Van Kampen diagrams are bicolimits in Span?

Tobias Heindel1 and Pawe l Sobocinski2

1 Abt. fur Informatik und angewandte kws, Universitat Duisburg-Essen, Germany2 ECS, University of Southampton, United Kingdom

Abstract. In adhesive categories, pushouts along monomorphisms areVan Kampen (vk) squares, a special case of a more general notion calledvk-diagram. Other examples of vk-diagrams include coproducts in exten-sive categories and strict initial objects. Extensive and adhesive categoriescharacterise useful exactness properties of, respectively, coproducts andpushouts along monos and have found several applications in theoreticalcomputer science. We show that the property of being vk is actuallyuniversal, not in C but in the bicategory of spans SpanC. This theoremof pure category theory sheds light on the nature of spans and suggestspromising generalisations of the theory of adhesive categories.

Introduction

The relationship between limits and colimits in various categories is a fundamentaland challenging topic with several applications in theoretical computer science,especially formal semantics. Interest in this topic has led to the study of severalclasses of categories that capture important connections between particular kindsof limits and colimits. Extensive categories [5] and adhesive categories [19] are twoexamples of such classes. The universe in which one can look for such relationshipsis the category Set of sets and functions. We can thus say that a particularkind of colimit in C is “well-behaved” if it relates to limits as it would in Set.However, Set is somewhat atypical (for instance, it is boolean); more generallyand usefully, we can look for relationships in any Grothendieck or elementarytopos [14]. Moreover, there are also “weak” universes such as quasitoposes [26]or categories of separated objects [6] which are not quite as “well-behaved” butcover several essential (non-topos) examples [4, 13].

Extensive categories [5] have coproducts which are “well-behaved” with respectto pullbacks: coproducts are disjoint and universal. They have been used bymathematicians [3] and computer scientists [22] alike. In the presence of productsextensive categories are distributive [5] and so can be used, for instance, to modelcircuits [25]. Many categories, including Set, the category of topological spaces,and toposes are extensive. Quasitoposes in general are not [26, 14].

Adhesive categories [18, 19] are categories in which pushouts along monos are“well-behaved” with respect to pullbacks. They have been used as a categoricalfoundation for double-pushout graph transformation [18, 8] and have found severalrelated applications [9, 24]. Toposes are adhesive [20], but quasitoposes are not.? Research partially supported by EPSRC grant EP/D066565/1.

The definition of adhesivity depends on the notion of Van Kampen (vk) square– adhesive categories are those where pushouts along monomorphisms exist andare vk-squares. vk-squares were inspired by the axiomatic presentation of theproperties satisfied by coproducts in extensive categories. In fact, vk-squaresare an instance of a more general notion called vk-diagram, a fact noticed byCockett and Guo, who call them vk-colimits [7]. Extensive categories are thosein which coproduct diagrams exist and are vk-cospans while strict initial objectsare exactly the empty vk-diagrams. It is expected that vk-ω-chains will benecessary in order to obtain a satisfactory theory of unfoldings for adhesiverewriting systems [1]. The property of a colimit diagram being a vk-diagram canthus be taken to be one measure of “good-behaviour”.

The definition of vk-diagrams is axiomatic and, while clearly very useful,its meaning and precise relationship with other concepts in category theory isnot entirely clear in general. In fact, there exist definitions of extensive andadhesive categories which do not refer to vk-diagrams. For instance, a categoryC is extensive iff the functor C/A×C/B → C/A+B that “takes coproducts”is an equivalence of categories for any A,B ∈ C [21, 5]. Adhesive categoriescan be characterised in a similar manner [19]. We would, however, argue that asatisfactory unifying explanation of the underlying mathematical meaning of theclass of all vk-diagrams has not been given.

We show that a fitting setting to talk about the meaning of vk-diagrams inC is the bicategory of spans SpanC. Spans are known to theoretical computerscientists through the work of Katis, Sabadini and Walters [15] who used them tomodel systems with boundary, see also [11]. This universe can be seen as containingC via a canonical embedding Γ which is an identity on objects and takes arrowsC

f−→ D of C to spans C ← Cf−→ D. Spans also generalise partial maps [23] (those

spans which have a monomorphism as their left-leg) and, moreover, relations3

(those spans in which the two arrows are jointly monomorphic). Additionally,these embeddings also take into account the two-dimensional structure, sincepartial maps and relations are intrinsically the arrows of 2-categories. The two-dimensional structure of these examples is less interesting than that of Span (C)since the former consists of preorders and the latter of categories.

It is at the level of the bicategory of spans that one can elegantly characterisethe meaning of vk-diagrams – our main result (cf. Theorem 17) is that vk-diagrams are exactly those diagrams in C which satisfy a universal property;not in C but rather when considered (via Γ ) as living in the larger universe ofSpanC. Moreover, the universal property considered is the canonical notion ofcolimit for bicategories, the bicolimit [16].

There is some interesting related recent work. Milius [22] showed that co-products are preserved (as a lax-adjoint-cooplimit) to the 2-category of relationsover C when C is extensive. Cockett and Guo [7] have investigated the generalconditions under which partial map categories are join-restriction categories andhave found that this requires that certain colimits in the underlying category bevk-diagrams.

3 In the presence of a factorisation system in C.

2

In hindsight, it could be argued that the lack of such a clear general picture wasone of the reasons behind the difficulties in generalising existing theories based onaxiomatic presentations of vk-diagrams. In particular, the class of quasiadhesivecategories was introduced in order to capture several interesting examples whichare not toposes but rather arise naturally as categories of separated objects, orquasitoposes. A quasiadhesive category is one in which only pushouts along regularmonomorphisms are required to be vk-squares. Recently it has transpired [13]that quasiadhesive categories have too much structure and as a result do nothave many interesting examples. We believe that interesting notions of “weak”adhesivity will come by considering the axiomatic conditions that characterisethe fact that colimits are preserved into a natural sub-universe of SpanC (forinstance, partial maps or relations). While this is future work and outside thescope of the present paper, we have observed connections with the existing notionof weak-adhesivity [10].

Structure of the paper. In §1 we set our notation and recall the relevant definitionsfrom the theory of bicategories. In §2 we recall the definition of bicolimits inLemma 12 show that there is a close relationship between the concepts related tobicolimits and certain conditions on the corresponding diagrams in C. In §3 wereap the fruits of our observations and, after recalling the notion of Van Kampendiagram, we prove our main result, Theorem 17, which states that the notionof Van Kampen diagram is actually a universal property in SpanC. We end thesection with some of the consequences of this theorem. We conclude with plansfor future work in §4.

1 Preliminaries

In this section we introduce the necessary background from the theory of bicat-egories [2] and explain our notational conventions. Throughout the paper weassume that our ambient category C has identity preserving chosen pullbacks– that is, for any pair of arrows f : X → Z and g : Y → Z there exists anobject X ×Z Y and arrows π1 : X ×Z Y → X, π2 : X ×Z Y → Y such thatthe resulting square is a pullback diagram. We assume that the pullback of anidentity morphism is an identity morphism. No further properties are assumed ofthe choice.

The assumption that the choice of pullbacks preserves identities reduces theamount of theoretical overhead in the remainder of the paper. Indeed, we will beable to work with a simpler notion of bicategory (cf. Definition 1) that has stricthorizontal identities. Moreover, we will consider strict homomorphisms betweensuch bicategories (cf. Definition 3). It is important to emphasise that we do notlose any generality by making the assumption: our results would hold without it,at the cost of more tedious proofs.

The functors for which we shall compute colimits will have as their domaina small category J. We will use i, j to range over the objects of J and u, v overits arrows. Given a functor F : J → C we will usually write Fi,Fu instead of

3

F(i), F(u). Given an object C ∈ C, ∆C : J→ C is the functor constant at C;i.e. (∆C)i = C for all i ∈ J and (∆C)u = idC for all arrows u in J. For C ∈ C,a cocone κ : F → C is a synonym for a natural transformation κ : F → ∆C.A colimit for a functor F : J → C is a pair 〈colF , κ〉 where colF ∈ C andκ : F → C is a cocone which satisfies the well-known universal property. Weshall refer to κ as a colimit diagram for F . We say that C has J-colimits if everyfunctor F : J → C has a colimit. Examples of J-colimits for particular finitechoices of J include initial objects, coproducts and pushouts.

The bicategories [2] of particular interest to us will be those of spans (cf.Example 2). When chosen pullbacks preserve identities, these bicategories havestrict (horizontal) identities and for this reason we are able to consider a slightsimplification of the usual notion of bicategory.

Definition 1 (Strictly unitary bicategories). A strictly unitary ( su) bicat-egory B consists of:

– A collection ob B of objects;– for each A,B ∈ ob B, a category B(A,B), the objects and arrows of which are

called, respectively, the arrows and the 2-cells of B. Composition in each ofthese categories will be denoted by • and referred to as vertical composition.Given an arrow f : A→ B of B, its identity 2-cell will be denoted 1f . Eachcategory B(A,A) contains a special object idA called the identity arrow;

– for each A,B,C ∈ ob B, a functor ◦A,B,C : B(B,C) × B(A,B) → B(A,C)called horizontal composition, which will be written infix with subscriptsomitted. For any f : A→ B we require that idB ◦ f = f = f ◦ idA;

– for each A,B,C,D ∈ ob B, a natural isomorphism called associativity :

αA,B,C,D : λ(h, g, f). h ◦ (g ◦ f)→ λ(h, g, f). (h ◦ g) ◦ f

where the domain and the codomain are functors B(C,D) × B(B,C) ×B(A,B)→ B(A,D). Associativity satisfies coherence: the following diagramcommutes for any composable p, q, r, s:

s ◦ (r ◦ (q ◦ p))α ��

s◦α // s ◦ ((r ◦ q) ◦ p)α��

(s ◦ r) ◦ (q ◦ p)α

// ((s ◦ r) ◦ q) ◦ p (s ◦ (r ◦ q)) ◦ p.α◦p

oo

We shall sometimes refer to the dimensions of a bicategory: the zero-dimensionaldata are the objects, the one-dimensional data are the arrows and the two-dimensional data are the 2-cells.

Any ordinary category C can be thought of as an (su-)bicategory with trivialtwo-dimensional structure. For our purposes, the most important example of ansu-bicategory is the bicategory of spans.

Example 2 (Span bicategory [2]). Let SpanC consist of the following:

– objects: the objects of C;

4

– arrows C → D: spans of arrows C e1←− Ee2−→ D in C. We shall usually

write 〈e1, e2〉 : C → D to denote such a span and refer to E as its carrier.Composition is defined by using the chosen pullbacks in C.

– 2-cells: α : 〈e1, e2〉 → 〈e′1, e′2〉 are arrows α : E → E′ in C such that e′1◦α = e1and e′2 ◦ α = e2.

It is an easy exercise to check, using the assumption on the choice of pullbacks,that SpanC is an su-bicategory.

We will need to consider only special kinds of functors to SpanC, those with anordinary category as domain and those that factor through the “embedding” Γ ofC into SpanC (cf. Example 4). All these are instances of strict homomorphisms.

Definition 3 (Strict homomorphisms [2]). Suppose that A and B are su-bicategories. A strict homomorphism F : A→ B consists of a function F : ob A→ob B and a family of functors F (A,B) : A(A,B)→ B(FA,FB) such that:

(i) for all A ∈ A, F (idA) = idFA;(ii) for all f : A→ B, g : B → C in A, F (g ◦ f) = F (g) ◦ F (f);(iii) FαA,B,C,D = αFA,FB,FC,FD.

Example 4. The following are some examples of strict homomorphisms whichwill be of interest to us:

– there is a well-known canonical strict homomorphism Γ : C→ SpanC whichacts as the identity on objects and takes an arrow f : C → D to the span〈idC , f〉 : C → D;

– given any functor F : J→ C, ΓF : J→ SpanC is a strict homomorphism;– given an arbitrary su-bicategory B and some B ∈ B, we shall abuse notation

and denote the constant strict homomorphism J→ B constant at B by ∆B.Note that in the case of B = SpanC, ∆ = Γ∆.

We shall now introduce transformations between homomorphisms. The def-inition of bicolimits (cf. Definition 10) relies on the notion of strong naturaltransformation.

Definition 5 (Natural transformations). Given strict homo-morphisms F,G : A→ B between su-bicategories, a (lax) naturaltransformation consists of arrows κA : FA→ GA for A ∈ A and2-cells κf : Gf ◦ κA → κB ◦ Ff for f : A→ B in A so that:

FA

κA

��

Ff // FB

κB

��GA

κf 6>uuu uuu

Gf// GB

(i) κidA= 1κA

for each A ∈ A;(ii) for any f : A → B, g : B → C in A, the following diagrams of 2-cells are

“equal up-to-associativity”:

FA

κA

��

Ff // FBFg //

κB

��

FC

κC

��GA

κf 19kkkkk kkkkk

Gf// GB

κg 19kkkkk kkkkk

Gg// GC

“=”

FA

κA

��

F (g◦f) // FC

κC

��GA

κg◦f /7hhhhhhh hhhhhhh

G(g◦f)// GC

5

More formally, the following diagram commutes:

Gg ◦ (Gf ◦ κA)Gg◦κf ��

α // (Gg ◦Gf) ◦ κA

κg◦f��

Gg ◦ (κB ◦ Ff)α

��

κC ◦ (Fg ◦ Ff)α

��(Gg ◦ κB) ◦ Ff

κg◦Ff// (κC ◦ Fg) ◦ Ff.

A natural transformation is said to be strong when all the κf are isomorphic 2-cells.Given B ∈ B and a homomorphismM : J→ B, a pseudo-cocone λ :M→ B isa synonym for a strong natural transformation λ :M→ ∆B.

Because bicategories have a second dimension, there are morphisms betweennatural transformations. They are called modifications and are very simple todefine.

Definition 6 (Modifications [2, 17]). Given κ, λ natural transformations fromF to G, a modification ξ : κ→ λ consists of 2-cells ξA : κA → λA for A ∈ A suchthat, for all f : A→ B in A, λf • (Gf ◦ ξa) = (ξB ◦ Ff) • κf ; diagrammatically:

FA

κA

��

λA

��

Ff // FB

λB

��GA

ξA +3λf

8@xxxxxxxxxxxx

Gf// GB.

=

FA

κA

��

Ff // FB

κB

��λB

��GA

κf8@zzzzzz

zzzzzz

Gf// GB

ξB +3

Given su-bicategories A and B, let Homl[A,B] denote the su-bicategory ofhomomorphisms, lax natural transformations and modifications. Let Hom[A,B]denote the corresponding su-bicategory with arrows the strong natural transfor-mations.

2 Spans and bicolimits

In this section we study the su-bicategory of spans SpanC, functors from J toSpanC which factor through Γ and the notion of bicolimits. Clearly any diagramin SpanC can be understood as a diagram in C – in fact, an arrow in SpanCis an object of C and two arrows. A 2-cell in SpanC is a particular arrow in C.We will start by making some additional observations along these lines. Roughly,because of the structure of SpanC, we are able to “drop a dimension” – natural-transformations of functors to SpanC are actually certain (spans of) functors toC – their modifications are correspondingly certain natural transformations offunctors to C.

First notice that [J,C] has an obvious choice of chosen pullbacks, using thechosen pullbacks in C. In particular, it follows that Span [J,C] is an su-bicategory.

6

The following lemma makes precise our comments in the preceding paragraph,in particular, it shows that for functors of the form ΓF : J→ SpanC there is acorrespondence:

– spans of natural transformations from F to G ↔ lax natural transformationsfrom ΓF to ΓG;

– morphisms of such spans ↔ modifications.

Lemma 7. There is a strict homomorphism

Γ : Span [J,C]→ Homl[J,SpanC]

defined on objects F 7→ Γ ◦F . It is full and faithful on both arrows and 2-cells. Forany functor F : J→ C, Γ defines a natural isomorphism between the followingtwo functors [J,C]→ Cat:

λX.Span [J,C](F , X) ∼= λX.Homl[J,SpanC](ΓF , ΓX).

Proof. We need to define the action of Γ on arrows and 2-cells. We start witharrows and prove that Γ is full and faithful. First we have to show that a spanof natural transformations F η←− H µ−→ G defines a lax-natural transformationλ : ΓF → ΓG. For objects i ∈ J let λi

def= 〈ηi, µi〉. Given u : i→ j in J, we needto define λu : Hi → Fi ×Fj Hj . But, using the naturality of η, we have thatηj ◦Hu = Fu◦ηi so λu is the unique arrow such that π1◦λu = ηi and π2◦λu = Hu.To check that λu is a 2-cell it remains to check µj ◦ π2 ◦ λu = µj ◦ Hu = Gu ◦ µi,which follows by the naturality of µ.

It follows from the definition that λidi= 1Hi

. To check the second requirementof Definition 5, it suffices to show that the following diagram commutes, which isstraightforward the by functoriality of H:

Hi

λu ��

λv◦u // Fi ×FkHk

�

Fi ×Fj Hj Fi×Fjλv

// Fi ×Fj (Fj ×FkHk).

Faithfulness on arrows is immediate. Conversely, it is easy to see that for anylax natural transformation λ : ΓF → ΓG, there exists a functor H and naturaltransformations η : H → F , µ : H → G such that Γ〈η, µ〉 = λ; simply let Hi bethe carrier of the span λi and Hu

def= π2 ◦ λu. Functoriality follows directly fromthe commutativity of the diagram above. Naturality of η and µ follows from thefact that each λu is a 2-cell.

On 2-cells, a natural transformation ξ : H → H′ such that η′ • ξ = η andµ′ • ξ = µ induces a modification ΓH → ΓH′, we need to check that the followingdiagram commutes:

Hiλu //

ξi��

Fi ×Fj Hj

Fi×Fjξj

��H′i

λ′u

// Fi ×Fj H′j

7

On the first projection, this follows because ξ commutes with the ηs and on thesecond because ξ is natural.

Conversely, any modification ξ : λ→ λ′ is actually a natural transformationbetween the corresponding functors H → H′ – naturality follows directly fromthe commutativity of the diagram above. ut

The above lemma can be adapted to talk about strong natural transformations,which will be important for us because they are required in the definition ofbicolimits. We will need the notion of cartesian natural transformation:

Definition 8 (Cartesian natural transformations). Let F ,G : C → D befunctors, and let σ : F → G be a natural transformation. Then σ is cartesian ifall naturality squares are pullback diagrams in D.

It is an easy exercise to show that cartesian natural transformations includeall natural isomorphisms and are closed under pullback. Thus let CSpan [J,C] bethe su-bicategory of spans of natural transformations where the left leg of thespan is cartesian.

Proposition 9. There is a strict homomorphism

Γ : CSpan [J,C]→ Hom[J,SpanC]

which is full and faithful on both arrows and 2-cells. For any functor F : J→ C, Γdefines a natural isomorphism between the following two functors [J,C]→ Cat:

λX. CSpan [J,C](F , X) ∼= λX.Hom[J,SpanC](ΓF , ΓX).�

We will now define the notion of bicolimit. We shall restrict ourselves to(conical) bicolimits [16] for functors with domain an ordinary small category J.

Given a homomorphism M : J → B, if we have for some object bicM ∈ Ban equivalence of categories natural in X (i.e. the right hand side is essentiallyrepresentable as a functor λX.Hom[J,B](M,∆X) : B→ Cat)

B(bicM, X) ' Hom[J,B](M,∆X) (1)

then the pair 〈bicM, κ〉 where the pseudo-cocone κ : M→ bicM is the unit ofthe above equivalence will be referred to as the bicolimit of M.

Because bicolimits are the canonical notion of colimit in the setting of bicate-gories, we shall say that a homomorphismM : C→ B, where C is an ordinarycategory, preserves a colimit diagram κ : F → C if MC = bicMF and Mκ isa bicolimit diagram of MF in B. In particular, we shall say that M preservesJ-colimits wheneverM preserves the colimit diagrams of arbitrary functors fromJ to C.

Using the fact that equivalences of categories can be characterised as full, faith-ful functors that are essentially surjective on objects, we can give an equivalentaxiomatic presentation:

8

Definition 10 (Bicolimits). Given a strict homomorphism M : J → B, abicolimit for M consists of:

– an object bicM∈ B;– a pseudo-cocone κ : M→ ∆ bicM: for each object i ∈ J

an arrow κi : Mi → bicM, and for each u : i→ j in J aninvertible 2-cell κu : κi → κj ◦Mu satisfying the axiomsrequired for κ to be a strong natural transformation, i.e. :κidi

= 1κiand α • κv◦u = (κv ◦Mu) • κu.

Miκu

5=ttttttttκi

��

Mu //Mj

κjrrbicM

The bicolimit satisfies a universal property: for any pseudo-cocone λ : M→ ∆X:

(i) there exists a (pseudo) mediating morphism from κ to λ: an arrow h : bicM→X in B together with an invertible modification ϕ : λi → (∆h) ◦ κ, i.e.∀u : i→ j in J, we have hκu • ϕi = ϕjMu • λu; diagrammatically:

Mi

κu

.6ffff ffffκi $$III

III

λi

��

Mu //Mj

κj��

bicM

hzztttttt

X

ϕi

+3 =

Miλu

�$AAAA

λi

��

Mu //Mj

κj��

λj

������

����

���

bicM

hzztttttt

X

ϕj +3

(ii) Given h, h′ : bicM→ X in B, any modification ψ : ∆h ◦ κ → ∆h′ ◦ κ is(∆ξ) ◦ κ for a unique ξ : h→ h′. In particular, ξ is invertible iff ψ is.

Condition (i) above holds iff (1) is essentially surjective on objects. Condition (ii)holds iff it is full and faithful.

Condition (ii) also ensures that mediating morphisms from a bicolimit to apseudo-cocone are essentially unique: any two such mediating morphisms {h, ϕ}and {h′, ϕ′} are isomorphic via a unique isomorphic 2-cell compatible with theϕs.

To relate the notion of bicolimit with the vk condition, we shall restate theabove axiomatic definition. Given a pseudo-cocone κ : M → C a morphismh : C → D is universal for κ if, given any other morphism h′ : C → D anda modification ψ : ∆h ◦ κ → ∆h′ ◦ κ, there exists a unique 2-cell ξ : h → h′

such that ψ = (∆ξ) ◦ κ. The reason for the unusual statement of the followingproposition will become apparent in §3. Its proof is a simple calculation.

Proposition 11. A pair 〈C, κ〉, where κ : M → C is a pseudo-cocone, is abicolimit for M iff both of the following hold:

(i) for any pseudo cocone λ : M → D there exists a mediating morphism{h : C → D, ϕ} from κ to λ where h is universal for κ;

(ii) all arrows h : C → D are universal for κ. ut

9

We are interested in J-bicolimits of strict homomorphisms in SpanC. Thedefining equivalence of bicolimits (1) specialises as follows:

SpanC(bicΓF , X) ' Hom[J,SpanC](ΓF ,∆X).

Using Proposition 9, this is equivalent to:

SpanC(bicΓF , X) ' CSpan [J,C](F ,∆X).

We shall exploit working in CSpan [J,C] in the following lemma which relatesthe concepts involved in the axiomatic definition of bicolimits with diagrams inC. It will serve as the technical backbone of our main result, Theorem 17.

Lemma 12. Given a cocone κ : F → C in C, and a pseudo-cocone λ : ΓF → Din SpanC:

Fi

κi

��

Fu // Fj

κjrrC

ΓFi

λu 5=ttt tttλi

��

ΓFu // ΓFj

λjqqD

(i) given that λi = 〈ηi, µi〉, to give a mediating morphism

〈C h1←− H h2−→ D,ϕ : λ→ ∆〈h1, h2〉 ◦ Γκ〉

from Γκ to λ is to give a functor H : J→ C, a cartesian natural transfor-mation η : H → F , a cocone µ : H → D and a cocone θ : H → H such thatthe resulting three-dimensional diagram (†) in C (below left) is commutativewith all side faces (HiHCFi) pullbacks;

Hi

θi 11ηi

��

Hu

// Hj

ηj��

θj &&LLLLLL Dµj //

h2

OO

µi

$$

H

h1

��

Fi

κi 22

Fu // Fjκj

&&LLLLLL

C

(†)

Fi ×C H

π2i 11π1i

��

Fu×CH // Fj ×C H

π1j��

π2j

&&NNNNNN

H

h1

��

Fi

κi 11

Fu

// Fjκj

&&NNNNNNN

C

(‡)

(ii) given spans 〈h1, h2〉, 〈h′1, h′2〉 : C → D, to give a modification ψ : ∆〈h1, h2〉◦Γκ → ∆〈h′1, h′2〉 ◦ Γκ is to give a natural transformation ψ : F ×C H →F ×C H ′ (obtain F ×C H by taking pullbacks, see diagram (‡)) such thatπ′1◦ψ = π1 and (∆h′2)◦π′2◦ψ = (∆h2)◦π2. Then to give a cell ξ : 〈h1, h2〉 →〈h1, h2〉 which satisfies ∆ξ ◦ Γκ = ψ is to give an arrow ξ : H → H ′ suchthat h′1 ◦ ξ = h1, h′2 ◦ ξ = h2 and (∆ξ) ◦ π2 = π′2 ◦ ψ;

(iii) given 〈h1, h2〉 : C → D, if H = colF ×C H and π2 is the correspondingcolimit diagram then 〈h1, h2〉 is universal for Γκ;

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(iv) if C has J-colimits then also conversely, 〈h1, h2〉 universal for Γκ impliesthat H = colF ×C H and π2 is a colimit diagram.

Proof. (i), (ii) Immediate from the conclusion of Proposition 9.(iii). We need to show that every modification ψ : ∆〈h1, h2〉◦Γκ→ ∆〈h′1, h′2〉◦Γκis∆ξ◦Γκ for a unique ξ : 〈h1, h2〉 → 〈h′1, h′2〉. By (ii), ψ is a natural transformationF ×C H → F ×C H ′. But, by naturality of ψ, we have that for all u : i→ j inJ, π′2j ◦ ψj ◦ (Fu ×C H) = π′2i ◦ ψi and since H = colF ×C H we have a uniqueξ : H → H ′ such that, ∀i ∈ J, ξ ◦ π2i = π′2i ◦ ψi. It follows from the properties ofψ that also h′1 ◦ ξ = h1 and h′2 ◦ ξ = h2.(iv). Let H ′ def= colF ×C H and θ′ be the colimit diagram. We obtain a uniquek : H ′ → H such that ∀i ∈ J, k ◦ θ′i = π2i. We also obtain unique h′1 : H ′ → Cand h′2 : H ′ → D such that ∀i ∈ J, h′1 ◦ θ′i = κi ◦π1i and h′2 ◦ θ′i = h2 ◦π2i. It alsofollows that h1 ◦ k = h′1 and h2 ◦ k = h′2. Taking pullbacks, we obtain a naturaltransformation ψ : F×CH → F×CH

′ which satisfies π′1◦ψ = π1 and π′2◦ψ = π2.This, by (ii), defines a modification ψ : ∆〈h1, h2〉 ◦ Γκ → ∆〈h′1, h′2〉 ◦ κ. Usinguniversality, we get a unique ξ : H → H ′ such that h′1 ◦ ξ = h1, h′2 ◦ ξ = h2 andξ ◦π2i = π′2i ◦ψi. It follows that ξ ◦k = idH′ by the universal property of colimitsand that k ◦ ξ = idH by universality wrt Γκ. ut

3 Van Kampen diagrams and spans

In this section we state the definition of Van Kampen diagrams and prove ourmain result, Theorem 17. Van Kampen diagrams are simply a generalisation ofthe notion of Van Kampen square [18] to diagrams of arbitrary shape [7].

Definition 13 (Van Kampen diagrams, axiomatically).Let F : J → C be a functor. A cocone κ : F → Cis said to be a Van Kampen (vk) diagram if givena functor F ′ : J → C, a cartesian natural transfor-mation γ : F ′ → F , an object C ′ ∈ C, a coconeκ′ : F ′ → ∆C ′ and an arrow c : C ′ → C such thatc ◦ κ′i = κi ◦ γi (the diagram illustrated is commuta-tive, ∀u : i → j in J), the following two conditionsare equivalent:

(i) the cocone κ′ : F ′ → C ′ is a colimit diagram;(ii) for all i, 〈γi, κ

′i〉 is a pullback of 〈c, κi〉.

F ′i

κ′i 11γi

��

F ′u // F ′j

γj��

κ′j

&&MMMMMM

C ′

c

��Fi

κi 22

Fu

// Fj κj

&&NNNNNN

C

We shall refer to the implication (ii) ⇒ (i) as universality and to the implica-tion (i)⇒ (ii) as converse universality. Thus, with this terminology, a cocone κ isa vk-diagram iff it satisfies universality and converse universality.

It follows immediately from the definition (take γ = idF and c = idC) that if κ isa vk-diagram then it is a colimit diagram for F in C.

Example 14. The following well-known concepts are examples of vk-diagrams:

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(i) a strict initial object is a vk-diagram for the functor from the emptycategory;

(ii) a coproduct diagram which satisfies the axioms required of coproducts inextensive categories [5] is vk-diagram for a functor from · ·; An extensivecategory is thus a category with coproducts in which coproduct diagramsare vk;

(iii) a vk-square [18] is a vk-diagram for a functor from · ← · → ·; An adhesivecategory [18, 19] is a category with pushouts along monos in which suchpushouts are vk-diagrams.

The rest of this section will be devoted to a characterisation of vk-diagrams. Infact, we will show that the property of a diagram being Van Kampen is a universalproperty: not in C but rather in SpanC. The proof can be understood as follows:in the presence of J-colimits, converse universality of κ : F → C in the sense ofDefinition 13 is equivalent to the existence of universal mediating morphisms fromΓκ (Lemma 15). Universality of κ in the sense of Definition 13 is equivalent to theuniversality wrt Γκ of all SpanC-morphisms from C (Lemma 16). Recall thatexistence of universal mediating morphisms and the universality of all morphismscharacterises bicolimits (cf. Proposition 11).

Lemma 15. Suppose that C has J-colimits. A cocone κ : F → C satisfiesconverse universality (cf. Definition 13) iff given an arbitrary pseudo-coconeλ : ΓF → D, there exists an mediating morphism {〈h1, h2〉, ϕi} from Γκ to λ inSpanC which is universal for Γκ.

Proof. (⇒) Suppose that λ : ΓF → D is a pseudo-cocone in SpanC.We obtain (cf. Proposition 9) a commutative di-agram, as illustrated. Let H def= colH and θ bethe corresponding colimit diagram, we obtainh1 : H → C making diagram (†) commute andh2 : H → D such that h2θi = µi. By converseuniversality, the side faces are pullbacks, usingLemma 12(i) we get the required invertible mod-ification ϕ. Universality wrt to Γκ follows fromLemma 12(iii).

Hi

µi 33ηi

��

Hu // Hj

ηj��

µj

''PPPPPP

D

Fi

κi 33Fu

// Fj κj

''PPPPPP

C

(⇐) If in diagram (†) we have that H = colH and θ is the corresponding colimitdiagram, we first use the assumption to obtain an mediating morphism 〈h′1, h′2〉where h′2 : H ′ → H such that h′2θ

′i = θi, which is universal for Γκ. But, by the

universal property of 〈H, θ〉, there is an arrow k : H → H ′ such that kθi = θ′i. Itis immediate that h2k = idH and kh2 = idH′ follows by universality. ut

Lemma 16. Suppose that C has J-colimits. A cocone κ : F → C satisfiesuniversality (cf. Definition 13) iff every morphism 〈h1, h2〉 : C → D in SpanCis universal wrt to Γκ.

Proof. (⇒) Any morphism 〈h1, h2〉 leads to a diagram (‡) where all the side-facesare pullbacks. By universality of κ, the top face is a colimit diagram, thus 〈h1, h2〉is universal wrt Γκ by Lemma 12(iii).

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(⇐) Suppose that in diagram (‡) the side faces are all pullbacks. By assumption〈h1, h2〉 is universal wrt Γκ, thus H = colF ×C H by Lemma 12(iv). ut

Theorem 17. Suppose that C has J-colimits. A cocone κ : F → C is VanKampen in C iff Γκ is a bicolimit diagram in SpanC.

Proof. Immediate by the conclusions of Lemmas 15 and 16, using Proposition 11.ut

Theorem 17 demonstrates that vk-diagrams are actually, in the presence ofpullbacks and the appropriate colimits, precisely those diagrams in C which aremapped to bicolimits in SpanC by Γ . In fact, we can immediately conclude:

Proposition 18. All of the following hold:

– C has a strict initial object iff C has and Γ preserves the initial object;– C is extensive iff C has and Γ preserves coproducts;– C is adhesive iff C has and Γ preserves pushouts along monomorphisms. ut

Our theorem also suggests an alternative general definition of vk-diagramswhich is not dependent on the actual colimit, nor on the existence of specificcolimits, namely:

Definition 19 (Van Kampen diagrams). Let C be a category with chosenpullbacks and F : J → C a functor. A cocone δ : F → C is said to be a VanKampen diagram if Γδ is a bicolimit for ΓF in SpanC

Of course, if C has J-colimits then the conclusion of Theorem 17 implies thatDefinitions 13 and 19 are equivalent.

4 Conclusion and future work

We have given an elegant general characterisation of Van Kampen diagrams in Cas a universal property in SpanC. We believe that this is a significant clarificationof the notion of “being Van Kampen” and as a consequence, a useful developmentin the theories of extensive and adhesive categories. We believe that it will alsoserve as a guide for generalising the notion of adhesivity.

Indeed, there have been several attempts [19, 8, 10] at generalising that theoryin order to classify and study further examples. One attempt, quasiadhesivecategories, requires that pushouts along regular monos be van Kampen diagramshas been recently shown [13] to be too restrictive in the sense that the moreinteresting examples fail to be quasiadhesive. In fact, it can be concluded thatthe definition of quasiadhesivity which stems from altering the “axiomatic” pre-sentation of van Kampen squares was misguided. It seems, instead, that thealternative, general Definition 19 suggests several natural extensions of the notionof Van Kampen diagram. For instance, one can study those colimit diagramswhich satisfy a universal property in the universe of partial functions or relations(instead of spans, which can be considered as multirelations). We plan to study

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the resulting axiomatic presentations in the future and we are confident of cap-turing the “good-behaviour” of various colimits in several examples relevant forcomputer scientists.

Furthermore, inspired by the treatment of descent via discrete fibrations overequivalence relations in [12], we plan to investigate which properties of effectiveregular categories already hold in categories in which coequalisers of equivalencerelations exist and give rise to VK-forks.

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