I I

J2

a2

J2

j3J2

I I

JI jn

I I

JI

aI

In this section we prove the following theorem.

Morphisms of Colimits: from Paths to Profunctors

Theorem 1.2 (1) Given a family aI as in (AE) and F : A B for which limF andlimF exist, then there exists a unique morphism tF such that

limF limFtF

FI

limFinjI

FI FJIFaI FJI

limFinjJI

commutes. The tF are natural in F , i.e. tF satisfies (MC).(2) Every family tF satisfying (MC) comes from a family aI satisfying (AE).(3) Two families aI and aI satisfying (AE) give the same tF if and only if for each I

there is a zigzag path joining aI to aI

JI J1j1

I

JI

aI

I II

J1

a1

J1 J2

j2

I

J1

I II

J2

a2

J2

j3

I

J2

I

J Ijn

J I J Ijn

II

J I

aI

Before giving the proof let us state our set theoretical conventions and some of theirconsequences.

In what follows we assume that our diagram categories I, J, K, ... are small, whereasthe categories where they take their values, A, B, C, ... are arbitrary (with small hom sets).We also consider only those functors F,G, ... for which the relevant colimits exist, oftenwithout mention.

Final functors will figure prominently in what follows. For completeness we recall herethe basic idea (see [4]).

Definition 1.3 A functor : I J is final if for every functor : J A, limexists if and only if lim exists, and when they do exist, the canonical morphism :lim lim is an isomorphism.

Proposition 1.4 : I J is final if and only if for every J in J there exist an I in I anda morphism j : J I , and any other such morphism j is connected to j by a path

I I1i1

J

I

j

J JJ

I1

j1

I1 I2i2

J

I1

J JJ

I2

j2

I2 i3

J

I2

J

I in

JJ

I j

In other words, for each J the comma category (J,) is nonempty and connected, i.e.0(J,) = 1.

The idea of the proof of part (2) is to take F to be the Yoneda embedding Y : A SetAop .But for large A, SetAop may well be illegitimate and so we cut down the codomain of Ydrastically. Let D be the class of diagrams : K A such that either K = 1 or =

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or , and AD the full subcategory of SetAop

determined by those presheaves which areisomorphic to colimits of representables of the form

limK

A(,K)

where is in D. So AD is equivalent to A with two extra objects adjoined which we thinkof as the formal colimits of and . What we say below holds for any class D of diagramscontaining all diagrams of the form 1 A so that AD contains the representables. LetZ : A AD be the corestriction of the Yoneda embedding. AD has small homs. In factfor any presheaf P the natural transformations

limK

A(,K) P

correspond to compatible families xK PKK indexed by the objects of the smallcategory K, and so form a (small) set.

These set theoretical considerations are more of a nuisance than anything else but theyare not the main reason for introducing AD . It is rather that for any F for which limFexists for all in D, its left Kan extension along Z exists and preserves the colimitslimKA(,K). This is a minor modification of the well-known fact that for small A,SetAop is its free colimit completion [9].

For completeness we state and prove this in the form we will use below.

Lemma 1.5 For F and Z as above, the left Kan extension LanZF of F along Z

A

BF

A AD

Z AD

BLanZF

exists and for any diagram : K A in D, the canonical morphismlimF LanZF(limZ)

is an isomorphism.

Proof From the general theory of Kan extensions (see [15] e.g.), if the colimit of thepossibly large diagram

El(P ) U A F B

exists, it will give the Kan extension

LanZF(P ) = limxPA

FA,

this for any presheaf P in SetAop . If P = limKA(,K), its category of elements El(P )has as objects equivalence classes

[A a K]

Morphisms of Colimits: from Paths to Profunctors

where the equivalence relation comes from the description of the colimit in Set. Thus a aif and only if there is a path

K K1k1

A

K

a

A AA

K1

a1

K1 K2

k2

A

K1

A AA

K2

a2

K2

k3

A

K2

A

K kn

K AA

K kn

AA

K a

A morphism

[A a K] [A a K ]is a morphism f : A A such that [af ] = [a].

There is a functor : K El(P )

given by (K) = [1K : K K]. This is final. Indeed if [a : A K] is anyobject of El(P ), a provides us with a morphism [a] (K). If a : [a] (K ) isanother such morphism, i.e. a : A K such that [a 1K ] = [a], then the above pathexpressing this gives a path

K K1

[a]

K

a

[a] [a][a]

K1

a1

K1 K2

[a]

K1

[a] [a][a]

K2

a2

K2 K2

[a][a] K

K K

[a][a]

K a

This shows that is final, so the colimit expressing the left Kan extension can be computedon K, i.e. limFU exists and the canonical morphism

limFU limFUis an isomorphism. The codomain is LanZF(P ) and P = limKA(,K) = limZ, andas U = we get that

limF LanZF(limZ)is an isomorphism.

Remark 1.6 This lemma may seem obvious but it actually has some content. This becomesclear when we note that the corresponding statement with colimits replaced by limits andleft Kan extension replaced by right, is in fact false.

Lemma 1.7 Let tF : limF limF be a natural family of morphisms, i.e. satisfying(MC). Then if a : I J is such that

limZ limZtZ

ZI

limZinjI

ZI ZJZa ZJ

limZinjJ

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commutes, so does

limF limFtF

FI

limFinjI

FI FJFa FJ

limFinjJ

for any F for which limF and limF exist.

Proof Let L be the left Kan extension LanZF , and : F LZ the canonical isomorphismA

BF

A AD

Z AD

BL

We must show that (a) in the diagram below commutes.

LlimZ LlimZLtZ

limF

LlimZl

limF limF limF

LlimZl

limF limFtF

FI

limFinjI

FI FJFa FJ

limFinjJ

=

LlimZ LlimZLtZ

LZI

LlimZLinjI

LZI LZJ LZJ

LlimZLinjJ

LZI LZJLZa

FI

LZI

I

FI FJFa FJ

LZJ

J

(a)

(b)

(c)

(d)

Note that limZ and limZ are in AF by its very definition. l and l are the canonicalcomparisons induced by so (b) commutes by naturality of t( ). By Lemma 1.5, l is anisomorphism so to prove that (a) commutes we only have to show that the rectangle madeup of (a) and (b) does, and it does because its equal to the rectangle on the right. In this(c) is naturality of and (d) is by definition of tZ . The vertical composites in the respectiverectangles are equal by definition of l and l.

Proof (Of theorem) Let aI and F be as in (1). Note, first of all, that if we have twomorphisms a1 : I J1 and a2 : I J2 connected by a j : J1 J2 as in

J1 J2j

I

J1

a1

I II

J2

a2

then commutativity of

FJ1

limFinjJ1

FJ1 FJ2 FJ2

limFinjJ2

FJ1 FJ2

Fj

FI

FJ1

Fa1

FI

FJ2

Fa2

Morphisms of Colimits: from Paths to Profunctors

shows that injJi Fai is independent of which a is chosen. Thus if a : I J anda : I J are connected by a zigzag path of as and j s, we will have injJ Fa =injJ Fa. It then follows by the connectedness property of (AE) that

FI FaI FJIinjJI limFI

is a cocone on F. This gives the existence and uniqueness of tF : limF limFsatisfying the condition of (1). If G : A C and H : B C with : G = HF as in(MC), then we want to show the commutativity of (b) in the diagram below

HFI H limFH injI

GI

HFI

I

GI limGinjI limG

H limFh

H limF H limFHtF

limG

H limF

limG limGtG limG

H limFh

(a) (b)

(a) commutes by definition of h and the injI are jointly epic so it suffices to show the outsiderectangle commutes, and this can be rewritten as

HFI HFJIHFaI

GI

HFI

I

GI GJIGaI GJI

HFJI

JI

HFJI H limFH injJI

GJI

HFJI

GJI limGinjJI limG

H limFh

(c) (d)

the horizontal composites in the respective rectangles being equal by definition of tF andtG. Now (c) is naturality of and (d) is by definition of h. This proves (1).

Let tF be a natural family as in (2). Consider the Z component tZ : limZ limZ,i.e.

tZ : limI A(, I) limJ A(,J ).For every I in I

A(, I) injI limI A(, I)tZ limJ A(,J )

corresponds to an element of limJ A(I,J), i.e. an equivalence class

[I aI JI ]where the equivalence relation is given by a zigzag path of as and j s of the sort consideredin condition (3). Naturality of the colimit injections say that for any i : I I ,

[I i I aI JI ] = [I aI JI ]which translates exactly into the connectedness condition of (AE).

Picking a representative aI from its class gives a factorization of tZ injI ,

limI A(, I) limJ A(,J )tz

A(, I)

limI A(, I)injI

A(, I) A(,JI )A(,aI ) A(,JI )

limJ A(,J )injJI

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i.e.

limZ limZtZ

ZI

limZinjI

ZI ZJIZaI ZJI

limZinjJI

so that tZ is indeed the morphism induced by the aI as in (1).Now let F : A B be such that limF and limF exist. Then by Lemma 1.7

limF limFtF

FI

limFinjI

FI FJIFaI FJI

limFinjJI

commutes for all I , so that tF is also induced by aI . This proves (2).As for (3), it was remarked at the start of the proof that if aI and aI are connected as

in the statement of the theorem, then injJI FaI = injJ I FaI so the cocones used in thedefinition of the induced morphisms, tF and t F , are the same, i.e. tF and t F are the same.Conversely, just from tZ we can recover the equivalence class of aI so obviously any twochoices of representatives will be connected.

If we return to our coequalizer-pushout example of the introduction, the theorem says wehave to choose a Bji for each Ai and a morphism Ai Bji . These we take to be

f2 : A0 B0f1 : A1 B1.

For each ai it is required that f1ai be connected to f2 which in the example is verified by

B1 B0b0

A1

B1

f1

A1

B0

A1

A0

A1

a0

A0 A0A0

B1 B0b0

A0

B1

A0 A0A0

B0

f2

and

B1 B0b0

A1

B1

f1

A1

B0

A1

A0

A1

a1

A0 A0A0

B1 B0b0

A0

B1

A0 A0A0

B0

f3

B0 B1

b1

A0

B0

A0 A0A0

B1

B1 B0b1

A0

B1

A0 A0A0

B0

f2

This explains the example.The condition (AE), although explicit and equational, is unwieldy and not very practical

if the context where it is to be applied is a bit involved. Compare it to the more canonicalsituation of a lax triangle

I

A

I J

F J

A

also mentioned in the introduction. No matter how complicated I and J are, we can easilywrite down the induced morphisms

limF limF.

Morphisms of Colimits: from Paths to Profunctors

The relation to condition (AE) is the following. For every I we have a J namely FI and amorphism I J namely I : I FI . For each i : I I the connectednessis achieved in one step

FI FI Fi

I

FI

I

I

FI

I

I

I

i

I I I

FI FI Fi

I

FI

I I I

FI

I

via naturality.In the case of (AE) for each I there is a JI although not a unique one, and for each

i : I I there is not a corresponding morphism JI JI but only a path. Nor can sensebe made of aI being natural. What we have is the categorical equivalent of a relation, aneverywhere defined one in fact. Categorical relations have been around since the 1960sand are variously called profunctors, distributors, bimodules, modules and even relators hasbeen suggested. (See [2, 10, 13, 19]). We use the term profunctor.

We will show that our theorem can be recast in terms of profunctors and thus becomemore canonical, more flexible, more useful. Along the way we will introduce severalinteresting concepts relating to profunctors and show how everything fits together nicely.

2 Profunctor Formulation

The theory of profunctors is well established. Early references are [3, 14, 20]. Easily acces-sible references are [5, 11, 12] for the enriched version. An axiomatic treatment is developedin [21, 22].

Here, in the interest of completeness, we give a brief outline to establish notation. Itshould be sufficient to make the paper self-contained, and to provide an intuitive guide tothe use of profunctors.

A profunctor P : A B is a functor Aop B Set. This is a two-dimensionalversion of a relation R : A B viewed as a function R : AB 2, where an elementa A is related by R to b B if and only if R(a, b) = 1. For categories A and B wecapitalize on the extra degree of freedom and specify not just if an object A in A is relatedto a B in B, but how it is related. This is achieved by giving formal morphisms from A to B ,even though they are not in the same category. It is a useful notation to write x : A Bto mean x P (A,B) and bx : A B and xa : A B for P (A, b)(x) andP (a,B)(x) respectively. Functoriality of P is thus the unit laws and associativity of thismultiplication.

There are two profunctors associated to a functor F : A B, F : A B given byF = B(F,) : Aop B Set

and F : B A given byF = B(, F) : Bop A Set.

The relationship between these is that F is right adjoint to F. If a profunctor P has a rightadjoint then it is representable by a functor provided B has split idempotents. However inthe enriched case, where Set is replaced by a monoidal category V, the gap between repre-sentability and having a right adjoint becomes much wider and gives rise to the notion ofCauchy completion of a V-category (see [13]). We mention this here because in the case

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of relations, adjointness does characterize functionality, and adjoint pairs of profunctors areoften taken as a replacement for functors within the world of profunctors. See for exam-ple [6] where such adjoint pairs are called, suggestively, maps. We present a differentperspective on representability in the last section.

The above discussion presupposes a bicategory of profunctors. If we are going to talk ofadjoints we need such a structure. The 2-cells are simply natural transformations betweenfunctors into Set. The composition is more interesting.

Given profunctors

A P B Q Cwe know how the As are related to the Bs and how these are related to the Cs. This isgiven to us in the form of formal arrows from As to Bs and also from Bs to Cs. We wouldlike to construct from this some arrows As to Cs. The obvious thing is to take formalcomposites

A xP

B y

Q C.

But we know how to compose these formal arrows with morphisms in B, so in the situation

A xP

Bb B

y

Q C,

associativity would dictate that the formal composites

A bxP

B y

Q C

and

A xP

B y bQ

C

be equal. It is thus that we are led to take as formal arrows from A to C, not pairs, butequivalence classes of pairs obtained by identifying these last two composites. Becausethis shifting of b from one side to the other is suggestive of the bilinearity of tensor product,we denote the equivalence class of

A xP

B y

Q C

by y x or y B x if necessary. Thus we have y bx = y b x. Of course the rela-tion (y , bx) (y b, x) is not an equivalence relation and we have to take the symmetrictransitive closure. So y x = y x if and only if there exist a path bs and pairs

C C

B

C

y

B B1 B1

C

y1

B B1

A

B

x

A AA

B1

x1

C C

B1

C

B1 B2 B2

C

y2

B1 B2

A

B1

A AA

B2

x2

C

B2

C

B2

B2

A

B2

A

C

B B

C

y

B

AA

B x

We see now that the connectedness conditions of 1, or something very similar, are codedin profunctor composition.

Morphisms of Colimits: from Paths to Profunctors

Thus the composite profunctor, denoted Q P , is given by equivalence classes of pairsand this can be presented as the coend

Q P (A,C) = B

Q(B,C) P (A,B).

This is the usual definition of profunctor composition and generalizes to the enrichedsetting.

The analogy with matrix multiplication should be noted. This is another aspect ofprofunctors. They can be thought of as A by B matrices of sets.

Using the y x notation it is not hard to check that profunctors do indeed form abicategory. The identity on A is the hom functor

A(,) : Aop A Set.There is one set theoretical annoyance. If B is large, the coend defining Q P may

not exist (i.e. it may be a proper class). So, unless otherwise stated, we will consider onlyprofunctors between small categories and then the composite Q P is always defined.

This is fine for our purposes because we will be interested in profunctors between colimitindexing categories which are always taken to be small, but it will sometimes be convenientto consider some profunctors between arbitrary categories. In this context it may be usefulto note that QP will exist if either P is representable or Q is corepresentable (isomorphicto F ).

The case we are interested in is the profunctor : I J associated with twodiagrams in A

I

A

I JJ

A

This composite is easily computed and is given by

(I, J ) = A(I,J).We can now state the main result of this section.

Theorem 2.1 Let tF : limF limF be a natural family of morphisms as in (MC) ofSection 1.

Define Pt(I, J ) to be the set of morphisms a : I J such that

limZ limZtz

ZI

limZinjI

ZI ZJZa ZJ

limZinjJ

commutes for Z : A AD the corestriction of the Yoneda embedding. Then(1) Pt is a subprofunctor of (2) limJ Pt (I, J ) = 1 for all I

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(3) tF is the unique morphism such that

limFP limFtF

FI

limFPinjI

FI FJFa FJ

limFinjJ

commutes for all a in Pt (I, J ).

Proof (1) Let a Pt(I, J ) and i : I I , j : J J be arbitrary morphisms. ThenZI

limZinjI

ZI ZIZi ZI

limZinjI

ZJ

limZinjJ

ZJ ZJ

Zj ZJ

limZinjJ

ZI ZJZa

limZ limZtZ

commutes by the definition of a and naturality of the colimit cocones, so j a i Pt(I

, J ) and Pt is a subprofunctor of .(2) The condition for a to be in Pt(I, J ) is the commutativity of

limI A(, I ) limJ A(,J )tZ

A(, I)

limI A(, I )injI

A(, I) A(,J )A(,a) A(,J )

limJ A(,J )injJ

which, by Yoneda, is equivalent to each of the composites having the same value at1I . This means that Pt is given by the pullback

1 limJ A(I,J )

Pt(I, J )

1

Pt(I, J ) A(I,J) A(I,J)

limJ A(I,J )injJ

where the bottom arrow picks out the image of 1I under tZ injI . If we take thecolimit along J of the top row we get a pullback again

1 limJ A(I,J )

limJ Pt (I, J )

1

limJ Pt (I, J ) limJ A(I,I) limJ A(I,I)

limJ A(I,J )=

so limJ Pt (I, J ) = 1.(3) By property (2) there is for every I a JI and an element aI Pt(I, JI ) and by Lemma1.7 we have that

limF limFtf

FI

limFinjI

FI FJIFaI FJI

limFinjJI

commutes. Because the injI are jointly epic, this uniquely determines tF .

Morphisms of Colimits: from Paths to Profunctors

Remark 2.2 The last part of the proof shows how a profunctor satisfying (2) gives rise to afamily aI satisfying (AE). Conversely, given such a family we get a profunctor PaI bytaking PaI (I, J ) to be the set of all morphisms a : I J connected to aI by a path

J J1j1

I

J

a

I II

J1

a1

J1 J2

j2

I

J1

I II

J2

a2

J2

j3

I

J2

I

JIjn

II

JI

aI

JI

jn

JI

This is beginning to look like the example of oplax triangles from the introduction

I

A

I J

F J

A

But the profunctor this produces, PI , is not F, a fact which becomes obvious if we takeA = 1. To capture the full generality of colimit friendly morphisms of diagrams we mustdistance ourselves somewhat from subprofunctors of . This involves two ideas. Thefirst is contained in part 3 of the above theorem.

Definition 2.3 A profunctor P : A B is called total if for every A in A,limB

P (A,B) = 1.

This is the profunctor version of a total or everywhere defined relation, well-known aspart of the definition of function.

Let T : A 1 be the unique functor into 1. (We use the same T for all categories A.This should not create confusion.) Then P is total if and only if TP T. This is nothingbut a reformulation of the definition. The following proposition gives the basic propertiesof total profunctors.

Proposition 2.4 (1) Total profunctors are closed under composition.(2) If P and P Q are total, then Q is total.(3) F is total for any functor F : A B.(4) F is total if and only if F is final.(5) For a span of functors

A B

C

A

F

C

B

G

G F is total if and only if F is final.(6) Total profunctors are closed under connected colimits and quotients.

Proof (1), (2) and (3) are obvious from the formulation in terms of the functors T . (4)follows from Proposition 1.4 once we note that limAB(B,FA) = 0(B,F ) and (5) followsimmediately from (1)(4).

R. Pare

For (6) note that

T limP = lim(T P)= limTand a colimit of T s will be T provided the diagram is connected. The other assertion followsfrom the fact that T preserves epis and a quotient of T has to be T itself.

The other notion we need is that of profunctor over A, i.e. profunctor in the slice categoryCat/A. For us this notion came out of double category theory. In [16], Theorem 3.14 weshow how it arises as a module between lax functors A Cat and in theorem 3.16 asthe vertical arrows in the double slice category Cat//A (see also [8]). Richard Wood knewof this definition independently, which is not surprising given his work on proarrows ([21,22]). We are not sure where this notion first appeared.

Definition 2.5 Let : I A and : J A. A profunctor from to or profunctorfrom I to J over A

I

A

I J

P J

A

is a profunctor P : I J together with a morphism : P .

The adjointness allows us to transform this into a lax triangle of profunctorsI

A

I JP J

A

We prefer the direct description. As = A(,), assigns to each elementx : I J of P a morphism x : I J in A.

Of course, profunctors over A compose, and the composite is the obvious one. If

J

A

J K

R K

A

then (R, ) (P ,) = (R P, ) where is defined on an element y x ofR P by

( )(y x) = (y)(x)which is easily seen to be well defined.

An oplax triangle

I

A

I J

F J

A

Morphisms of Colimits: from Paths to Profunctors

produces a profunctor over A

I

A

I J

F J

A

(j ) = (I I FI j J)for j : FI J an element of F(I, J ).

We remark here that a lax triangle

J

A

J I

G I

A

also induces a profunctor over A, namely

I

A

I JG J

A

with (i) = J i.

Theorem 2.6 LetI

A

I JP J

A

be a total profunctor over A. Then for every F : A B for which limFP and limF exist,there is a unique morphism limF : limF limF such that for every x P (I, J )

limF limFlimF

FI

limFinjI

FI FJF(x) FJ

limFinjJ

commutes. If (R, ) : is another total profunctor over A, we havelimF( ) = (limF)(limF).

Proof The proof of the first part is very much the same as that for the Theorem 1.2. BecauseP is total, for every I there exists a JI and x : I JI . Totality also means that any otherx : I J I is connected to x by a zigzag path of J s and xs from which it follows that

FIF(x) FJI

injJI limF

R. Pare

is independent of the choice of x. If we let I be the common value of these composites, itis easily seen to give a cocone on F, whence the existence and uniqueness of limF .

For functoriality, limF( ) is the unique morphism such that

limF limFlimF()

FI

limFinjI

FI FKF(()(yx)) FK

limFinjK

commutes for each element of (RP )(I,K), necessarily of the form yx for x P (I, J )and y R(J,K). If we compare this square with

limF limFlimF

FI

limFinjI

FI FJF(x) FJ

limFinjJ

limF limFlimF

FJ

limF

FJ FKF(y) FK

limFinjK

whose top rows are equal by definition of , it becomes apparent that limF( ) =(limF)(limF).

We consider this, total profunctors over A, to be the right notion of morphism of diagramsfor colimit purposes. It encompasses oplax triangles by the remark preceding the theorem.In fact it covers all naturally induced morphisms between colimits by Theorems 1.2 and 2.1.The appropriateness of this notion is further borne out by Theorems 3.4 and 2.12 below.

Remark 2.3 By a total profunctor over A we mean a profunctor over A as in Definition 2.5with P total. This should not be confused with the stronger condition

I JP I

A

J

A

J A J

A

A

A

=I

A

I A

A

A

which might be called a profunctor total over A.

We end this section with a characterization of when two total profunctors inducethe same morphisms between colimits. It may be a bit surprising that lim( ) in Theorem3.4 below is actually functorial even though the system of profunctors is merely lax. Thatis, even though the canonical morphism

(Pi , i ) (Pi, i) (Pi i , i i )is not in general an isomorphism, we still have

(limi )(limi) = limi i .This is explained by the following proposition.

Morphisms of Colimits: from Paths to Profunctors

Proposition 2.8 Let (P , ) and (P , ) be two total profunctors from to , and t :P P a morphism of profunctors over A. Then for any F : A B for which limFand limF exist,

limF = limF.

Proof By Theorem 2.6, limF and limF are the unique morphisms such that

limF limFlimF

FI

limFinjI

FI FJF(x) FJ

limFinjJ

(1)

limF limFlimF

FI

limFinjI

FI FJF (x ) FJ

limFinjJ

(2)

commute for all x : I P

J and x : I P

J . That t : P P is a morphism over Ameans that for every x : I

P J , (t (x)) = (x). If we replace x by t (x) in (2) we see

that limF satisfies the same commutativity as (1), thus limF = limF .

Now given a total profunctor over A

I

A

I J

P J

A

we can take the image of

PP P P

and, as a quotient of a total profunctor is again total, P is total, and (P , ) induces thesame morphisms of colimits as (P ,). So we could, if we wished, consider the total sub-profunctors of , which cuts down considerably on the numbers of them. However,the more general notion is the one that comes up in practice.

From Theorem 2.1, every natural family comes from a total subprofunctor of but being a subprofunctor of is still not enough to ensure uniqueness. For this weneed another concept.

Definition 2.9 Let Q : I J be a profunctor and P Q a subprofunctor. We say P issaturated in Q if for every x Q(I, J ) for which jx P (I, J ) for some j : J J , wehave x P (I, J ).

Equivalently, P is saturated in Q if and only if P (I,) Q(I,) is complemented inSetJ for every I . This is not the same as saying that P Q is complemented as one caneasily see when J = 1.

Proposition 2.10 Every P Q has a saturation, i.e. a smallest saturated P with P P Q. If P is total so is P.

R. Pare

Proof Let P(I, J ) consist of all those x Q(I, J ) which are connected to an element ofP by a path

J J1j1

I

J

x

I II

J1

x

J1 J2j2

I

J1

I II

J2

x2

J2 j3

I

J2

I

II

Jnjn

II

Jn

xn

where x Q(I, J) and xn P (I, Jn). P is functorial because, if x is connected to xnas above then xi is connected to xni by the path of xi, whereas jx is connected to x1 by

J J1jj1

I

J jx

I II

J1

x1

which is then connected to xn. P is clearly saturated because if jx P (I, J ), it isconnected to some xn P (I, Jn), but x itself is connected to jx by

J J j

I

J

x

I II

J jx

so to xn as well. If P P Q with P saturated, then for x P(I, J ) with a path asabove, xn P (I, Jn) P (I, Jn) so xn1 P (I, Jn1) by saturation, and then xn2 P (I, Jn2) by functoriality, and so on until we get x P (I, J ). I.e., P P .

Finally, we have a morphism

: limJ P (I, J ) limJ P(I, J )

which takes an element of limJ P (I, J ), which is an equivalence class [x] of elements x P (I, J ) to the corresponding equivalence class [x] in limJP(I, J ). is onto as every classin limJP(I, J ) has a representative from P by the very construction of P. If P is total,limJ P (I, J ) = 1 so we must also have limJ P(I, J ) = 1.

We can now characterize when two total profunctors over A induce the same morphismsbetween colimits.

Theorem 2.11 Let and be diagrams in A, and

I

A

I JP J

A

and

I

A

I JR J

A

be total profunctors over A. Then, the morphisms limF and limF are equal for everyF : A B for which limF and limF exist if and only if the images of

: P and : R have the same saturation in .

Morphisms of Colimits: from Paths to Profunctors

Proof There is a morphism of profunctors over A from P onto its image, which willtherefore also be total by Proposition 2.4, and of course another morphism over A intothe saturation of the image, also total by the previous proposition. It follows, by Propo-sition 2.8 that (P , ) will induce the same morphisms of colimits as the saturation of itsimage. The same of course goes for (R, ), so if im() = im() , we havelimF = limF.

Conversely, if tF : limF limF is a natural family of morphisms as in Theorem 2.1then the profunctor constructed from this

Q is given by taking F = Z, the corestriction of the Yoneda embedding, and Q(I, J ) the setof all a : I J such that

limZ limZtZ

A(, I)

limZinjI

A(, I) A(,J )A(,a) A(,J )

limZinjJ

If t comes from (P , ), then tZ is the unique morphism such that

limZ limZtZ

A(, I)

limZinjI

A(, I) A(,J )A(,(x)) A(,J )

limZinjJ

for all x P (I, J ). Thus Q(I, J ) consists of all those a : I J such thatinjJ A(, a) = injJ A(, (x)) which, by Yoneda, means that [a] = [(x)] inlimJ A(I,J). Consequently a is connected to (x) by a path

J J1j1

I

J

a

I II

J1

a1

J1 J2

j2

I

J1

I II

J2

a2

J2

j3

I

J2

I

I I

J jn

I I

J

(x)

for all x, or equivalently to some x, as they are all connected. This describes exactly thesaturation of the image of in . So if (P ,) and (R, ) induce the same morphismsof colimits, their images have the same saturation.

We summarize the results of this section in the following theorem. It is understood thatonly those F for which limF and limF exist, are considered.

Theorem 2.12 (1) Every total profunctor from to (i.e. a total profunctor I Jover A) induces a natural family of morphisms limF limF.(2) Every natural family of morphisms limF limF comes from a total profunctorfrom to .

(3) Two total profunctors from to induce the same natural family limF limFif and only if their images in have the same saturation.

R. Pare

Remark 2.13 As mentioned in the introduction, the referee has sugested that most of theexplicit calculations involving paths in this section and characteristic of section 1 couldbe eliminated by judicious use of Kan extensions, had the following proposition beenintroduced right at the beginning.

Let W be the weight associated to , i.e. the presheaf limI A(, I).

Proposition 2.14 Families of morphisms satisfying (MC) are in bijective correspondencewith natural transformations t : W W.

This follows easily from basic facts about Kan extensions and Lemma 1.5, and involvesno zig-zag type arguments. We leave the details to the reader comfortable with the calculusof Kan extensions.

Now the main result of this section can be stated as follows. The proof involvesprofunctor manipulation and the observation that W = T .

Theorem 2.15 Natural transformations W W are in bijective correspondence withsaturated subprofunctors of which are total.

3 Functoriality

It was remarked by Benabou [1], as early as the 70s, that a category over I

I

K

corresponds to a lax normal functor from I to Prof , the bicategory of categories andprofunctors. This puts in a wider context Grothendiecks correspondence between pseudo-functors I Cat and opfibrations over I. For an arbitrary category K over I, an object Iis taken to KI , the category of objects over I. So KI has objects K in K such that K = Iand morphisms k in K such that k = 1I . For a morphism i : I I in I, we get theprofunctor Pi : KI KI defined by

Pi(K,K) = {k : K K | k = i}.

This is explained in [2] where it is also remarked that interesting subbicategories of Profwill give rise, via this correspondence, to interesting conditions on categories over I. Andone such subbicategory is that of total profunctors. The corresponding categories over I arecalled homotopy fibrations there.

Definition 3.1 : K I is a homotopy opfibration if each of the profunctors Pi is total.

In elementary terms, is a homotopy opfibration if for every K in K and morphismi : K I , there is a lifting of i to k : K K such that k = i,

K K k

K I i

and any two liftings are connected by a path of liftings over i.

Morphisms of Colimits: from Paths to Profunctors

We list some of the basic properties of homotopy opfibrations in the following proposi-tion. (1) and (2) are obvious as is the lifting property for (3). The connectedness propertyfor (3) isnt. It follows from Theorem 3.5 below.

Proposition 3.2 (1) Opfibrations are homotopy opfibrations.(2) Homotopy opfibrations are stable under pullback.(3) Homotopy opfibrations are closed under composition.

Our reason for mentioning homotopy opfibrations here is to introduce the concept ofa cohesive family of diagrams used in the following theorem to express in more globalterms the functoriality of lim.

Definition 3.3 A cohesive family of diagrams in A is a span

I

K

I

K A A

with a homotopy opfibration.

For each I in I we get a diagram in A

I : KI Aby restricting to the subcategory KI of K. For all i : I I , the profunctor Pi :KI KI of morphisms over i is part of a profunctor over A,

KI

AI

KI KI

Pi KI

AI

i

If k Pi(K,K ), i.e. k : K K and k = i, then i(k) = (k) : IK I K . Andso, the scene for the following theorem has been set.

Theorem 3.4 If (,) is a cohesive family of diagrams in A and each limI exists, thenlimI extends to a unique functor lim( ) : I A such that for all k : K K overi : I I ,

limI limI limi

K

limIinjK

K K k K

limI injK

commutes. lim( ) is the left Kan extension of along .

Proof That limi exist and are unique satisfying the given commutativity follows imme-diately from Theorem 2.6 and the definition of cohesive family. Functoriality doesntfollow from that theorem but rather from Proposition 2.8 as we have the laxity morphisms

R. Pare

(Pi , i ) (Pi, i) (Pi i , i i ). It is also a consequence of the functoriality of the Kanextensions.

As mentioned in section 1, the general theory of Kan extensions says that if for every I ,the colimit of the diagram

(, I)U K A

exists, then it gives the value of the Kan extension at I ,

Lan(I) = limKI

K.

There is a functor : KI (, I) given by K = (K 1I I) which is final. This isa reformulation of homotopy opfibration. Indeed, for any object i : K I of (, I), amorphism k : (i) K ,

K

I

i

K Kk K

I

1I

is the same as a lifting of i. The existence and connectedness in the definition of homotopyopfibration correspond exactly to the existence and connectedness conditions characterizingfinal functors. It follows then that the colimit for the Kan extension exists and is equal tothe colimit of

KI (, I)

U K Awhich is just I .

This result is more than just saying that the Kan extension exists. Its saying that it isfibrewise, the fibres being those of the homotopy opfibration . A more precise way offormulating this is as follows.

Theorem 5 : K I is a homotopy opfibration if and only if for every pullback diagram

J IF

P

J

P KG K

I

and every cocomplete A, the canonical morphism

AJ AIF

AP

AJ

Lan

AP AK G

AK

AI

Lan

is an isomorphism.

Proof First of all, the can be described as follows. Let be in AK and J in J. Then()(J ) : Lan(G)(J ) Lan()(FJ )

Morphisms of Colimits: from Paths to Profunctors

is the morphismlimGV limU

induced by the morphism of the diagrams

A

K

(, J )

K

(, J ) (,FJ )F (,FJ )

K

U

(, J )

PV

P

KG

used in the calculation of the Kan extensions. F takes (Pj J ) to ((GP ) =

FPFj FJ).

If is a homotopy opfibration, then by Proposition 3.2 so is . As came out in the proofof 3.4 these colimits can be computed by restricting to the fibres

(, J ) (,FJ )F

PJ

(, J )

PJ KFJF KFJ

(,FJ )

But because P is the pullback, F is an isomorphism. So ()(J ) is an isomorphism.Conversely suppose is an isomorphism for all pullbacks P and cocomplete A. Take

J = 1 and F = I the functor with value I . Then P = KI and Lan is lim. Now takeA = Set and : K Set the representable K(K,). Then Lan = I(K,). Thus is given by

() : limK KI

K(K,K ) I(K, I)

[K k K ]K (K k K = I).This is an isomorphism if and only if for each i : K I there exists K and k : K K over i, unique up to connectedness over i. That is, is a homotopy fibration.

It is clear from this theorem that homotopy opfibrations are closed under composition,which completes the proof of part (3) of Proposition 3.2.

4 Deterministic Profunctors

We round out our study of total profunctors with a section on a complementary notion,which together with totality, will be equivalent to representability. This property, which wecall deterministic, is analogous to single valuedness, a condition well-known as part of thedefinition of function.

The plan for generalizing single valuedness is this. A relation R : A B is singlevalued if for every a A there is either zero or one b B such that aRb. We generalizethis to profunctors P : A B by saying, more or less, that for every A there should be a

R. Pare

(discrete) set of Bs related to A. It may not be obvious at first glance that we have the rightnotion but its equivalence to other concepts that have come up in quite different contexts,notably partial functors, multivalued functors and Mealy machines, is empirical evidencein its favour. A detailed study of this will appear elsewhere [18]. We thank Jeff Egger forsuggesting the name deterministic which is sometimes used in computer science circlesfor partial function.

Definition 4.1 A profunctor P : A B is deterministic if for every A in A there are aset I and a family of elements of P , A xi BiiI , such that for every element A x Bthere is a unique i I and a unique b : Bi B such that x = bxi .

This is a generalization of Diers familles universelles de morphismes [7] to profunc-tors. In fact we can reformulate the definition in terms of his multirepresentability.

Definition 4.2 (Diers) A functor : B Set is multirepresentable if it is isomorphic toa sum of representables

=iI

B(Bi,).

This is equivalent to saying that is free, i.e. is freely generated by a set of elements{xi Bi |i I }. Thus for every element x B there is a unique i and a uniqueb : Bi B such that x = bxi . The I and xi Bi are unique up to isomorphism.

It is clear how to reformulate our definition.

Definition 4.3 A profunctor P : A B is deterministic if P (A,) is multirepre-sentable for every A in A.

An object A of A determines a functor 1 A which we also call A, and a profunctorA : 1 A. A is the representable A(A,) and P A is P (A,). So to say P isdeterministic means that for every A

P A =iI

Bi.

Proposition 4.4 (1) For every functor F : A B, F is deterministic.(2) If P : A B is deterministic and Q : B C arbitrary, then QP exists (i.e.

is small).(3) If P and Q are deterministic then so is Q P .(4) A coproduct of deterministic profunctors is deterministic.

Proof (1) F A = (FA).(2) If P is deterministic, we have

Q P A = Q (iI

Bi)

=iI

Q Bi

Morphisms of Colimits: from Paths to Profunctors

which means that(Q P )(A,C) =

iI

Q(Bi, C).

(3) If Q is also deterministic we further haveQ P A =

iI

Q Bi =iI

jJi

Cj.

(4) (iI Pi) A = iI (Pi A) = iI jJi Bj.

The following result, although trivial, is at the heart of the matter.

Theorem 4.5 P : A B is representable if and only if it is total and deterministic.

Proof If P is representable then it is total by Proposition 2.4 and deterministic byProposition 4.4 (1). Conversely, if P is total and deterministic, then

P (A,) =iI

B(Bi,)

and

1 = limBP (A,B) = limBiI

B(Bi, B) =iI

limBB(Bi, B) =iI

1 = I.

Therefore P (A,) = B(B0,), i.e. is representable.

Acknowledgments Research supported by an NSERC grant.

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Morphisms of Colimits: from Paths to ProfunctorsAbstractIntroductionArrows and Equations SolutionProfunctor FormulationFunctorialityDeterministic ProfunctorsAcknowledgmentsReferences