INTERNAL CATEGORIES, ANAFUNCTORS AND LOCALISATIONS

In memory of Luanne Palmer (1965-2011)

DAVID MICHAEL ROBERTS

Abstract.

TO DO: Rewrite abstract. Write introduction. Finish examples section, in particular,track down references to literature where the main theorem is used (Pronk, Carchedi,Colman...) Make sure all references in bibliography are used, delete if not. Use hyperrefpackage?

Contents

1 Introduction 1

2 Sites and internal categories 2

3 Internal categories 5

4 Anafunctors 10

5 Localising bicategories at a class of 1-cells 17

6 2-categories of internal categories admit bicategories of fractions 20

7 Examples 24

A Superextensive sites 29

1. Introduction

It is a well-known result of classical category theory that a functor is an equivalence ifand only if it is fully faithful and essentially surjective. The proof of this fact requires theaxiom of choice. Whatever one’s position on this axiom is, one situation where Choice islacking is when dealing with internal categories and groupoids. We can recover this resultby localising at (i.e. formally inverting) the class of fully faithful and essentially surjectivefunctors. Since surjectivity is a concept that doesn’t generalise straightforwardly fromSet, we consider instead a class of arrows E which plays the role of the surjections. Weshall call these weak equivalences (see [Bunge-Pare, 1979]). Note however that what isneeded here is not localisation as given in [Gabriel-Zisman, 1967], but a 2-categoricallocalisation.

An Australian Postgraduate Award provided financial support during part of the time the materialfor this paper was written.

2000 Mathematics Subject Classification: Primary 18D99;Secondary 18F10, 18D05, 22A22.Key words and phrases: internal categories, anafunctors, localization, bicategory of fractions.c© David Michael Roberts, 2011. Permission to copy for private use granted.

1

2

In [Bunge-Pare, 1979] the interest in weak equivalences is to do with the ‘stack com-pletion’ of an internal category or groupoid, but it is from the theory of geometric stacksthat we find the necessary construction to localise a 2-category. It is a common idea thatalgebraic stacks are internal groupoids up to weak equivalence. This was made precice in[Pronk, 1996] where Pronk proved that the 2-category of algebraic stacks is a localisationof a certain 2-category of groupoid schemes, as well as analogous results for topologicaland smooth stacks.

It is for this reason that we are interested not only in the 2-categories Cat(S) andGpd(S) of all internal categories or groupoids in S respectively, but certain sub-2-categories Cat′(S) ⊂ Cat(S), which may or may not consist entirely of groupoids. Notethat because we wish to recover Lie groupoids as a special case of the results here, wewill not assume the ambient category is finitely complete. This does not introduce anycomplications.

This article started out based on material from the author’s PhD thesis. Many thanksare due to Michael Murray, Mathai Varghese and Jim Stasheff, supervisors to the author.The patrons of the n-Category Cafe and nLab, especially Mike Shulman, with whom thiswork was shared in development, provided helpful input.

2. Sites and internal categories

The idea of localness is inherent in many constructions in algebraic topology and algebraicgeometry. For an abstract category the concept of ‘local’ is encoded by a Grothendieckpretopology. Localness is needed to be able to talk about local sections of a map in acategory – a concept that will replace surjectivity when moving from Set to more generalcategories. This section gathers definitions and notations for later use.

2.1. Definition. A Grothendieck pretopology (or simply pretopology) on a category S isa collection J of families

{(Ui // A)i∈I}A∈Obj(S)

of morphisms for each object A ∈ S satisfying the following properties

1. (A′∼ // A) is in J for every isomorphism A′ ' A.

2. Given a map B //A, for every (Ui //A)i∈I in J the pullbacks B ×A Ai exist and(B ×A Ai //B)i∈I is in J .

3. For every (Ui // A)i∈I in J and for a collection (V ik

// Ui)k∈Kifrom J for each

i ∈ I, the family of composites

(V ik

// A)k∈Ki,i∈I

are in J .

3

Families in J are called covering families. A category S equipped with a pretopology Jis called a site, denoted (S, J).

The pretopology J is called a singleton pretopology if every covering family consistsof a single arrow (U // A). In this case a covering family is called a cover and we call(S, J) a unary site.

We do not assume that the collection of covering families for each object is a set (orsmall set, depending on one’s choice of foundations).

Very often, one sees the definition of a pretopology as being an assignment of a setcovering families to each object. However, when dealing with singleton pretopologies,one can define it as a subcategory with certain properties, and there is not necessarilya set of covers for each object. One example is the category of groups with surjectivehomomorphisms as covers. This distinction will be important later.

2.2. Example. The prototypical example is the pretopology O on Top, where a coveringfamily is an open cover. The class of numerable open covers (i.e. those that admit asubordinate partition of unity [Dold, 1963]) also forms a pretopology on Top. Much oftraditional bundle theory is carried out using this site; for example the Milnor classifyingspace classifies bundles which are locally trivial over numerable covers.

2.3. Definition. A covering family (Ui //A)i∈I is called effective if A is the colimit ofthe following diagram: the objects are the Ui and the pullbacks Ui×A Uj, and the arrowsare the projections

Ui ← Ui ×A Uj // Uj.

If the covering family consists of a single arrow (U // A), this is the same as sayingU // A is a regular epimorphism.

2.4. Definition. A site is called subcanonical if every covering family is effective.

2.5. Example. On Top, the usual pretopology O of opens, the pretopology of numerablecovers and that of open surjections are subcanonical.

2.6. Example. In a regular category, the class of regular epimorphisms forms a sub-canonical singleton pretopology.

2.7. Definition. Let (S, J) be a site. An arrow P //A in S is called a J-epimorphismif there is a covering family (Ui // A)i∈I and a lift

P

��Ui

??

// A

for every i ∈ I. A J-epimorphism is called universal its pullback along an arbitrary mapexists. We denote the class of universal J-epimorphisms by Jun.

4

The definition of J-epimorphism is equivalent to the definition in III.7.5 in [Mac Lane-Moerdijk, 1992]. The dotted maps in the above definition are called local sections, afterthe case of the usual open cover pretopology on Top. If J is a singleton pretopology, itis clear that J ⊂ Jun.

2.8. Example. The universal O-epimorphisms for the pretopology O of open covers onDiff is Subm, the pretopology of surjective submersions.

Sometimes a pretopology J contains a smaller pretopology that still has enough coversto compute the same J-epimorphisms.

2.9. Definition. If J and K are two singleton pretopologies with J ⊂ K, such thatK ⊂ Jun, then J is said to be cofinal in K.

Clearly J is cofinal in Jun for any singleton pretopology J .

2.10. Lemma. If J is cofinal in K, then Jun = Kun.

As mentioned earlier, one may be given a singleton pretopology such that each objecthas more than a set’s worth of covers. If such a pretopology contains a cofinal pretopologywith set-many covers for each object, then we can pass to the smaller pretopology andrecover the same results. In fact, we can get away with something weaker: one could askonly that the category of all covers of an object (see definition 2.13 below) has a set ofweakly initial objects, and such set may not form a pretopology. This is the content ofthe axiom WISC below. We first give some more precise definitions.

2.11. Definition. A category C has a weakly initial set I of objects if for every objectA of C there is an arrows O // A from some object O ∈ I.

Every small category has a weakly initial set, namely its set of objects.

2.12. Example. The large category Fields of fields has a weakly initial set, consistingof the prime fields {Q,Fp|p prime}. To contrast, the category of sets with surjections forarrows doesn’t have a weakly initial set of objects.

We pause only to remark that the statement of the adjoint functor theorem can beexpressed in terms of weakly initial sets.

2.13. Definition. Let (S, J) be a site. For any object A, the category of covers ofA, denoted J/A has as objects the covering families (Ui // A)i∈I and as morphisms(Ui // A)i∈I // (Vj // A)j∈J tuples consisting of a function r : I // J and arrowsUi // Vr(i) in S/A.

When J is a singleton pretopology this is simply a full subcategory of S/A. Wenow define the axiom WISC (Weakly Initial Set of Covers), due independently to MikeShulman and Thomas Streicher.

5

2.14. Definition. A site (S, J) is said to satisfy WISC if for every object A of S, thecategory J/A has a weakly initial set of objects.

A site satisfying WISC is in some sense constrained by a small amount of data for eachobject. Any small site satisfies WISC, for example, the site of finite-dimensional smoothmanifolds and open covers. Any pretopology J containing a cofinal pretopology K suchthat K/A is small satisfies WISC.

2.15. Example. Any regular category (for example a topos) with enough projectives,equipped with the canonical pretopology, satisfies WISC. In the case of Set ’enoughprojectives’ is the axiom COSHEP, also known as the presentation axiom (PAx), studied,for instance, by Aczel [Aczel 1978] in the context of constructive set theory.

2.16. Example. (Top,O) satisfies WISC (using AC in Set).

Choice may be more than is necessary here; it would be interesting to see if weakerchoice principles in the site (Set, surjections) are enough to prove WISC here.

2.17. Lemma. If (S, J) satisfies WISC, then so does (S, Jun).

It is instructive to consider an example where WISC fails in a non-artificial way. Thecategory of sets and surjections with all arrows covers clearly doesn’t satisfy WISC, but iscontrived and not a ‘useful’ sort of category. For the moment, assume the existence of aGrothendieck universe U with cardinality λ, and let SetU refer to the category of U-smallsets. Clearly we can discuss WISC relative to U. Let G be a U-large group and BG theU-large groupoid with one object associated to G. The Boolean topos SetBGU of U-smallG-sets is a unary site with the class epi of epimorphisms for covers. One could considerthis topos as being an exotic sort of forcing construction

2.18. Proposition. If G has at least λ-many conjugacy classes of subgroups, then (SetBGU , epi)does not satisfy WISC.

Recently, van den Berg showed [van den Berg, 2012] that it is consistent, relative tosome large cardinal axioms, that the category of ZF-sets fails to satisfy WISC.1

Perhaps of independent interest is a form of WISC with a bound: the weakly initialset for each category J/A has cardinality less than some (regular, say) cardinal κ. Thenone could consider, for example, sites where each object has a weakly initial finite set ofcovers.

3. Internal categories

Internal categories were introduced in [Ehresmann, 1963], starting with differentiableand topological categories (i.e. internal to Diff and Top respectively). We collect here

1As an aside, what we are considering is external WISC. There is a form of WISC suitable for theinternal logic of a (pre)topos, but would take us too far afield to cover. An internal version of WISCis considered for pretoposes as models of CZF in [van den Berg-Moerdijk, 2012], but is there called theAxiom of Multiple Choice.

6

the necessary definitions, terminology and notation. For a thorough recent account, see[Baez-Lauda, 2004] or the encyclopedic [Johnstone, 2002].

Fix a category S, referred to as the ambient category. We will assume throughout thatS has binary products.

3.1. Definition. An internal category X in a category S is a diagram

X1 ×X0 X1 ×X0 X1 ⇒ X1 ×X0 X1m−→ X1

s,t

⇒ X0e−→ X1

in S such that the multiplication m is associative (we demand the limits in the diagramexist), the unit map e is a two-sided unit for m and s and t are the usual source andtarget. An internal groupoid is an internal category with an involution

(−)−1 : X1//X1

satisfying the usual diagrams for an inverse.

Since multiplication is associative, there is a well-defined map X1×X0X1×X0X1//X1,

which will also be denoted by m. The pullback in the diagram in definition 3.1 is

X1 ×X0 X1//

��

X1

s

��X1 t

// X0 .

and the double pullback is the limit of X1t→ X0

s← X1t→ X0

s← X0. These, and pullbackslike these (where source is pulled back along target), will occur often. If confusion canarise, the maps in question will be explicity written, as in X1×s,X0,tX1. One usually seesthe requirement that S is finitely complete in order to define internal categories. This isnot strictly necessary, and not true in the well-studied case of S = Diff , the category ofsmooth manifolds.

Often an internal category will be denoted X1 ⇒ X0, the arrows m, s, t, e (and (−)−1)will be referred to as structure maps and X1 and X0 called the object of arrows and theobject of objects respectively. For example, if S = Top, we have the space of arrows andthe space of objects, for S = Grp we have the group of arrows and so on.

3.2. Example. If X // Y is an arrow in S admitting iterated kernel pairs, there isan internal groupoid C(X) with C(X)0 = X, C(X)1 = X ×Y X, source and target areprojection on first and second factor, and the multiplication is projecting out the middlefactor in X ×Y X ×Y X. This groupoid is called the Cech groupoid of the map X // Y .The origin of the name is that in Top, for maps of the form

∐I Ui

// Y (arising from anopen cover), the Cech groupoid C(

∐I Ui) appears in the definition of Cech cohomology.

7

3.3. Example. Let S be a category. For each object A ∈ S there is an internal groupoiddisc(A) which has disc(A)1 = disc(A)0 = A and all structure maps equal to idA. Such acategory is called discrete. There is also an internal groupoid codisc(A) with

codisc(A)0 = A, codisc(A)1 = A× A

and where source and target are projections on the first and second factor respectively.Such a groupoid is called codiscrete.

3.4. Definition. Given internal categoriesX and Y in S, an internal functor f : X // Yis a pair of maps

f0 : X0// Y0 and f1 : X1

// Y1

called the object and arrow component respectively. Both components are required tocommute with all the structure maps.

3.5. Example. If A // C and B // C are maps admitting iterated kernel pairs, andA //B is a map over C, there is a functor C(A) // C(B).

3.6. Example. If (S, J) is a subcanonical unary site, and U // A is a cover, a functorC(U) // disc(B) gives a unique arrow A // B. This follows immediately from the factA is the colimit of the diagram underlying C(U).

3.7. Definition. Given internal categories X, Y and internal functors f, g : X //Y , aninternal natural transformation (or simply transformation)

a : f ⇒ g

is a map a : X0//Y1 such that s◦a = f0, t◦a = g0 and the following diagram commutes

X1(g1,a◦s) //

(a◦t,f1)

��

Y1 ×Y0 Y1

m

��Y1 ×Y0 Y1

m // Y1

(1)

expressing the naturality of a.

Internal categories (resp. groupoids), functors and transformations form a locally small2-category Cat(S) (resp. Gpd(S)) [Ehresmann, 1963]. There is clearly an inclusion 2-functor Gpd(S) //Cat(S). Also, disc and codisc, described in example 3.3, are 2-functorsS //Gpd(S), whose underlying functors are left and right adjoint to the functor

Obj : Cat(S) // S, (X1 ⇒ X0) 7→ X0.

Here Cat(S) is the 1-category underlying the 2-category Cat(S). Hence for an internalcategory X in S, there are functors disc(X0) // X and X // codisc(X0), the arrowcomponent of the latter being (s, t) : X1

//X20 .

8

We say a natural transformation is a natural isomorphism if it has an inverse withrespect to vertical composition. Clearly there is no distinction between natural trans-formations and natural isomorphisms when the codomain of the functors is an internalgroupoid. We can reformulate the naturality diagram (1) in the case that a is a naturalisomorphism. Denote by −a the inverse of a. Then the diagram (1) commutes if and onlyif the diagram

X0 ×X0 X1 ×X0 X0−a×f1×a //

'��

Y1 ×Y0 Y1 ×Y0 Y1

m

��X1 g1

// Y1

(2)

commutes, a fact we will use several times.

3.8. Example. If X is a category in S, A is an object of S and f, g : X // codisc(A)are functors, there is a unique natural isomorphism f

∼⇒ g.

3.9. Definition. An internal or strong equivalence of internal categories is an equiva-lence in the 2-category of internal categories. That is, an internal functor f : X //Y suchthat there is a functor f ′ : Y //X and natural isomorphisms f ◦ f ′ ⇒ idY , f ′ ◦ f ⇒ idX .

3.10. Remark. In all that follows, ‘category’ will mean object of Cat′(S) and similarlyfor ‘functor’ and ‘natural transformation/isomorphism’.

The strict pullback of internal categories

X ×Y Z //

��

Z

��X // Y

when it exists, is the internal category with objects X0 ×Y0 Z0, arrows X1 ×Y1 Z1, and allstructure maps given componentwise by those of X and Z. Often we will be able to provethat certain pullbacks exist because of conditions on various component maps in S. Wedo not assume that all strict pullbacks of internal categories exists in our chosen Cat′(S).

3.11. Definition. For a category X and a map p : M // X0 in S the base change ofX along p is any category X[M ] with object of objects M and object of arrows given bythe pullback

M2 ×X20X1

//

��

X1

(s,t)

��M2

p2// X2

0

If the base change along any map in a given class K of maps exists for all objects ofCat′(S), then we say Cat′(S) admits base change along maps in K.

9

It follows immediately from the definition that given maps N //M and M // X0,there is a canonical isomorphism

X[M ][N ] ' X[N ]. (3)

with object component the identity map, when these base changes exist.

3.12. Remark. If we agree to follow the convention that M ×N N = M is the pullbackalong the identity arrow idN , then X[X0] = X. This also simplifies other results of thispaper, so will be adopted from now on.

One consequence of this assumption is that the iterated fibre product

M ×M M ×M . . .×M M,

bracketed in any order, is equal to M . We cannot, however, equate two bracketings of ageneral iterated fibred product; they are only canonically isomorphic.

3.13. Lemma. Let Y //X be a functor in the unary site (S,E) and U //X0 a cover.When base change along arrows in E exist, the following square is a strict pullback

Y [Y0 ×X0 U ] //

��

X[U ]

j��

Y // X

Proof. Since base change along arrows in E exists, we know that we have the functorY [Y0×X0 U ] //Y , we just need to show it is a strict pullback of j. On the level of objectsthis is clear, and on the level of arrows, we have

(Y0 ×X0 U)2 ×Y 20Y1 ' U2 ×X2

0Y1

' (U2 ×X20X1)×X1 Y1

' X[U ]1 ×X1 Y1

so the square is a pullback.

We are interested in 2-categories Cat′(S) which admits base change along a givenpretopology E on S. Examples sufficient for our purposes are when S is one of: Top(a category of topological spaces, convenient or otherwise), Diff , a category of manifolds(finite or infinite dimensional – the latter Frechet for choice), Grp (or more generally asemi-abelian category), Sch (some category of schemes) or an abelian category A, or atopos E (or more generally a regular category). Section 7 will cover some of these in moredetail.

Equivalences in Cat—assuming the axiom of choice—are precisely the fully faithful,essentially surjective functors. For internal categories, however, this is not the case. Inaddition, we need to make use of a pretopology to make the ‘surjective’ part of ‘essentiallysurjective’ meaningful.

10

The first definition of weak equivalence of internal categories along the lines we areconsidering appeared in [Bunge-Pare, 1979] for S a regular category, and E the class ofregular epimorphisms, in the context of stacks and indexed categories. We will in factuse a different definition which is equivalent to the one in [Bunge-Pare, 1979] in the sitesthey consider (see corollary 7.6).

3.14. Definition. Let (S,E) be a unary site. An internal functor f : X // Y in S iscalled

1. fully faithful if

X1f1 //

(s,t)��

Y1

(s,t)��

X0 ×X0 f0×f0// Y0 × Y0

is a pullback diagram

2. E-locally split if there is a an E-cover U // Y0 and a diagram

Y [U ]

f��

u

��X

f// Y

��

commuting up to a natural isomorphism

3. an E-equivalence if it is fully faithful and E-locally split.

The class of E-equivalences will be denoted WE. If mention of E is suppressed, they willbe called weak equivalences.

3.15. Remark. There is another defintion of full faithfulness for internal categories,namely that of a functor f : Z // Y being representably fully faithful. This means thatfor all categories Z, the functor

f∗ : Cat(S)(Z,X) //Cat(S)(Z, Y )

is fully faithful. It is a well-known result that these two notions coincide, so we shall useeither characterisation as needed.

3.16. Lemma. If f : X // Y is a functor such that f0 is in E, then f is E-locally split.Thus the functor X[U ] //X is an E-equivalence. whenever the base change exists.

Notice that E is not required to be subcanonical. We record here a useful lemma.

3.17. Lemma. Given a fully faithful functor f : X // Y in Cat′(S) and a natural iso-morphism f ⇒ g, the functor g is also fully faithful. In particular, an internal equivalenceis fully faithful.

Proof. This is a simple application of the definition of representable full faithfulness andthe fact the result is true in Cat.

11

4. Anafunctors

We now let J be a subcanonical singleton pretopology on the ambient category S. In thissection we assume that Cat′(S) admits base change along arrows in the given pretopologyJ . This is a slight generalisation of what is considered in [Bartels, 2006], where onlyCat′(S) = Cat(S) is considered.

4.1. Definition. [Makkai, 1996, Bartels, 2006] An anafunctor in (S, J) from a categoryX to a category Y consists of a cover (U //X0) and an internal functor

f : X[U ] // Y.

Since X[U ] is an object of Cat′(S), an anafunctor is a span in Cat′(S), and can bedenoted

(U, f) : X−7→ Y.

4.2. Example. For an internal functor f : X //Y in S, define the anafunctor (X0, f) : X−7→Y as the following span

X=←− X[X0]

f−→ Y.

We will blur the distinction between these two descriptions. If f = id : X // X, then(X0, id) will be denoted simply by idX .

4.3. Example. If U //A is a cover in (S, J) and BG is a groupoid with one object in S(i.e. a group), an anafunctor (U, g) : disc(A)−7→ BG is the same thing as a Cech cocycle.

4.4. Definition. [Makkai, 1996, Bartels, 2006] Let (S, J) be a site and let

(U, f), (V, g) : X−7→ Y

be anafunctors in S. A transformation

α : (U, f)⇒ (V, g)

from (U, f) to (V, g) is a natural transformation

X[U ×X0 V ]

xx &&X[U ]

f&&

α⇒ X[V ]

gxx

Y

If α is a natural isomorphism, then α will be called an isotransformation. In that casewe say (U, f) is isomorphic to (V, g). Clearly all transformations between anafunctorsbetween internal groupoids are isotransformations.

12

4.5. Example. Given functors f, g : X // Y between categories in S, and a naturaltransformation a : f ⇒ g, there is a transformation a : (X0, f) ⇒ (X0, g) of anafunctors,given by the component X0 ×X0 X0 = X0

a−→ Y1.

4.6. Example. If (U, g), (V, h) : disc(A)−7→ BG are two Cech cocycles, a transformationbetween them is a coboundary on the cover U ×A V // A.

4.7. Example. Let (U, f) : X−7→ Y be an anafunctor in S. There is an isotransfor-mation 1(U,f) : (U, f) ⇒ (U, f) called the identity transformation, given by the naturaltransformation with component

U ×X0 U ' (U × U)×X20X0

id2U×e−−−→ X[U ]1f1−→ Y1 (4)

4.8. Example. [Makkai, 1996] Given anafunctors (U, f) : X //Y and (V, f ◦k) : X //Ywhere k : V // U is a cover (over X0), a renaming transformation

(U, f)⇒ (V, f ◦ k)

is an isotransformation with component

1(U,f) ◦ (k × id) : V ×X0 U // U ×X0 U // Y1.

If k is an isomorphism, then it will itself be referred to as a renaming isomorphism.

We define (following [Bartels, 2006]) the composition of anafunctors as follows. Let

(U, f) : X−7→ Y and (V, g) : Y−7→ Z

be anafunctors in the site (S, J). Their composite (V, g) ◦ (U, f) is the composite spandefined in the usual way. It is again a span in Cat′(S).

X[U ×Y0 V ]

xx

fV

&&X[U ]

|| f&&

Y [V ]

xx

g

!!X Y Z

The square is a pullback by lemma 3.13 (which exists because V // Y0 is a cover), andthe resulting span is an anafunctor because V // Y0, and hence U ×Y0 V // X0, is acover, and using (3). We will sometimes denote the composite by (U ×Y0 V, g ◦ fV ).

Consider the special case when V = Y0, and hence (Y0, g) is just an ordinary functor.Then there is a renaming transformation (the identity transformation!) (Y0, g) ◦ (U, f)⇒(U, g ◦ f), using the equality U ×Y0 Y0 = U (by remark 3.12). If we let g = idY , thenwe see that (Y0, idY ) is a strict unit on the left for anafunctor composition. Similarly,considering (V, g) ◦ (Y0, id), we see that (Y0, idY ) is a two-sided strict unit for anafunctorcomposition. In fact, we have also proved

13

4.9. Lemma. Given two functors f : X // Y , g : Y // Z in S, their composition asanafunctors is equal to their composition as functors:

(Y0, g) ◦ (X0, f) = (X0, g ◦ f).

4.10. Example. As a concrete and relevant example of a renaming transformation wecan consider the triple composition of anafunctors

(U, f) : X−7→ Y,

(V, g) : Y−7→ Z,

(W,h) : Z−7→ A.

The two possibilities of composing these are((U ×Y0 V )×Z0 W,h ◦ (gfV )W

),(U ×Y0 (V ×Z0 W ), h ◦ gW ◦ fV×Z0

W)

The unique isomorphism (U ×Y0 V ) ×Z0 W ' U ×Y0 (V ×Z0 W ) commuting with thevarious projections is then the required renaming isomorphism. The isotransformationarising from this renaming transformation is called the associator.

A simple but useful criterion for describing isotransformations where one of the ana-functors involved is a functor is as follows.

4.11. Lemma. An anafunctor (V, g) : X−7→ Y is isomorphic to a functor (X0, f) : X−7→Y if and only if there is a natural isomorphism

X[V ]

}}

g

!!X

f

99∼⇒ Y

Just as there is composition of natural transformations between internal functors, thereis a composition of transformations between internal anafunctors [Bartels, 2006]. This iswhere the effectiveness of our covers will be used in order to construct a map locally oversome cover. Consider the following diagram

X[U ×X0 V ×X0 W ]

uu ))X[U ×X0 V ]

yy ))

X[V ×X0 W ]

uu &&X[U ]

f

++

a⇒ X[V ]

g

��

b⇒ X[W ]

h

ssY

14

from which we can form a natural transformation between the leftmost and the rightmostcomposites as functors in S. This will have as its component the arrow

ba : U ×X0 V ×X0 Wid×∆×id−−−−−→ U ×X0 V ×X0 V ×X0 W

a×b−−→ Y1 ×Y0 Y1m−→ Y1

in S. Notice that the Cech groupoid of the cover

U ×X0 V ×X0 W // U ×X0 W (5)

is

U ×X0 V ×X0 V ×X0 W ⇒ U ×X0 V ×X0 W,

with source and target arising from the two projections V ×X0 V // V . Denote this pairof parallel arrows by s, t : UV 2W ⇒ UVW for brevity. In [Bartels, 2006], section 2.2.3,we find the commuting diagram

UV 2W t //

s��

UVW

ba��

UVWba

// Y1

(6)

(this can be checked by using elements) and so we have a functor C(U ×X0 V ×X0

W ) // disc(Y1). Our pretopology J is assumed to be subcanonical, and using remark 3.6this gives us a unique arrow ba : U ×X0 W // Y1, the composite of a and b.

4.12. Remark. In the special case that U ×X0 V ×X0 W //U ×X0 W is split (e.g. is anisomorphism), the composite transformation has

U ×X0 W // U ×X0 V ×X0 Wba−→ Y1

as its component arrow. In particular, this is the case if one of a or b is a renamingtransformation.

4.13. Example. Let (U, f) : X−7→ Y be an anafunctor and U ′′j′−→ U ′

j−→ U succes-sive refinements of U // X0 (e.g isomorphisms). Let (U ′, fU ′) and (U ′′, fU ′′) denote thecomposites of f with X[U ′] //X[U ] and X[U ′′] //X[U ] respectively. The arrow

U ×X0 U′′ idU×j◦j′−−−−−→ U ×X0 U // Y1

is the component for the composition of the isotransformations (U, f) ⇒ (U ′, fU ′),⇒(U ′′, fU ′′) described in example 4.8. Thus we can see that the composite of renamingtransformations associated to isomorphisms φ1, φ2 is simply the renaming transformationassociated to their composite φ1 ◦ φ2.

15

4.14. Example. If a : f ⇒ g, b : g ⇒ h are natural transformations between functorsf, g, h : X // Y in S, their composite as transformations between anafunctors

(X0, f), (X0, g), (X0, h) : X−7→ Y.

is just their composite as natural transformations. This uses the equality

X0 ×X0 X0 ×X0 X0 = X0 ×X0 X0 = X0,

which is due to our choice in remark 3.12 of canonical pullbacks

COMMENTS ON FOLLOWING LEMMA HERE

MIGHT NEED SMALL REWRITE

4.15. Lemma. Let f : X // Y be a J-equivalence in S. There is an anafunctor

(U, f) : Y−7→ X

and isotransformations

ι : (X0, f) ◦ (U, f)⇒ idY

ε : (U, f) ◦ (X0, f)⇒ idX

Proof. We have the anafunctor (U, f) from lemma ??. Since the anafunctors idX , idY areactually functors, we can use lemma 4.11. Using the special case of anafunctor compositionwhen the second is a functor, this tells us that ι will be given by a natural isomorphism

Xf

����Y [U ] //

f<<

Y

This has component ι : U // Y iso1 , using the notation from the proof of the previous

lemma. Notice that the composite f1 ◦ f1 is just

Y [U ]1 ' U ×Y0 Y1 ×Y0 Uι×id×−ι−−−−−→ Y iso

1 ×Y0 Y1 ×Y0 Y iso1 ↪→ Y3

m−→ Y1.

Since the arrow component of Y [U ] // Y is U ×Y0 Y1 ×Y0 Upr2−−→ Y1, ι is indeed a natural

isomorphism using the diagram (2).

The other isotransformation is between (X0×Y0 U, f ◦ pr2) and (X0, idX), and is givenby the arrow

ε : X0 ×X0 X0 ×Y0 U ' X0 ×Y0 Uid×(s′,a)−−−−−→ X0 ×Y0 (X0 ×Y0 Y1) ' X2

0 ×Y 20Y1 ' X1

16

This has the correct source and target, as the object component of f is s′, and the sourceis given by projection on the first factor of X0 ×Y0 U . This diagram

(X0 ×Y 20U)2 ×X2

0X1

'��

pr2 // X1

'

��

U ×Y0 X1 ×Y0 U−ι×f×ι

��(X0 ×Y0 Y iso

1 )×Y0 Y1 ×Y0 (Y iso1 ×Y0 X0)

id×m×id// X0 ×Y0 Y1 ×Y0 X0

commutes (a fact which can be checked using elements), and using (2) we see that ε isnatural.

The first half of the following theorem is proposition 12 in [Bartels, 2006], and thesecond half follows because all the constructions of categories involved in dealing withanafunctors outlined above are still objects of Cat′(S).

4.16. Theorem. [Bartels, 2006] For a site (S, J) where J is a subcanonical single-ton pretopology, internal categories, anafunctors and transformations form a bicategoryCatana(S, J). If we restrict attention to a full sub-2-category Cat′(S) which admits basechange for arrows in J , we have an analogous full sub-bicategory Cat′ana(S, J).

There is a strict 2-functor Cat′ana(S, J) //Catana(S, J) which is an inclusion on objectsand is the identity on hom-categories. The following is the main result of this section,and allows us to relate anafunctors to the localisations considered in the next section.

4.17. Proposition. There is a strict 2-functor

αJ : Cat′(S) //Cat′ana(S, J)

sending J-equivalences to equivalences, and commuting with the respective inclusions intoCat(S) and Catana(S, J).

Proof. We define αJ to be the identity on objects, and as described in examples 4.2, 4.5on 1-arrows and 2-arrows (i.e. functors and transformations). We need first to show thatthis gives a functor Cat′(S)(X, Y ) //Cat′ana(S, J)(X, Y ). This is precisely the contentof example 4.14. Since the identity 1-cell on a category X in Cat′ana(S, J) is the image ofthe identity functor on S in Cat′(S), αJ respects identity 1-cells. Also, lemma 4.9 tellsus that αJ respects composition. That αJ sends J-equivalences to equivalences is thecontent of lemma 4.15.

The 2-category Cat′(S) is clearly locally small, but a priori the collection of anafunc-tors X−7→ Y do not constitute a set for S a large category.

4.18. Proposition. Let (S, J) be a site with a subcanonical singleton pretopology J ,satisfying WISC and let Cat′(S) admit base change along arrows in J . Then Cat′ana(S, J)is locally essentially small.

17

Proof. Given an object A of S, let I(A) be a weakly initial set for J/A. Considerthe locally full sub-2-category of Cat′ana(S, J) with the same objects, and arrows thoseanafunctors (U, f) : X−7→ Y such that U // X0 is in I(X0). Every anafunctor is thenisomorphic, by the generalisation of example 4.8, to one in this sub-2-category.

Examples of sites (S, J) where Cat′ana(S, J) is not known to be locally essentially smallare the category of sets from Gitik’s model as used in [van den Berg, 2012] and the toposfrom proposition 2.18.

5. Localising bicategories at a class of 1-cells

Ultimately we are interesting in inverting all E-equivalences in Cat′(S) and so need todiscuss what it means to add the formal pseudoinverses to a class of 1-cells in a 2-category– a process known as localisation. This was done in [Pronk, 1996] for the more generalcase of a class of 1-cells in a bicategory, where the resulting bicategory is constructed andits universal properties (analogous to those of a quotient) examined. The application inloc. cit. is to show that the equivalence of various bicategories of stacks to localisationsof 2-categories of smooth, topological and algebraic groupoids. The results of this articlecan be seen as one-half of a generalisation of these results to more general sites.

5.1. Definition. [Pronk, 1996] Let B be a bicategory and W ⊂ B1 a class of 1-cells. Alocalisation of B with respect to W is a bicategory B[W−1] and a weak 2-functor

U : B //B[W−1]

such that U sends elements of W to equivalences, and is universal with this property i.e.composition with U gives an equivalence of bicategories

U∗ : Hom(B[W−1], D) //HomW (B,D),

where HomW denotes the sub-bicategory of weak 2-functors that send elements of W toequivalences (call these W -inverting, abusing notation slightly).

The universal property means that W -inverting weak 2-functors F : B // D factor,up to a transformation, through B[W−1], inducing an essentially unique weak 2-functor

F : B[W−1] //D.

5.2. Definition. [Pronk, 1996] Let B be a bicategory B with a class W of 1-cells. W issaid to admit a right calculus of fractions if it satisfies the following conditions

2CF1. W contains all equivalences

2CF2. a) W is closed under compositionb) If a ∈ W and there is an isomorphism a

∼⇒ b then b ∈ W

18

2CF3. For all w : A′ // A, f : C // A with w ∈ W there exists a 2-commutative square

P

v

��

g // A′

w

��C

f // A

'z�

with v ∈ W .

2CF4. If α : w ◦ f ⇒ w ◦ g is a 2-arrow and w ∈ W there is a 1-cell v ∈ W and a 2-arrowβ : f ◦ v ⇒ g ◦ v such that α ◦ v = w ◦ β. Moreover: when α is an isomorphism,we require β to be an isomorphism too; when v′ and β′ form another such pair,there exist 1-cells u, u′ such that v ◦ u and v′ ◦ u′ are in W , and an isomorphismε : v ◦ u⇒ v′ ◦ u′ such that the following diagram commutes:

f ◦ v ◦ u β◦u +3

f◦ε '

��

g ◦ v ◦ u

g◦ε'

��f ◦ v′ ◦ u′

β′◦u′+3 g ◦ v′ ◦ u′

(7)

For a bicategory B with a calculus of right fractions, [Pronk, 1996] constructed alocalisation of as a bicategory of fractions; the 1-arrows are spans and the 2-arrows areequivalence classes of 2-categorical spans-of-spans diagrams.

From now on we shall refer to a calculus of right fractions as simply a calculus offractions, and the resulting localisation constructed by Pronk as a bicategory of fractions.Since B[W−1] is defined only up to equivalence, it is of great interest to know when abicategory D in which elements of W are sent to equivalences by a 2-functor B // Dis itself equivalent to B[W−1]. In particular, one would be interested in finding such anequivalent bicategory with a simpler description than that which appears in [Pronk, 1996].

5.3. Proposition. [Pronk, 1996] A weak 2-functor F : B // D which sends elementsof W to equivalences induces an equivalence of bicategories

F : B[W−1]∼−→ D

if the following conditions hold

EF1. F is essentially surjective,

EF2. For every 1-cell f ∈ D1 there are 1-cells w ∈ W and g ∈ B1 such that Fg∼⇒ f ◦Fw,

19

EF3. F is locally fully faithful.

Thanks are due to Matthieu Dupont for pointing out (in personal communication) thatproposition 5.3 actually only holds in one direction, not in both, as claimed in loc. cit.

The following is useful in showing a weak 2-functor sends weak equivalences to equiv-alences, because this condition only needs to be checked on a class that is in some sensecofinal in the weak equivalences.

5.4. Proposition. In the bicategory B and let V ⊂ W be two classes of 1-cells such thatfor all w ∈ W , there exists v ∈ V and s ∈ W such that there is an isomorphism

a

w

��b v

//

s

??

c .

'��

Then a weak 2-functor F : B // D that sends elements of V to equivalences also sendselements of W to equivalences.

Proof. In the following the coherence arrows will be present, but unlabelled. It is enoughto prove that in a bicategory D with a class of maps M (in our case, the image of W )such that for all w ∈M there is an equivalence v and an isomorphism α

a

w

��b v

//

s

??

c .

'�

with s ∈M , then w is also an equivalence. Let v be a pseudoinverse for v and let j = s◦ v.Then there is the following invertible chain of isomorphisms

w ◦ j ⇒ (w ◦ s) ◦ v ⇒ v ◦ v ⇒ I.

Since s ∈ M , there is an equivalence u and t ∈ M and an isomorphism β giving thefollowing diagram

d

t

��

u // a

w

��b v

//

s

??

c .

�

�

20

Let u be a pseudoinverse of u. We know from the first part of the proof that we have apseudosection k = t ◦ u of s, with an isomorphism s ◦ k ⇒ I. We then have the followingchain of isomorphisms:

j ◦w = (s ◦ v) ◦w ⇒ ((s ◦ v) ◦w) ◦ (s ◦ k)⇒ s ◦ ((v ◦ v) ◦ (t ◦ u))⇒ (s ◦ t) ◦u⇒ u ◦u⇒ I.

Hence all elements of M are equivalences.

6. 2-categories of internal categories admit bicategories of fractions

In this section we prove the result that Cat′(S) admits a calculus of fractions for the aclass of maps called E-equivalences, where E is a singleton pretopology on S.

The following is the first main theorem of the paper, and subsumes a number of other,similar theorems throughout the literature (see section 7 for details).

6.1. Theorem. Let S be a category with a singleton pretopology E. Assume the 2-category Cat′(S) admits base change along maps in E. Then Cat′(S) admits a rightcalculus of fractions for the class WE of E-equivalences.

Proof. We show the conditions of definition 5.2 hold.2CF1. An internal equivalence is clearly E-locally split. Lemma 3.17 gives us the rest.2CF2 a). That the composition of fully faithful functors is again fully faithful is trivial.

Consider the composition g ◦ f of two E-locally split functors,

Y [U ]

��

u

��

Z[V ]

��

v

��X

f// Y g

// Z�� ��

By lemma 3.13 the functor u pulls back to a functor Z[U ×Y0 V ] //Z[V ]. The compositeZ[U ×Y0 V ] // Z is fully faithful with object component in E, hence g ◦ f is E-locallysplit.

2CF2 b). Lemma 3.17 tells us that fully faithful functors are closed under isomorphism,so we just need to show E-locally split functors are closed under isomorphism.

Let w, f : X // Y be functors and a : w ⇒ f be a natural isomorphism. First, let wbe E-locally split. It is immediate from the diagram

Y [U ]

��

u

��X

w

''

f

77 Y

�

a��

21

That f is also E-locally split.2CF3 Let w : X // Y be an E-equivalence, and let f : Z // Y be a functor. From

the definition of E-locally split, we have the diagram

Y [U ]

��

u

��X w

// Y��

We can use lemma 3.13 to pull u back along f to get a 2-commuting diagram

Z[U ×Y0 Z0]v

%%xxY [U ]

��

u

!!

Z

fyy

X w// Y

�

with v ∈ WE as required.2CF4. Since E-equivalences are representably fully faithful, given

Yw

X

f==

g!!

⇓ a Z

Y

w

>>

where w ∈ WE, there is a unique a′ : f ⇒ g such that

Yw

X

f==

g!!

⇓ a Z

Y

w

>>= X

f

$$

g

::⇓ a′ Y w // Z .

The existence of a′ is the first half of 2CF4, where v = idX . If v′ : W // X ∈ WE suchthat there is a transformation

Xf

W

v′==

v′ !!

⇓ b Y

X

g

>>

22

satisfying

Xf

W

v′==

v′ !!

⇓ b Y w // Z

X

g

>>=

Yw

W v′ // X

f==

g!!

⇓ a Z

Y

w

>>

= Wv′ // X

f

$$

g

::⇓ a′ Yw // Z , (8)

then as v′ is anE-equivalence we have a functor u′ : X[U ] //W for some u = u′0 : U //X0 ∈E and a natural isomorphism ε:

X[U ]u′

||

u

""W

v′

99⇐ ε X

where u ∈ WE, and since v′ ◦ u′ '⇒ u, we have v′ ◦ u′ ∈ WE by 2CF2 a) above as requiredby 2CF4. The uniqueness of a′ from the earlier, together with equation (8) gives us

Xf

W

v′==

v′ !!

⇓ b Y

X

g

>>= W v′ // X

f

$$

g

::⇓ a′ Y .

We paste this with ε,

X[U ]

u

%%

u′ ""

ε ⇓ Xf

��W

v′>>

v′ !!

⇓ b Y

X

g

>>=

X[U ]

u′

��

u

��W

v′// X

f

$$

g

::⇓ a′ Y

� ,

which is precisely the diagram (7) with v = idX . Hence 2CF4 holds.

23

The second main result of the paper is that we want to know when this bicategory offractions is equivalent to a bicategory of anafunctors.

6.2. Theorem. Let (S,E) be a unary site, let J be a subcanonical pretopology cofinal inE and let Cat′(S) admit base change along arrows in E. Then there is an equivalence ofbicategories

Cat′ana(S, J) ' Cat′(S)[W−1E ]

Proof. Let us show the conditions in proposition 5.3 hold. First, αJ sends E-equivalencesto equivalences by proposition 5.4 and proposition 4.17.

EF1. αJ is the identity on 0-cells, and hence surjective on objects.EF2. This is equivalent to showing that for any anafunctor (U, f) : X−7→ Y there are

functors w, g such that w is in WE and

(U, f)∼⇒ αJ(g) ◦ αJ(w)−1

where αJ(w)−1 is some pseudoinverse for αJ(w).Let w be the functor X[U ] // X (this has object component in J ⊂ E, hence is an

E-equivalence) and let g = f : X[U ] // Y . First, note that

X[U ]

}}

=

##X X[U ]

is a pseudoinverse for

αJ(w) =

X[U ][U ]=

yy ##X[U ] X

.

Then the composition αJ(f) ◦ αJ(w)−1 is

X[U ×U U ×U U ]

ww ''X Y

which is isomorphic to (U, f) by the renaming transformation arising from the isomor-phism U ×U U ×U U ' U .

EF3. If a : (X0, f)⇒ (X0, g) is a transformation of anafunctors for functors f, g : X //Y ,it is given by a natural transformation

f ⇒ g : X ' X[X0 ×X0 X0] // Y.

Hence we get a unique natural transformation a : f ⇒ g such that a is the image of a′

under αJ .

24

We now give a series of results following from this theorem, using basic propertiesof pretopologies from section 2. The first, and most obvious observation, is that if E issubcanonical to start with, we can simply consider anafunctors using the pretopology E,and we get an equivalence

Cat′(S)[W−1E ] ' Cat′ana(S,E).

6.3. Corollary. Assume Cat′(S) and E are as in theorem 6.1. If E contains a co-final subcanonical pretopology J such that (S, J) satisfies WISC, then any localisationCat′(S)[W−1

Jun] is locally essentially small. Moreover, this localisation can be chosen such

that the class of objects if the same size as the class of objects of Cat′(S).

6.4. Corollary. When J and K are two subcanonical singleton pretopologies on S suchthat Jun = Kun, there is an equivalence of bicategories

Cat′ana(S, J) ' Cat′ana(S,K).

The class of maps in Top of the form∐Ui // X for an open cover {Ui} of X

form a singleton pretopology. This is because O is a superextensive pretopology (see theappendix).

Given a site with a superextensive pretopology J , we have the following result whichis useful when J is not a singleton pretopology.

6.5. Corollary. Let (S, J) be a superextensive site where J is a subcanonical pretopol-ogy. Then

Cat′(S)[W−1Jun

] ' Cat′ana(S,qJ).

Proof. This essentially follows from the corollary to lemma A.9.

Obviously this can be combined with previous results, for example if K ≤ qJ , for J anon-singleton pretopology, K-anafunctors localise Cat′(S) at the class of J-equivalences.

7. Examples

The easiest case to start with is that of a regular category with the canonical singletonpretopology, which is the class of regular epimorphisms. This is the setting of (Bunge-Pare). However they use an a priori different notion of weak equivalence, which is alsoused in (Everaert et al).

When S is finitely complete, Gpd(S) ↪→ Cat(S)(2,1) is a coreflective, full sub-2-category [Bunge-Pare, 1979], where C(2,1) denotes the maximal (2,1)-category underlyingthe 2-category C, with the coreflector an identity-on-objects 2-functor. Hence for everyinternal category X1 ⇒ X0 there is a maximal subobject X iso

1 ↪→ X1 such that X iso1 ⇒ X0

is an internal groupoid. The object X iso1 of isomorphisms can in this case be constructed

as a finite limit, and in the case when X is a groupoid then X iso1 ' X1.

25

7.1. Definition. [Bunge-Pare, 1979, Everaert et al, 2005] For a finitely complete unarysite (S,E) a functor f is called essentially E-surjective if the arrow labelled ~ below isin E.

X0 ×Y0 Y iso1

yy ~

��

��X0

f0��

Y iso1

syy

t%%

Y0 Y0

A functor is called a Bunge-Pare E-equivalence if it is fully faithful and essentially E-surjective. Denote by WBP

E .

We can relate E-equivalences as we have defined them in this paper and Bunge-PareE-equivalences as follows, where we work with the 2-categories Cat(S) and Gpd(S) andnot with other sub-2-categories.

7.2. Proposition. If (S,E) is a finitely complete unary site, then every Bunge-PareE-equivalence in Cat(S) is an E-equivalence. In particular, a WBP

E -inverting 2-functorCat(S) //D is a WE-inverting 2-functor. Likewise, for an arbitrary site (S,E), everyBunge-Pare E-equivalence in Gpd(S) is an E-equivalence, and WBP

E -inverting 2-functorGpd(S) //D is a WE-inverting 2-functor.

Proof. Let f : X // Y be a Bunge-Pare E-equivalence, and choose an E-cover U // Y0

and a local section (s0, ι) : U // X0 ×Y0 Y iso1 . The composite pr1 ◦ s will be the object

component of a functor s : Y [U ] //X, we need to define the arrow component. Considerthe composite

Y [U ]1 ' U ×Y0 Y1 ×Y0 U(s0,ι)×id×(−ι,s0)−−−−−−−−−−→ (X0 ×Y0 Y iso

1 )×Y0 Y1 ×Y0 (Y iso1 ×Y0 X0)

↪→ X0 ×Y0 Y3 ×Y0 X0id×m×id−−−−−→ X0 ×Y0 Y1 ×Y0 X0 ' X1

where the last isomorphism arises from f being full faithful. It is clear that this commuteswith source and target, because these are projection on the first and last factor at eachstep. To see that it respects identities and composition, just use the fact that the ιcomponent will cancel with the −ι component.

We define the natural isomorphism f ◦s⇒ j (here j : Y [U ] //Y is the canonical func-tor) as having component (WHAT’S THE RIGHT WORD HERE?) as having componentι as denoted above. Notice that the composite f1 ◦ f1 is just

Y [U ]1 ' U ×Y0 Y1 ×Y0 Uι×id×−ι−−−−−→ Y iso

1 ×Y0 Y1 ×Y0 Y iso1 ↪→ Y3

m−→ Y1.

Since the arrow component of Y [U ] // Y is U ×Y0 Y1 ×Y0 Upr2−−→ Y1, ι is indeed a natural

isomorphism using the diagram (2).

26

7.3. Corollary. Given a finitely complete unary site (S,E), there is an equivalence ofbicategories

Cat(S)[(WBPE )−1] ' Cat(S)[W−1

E ].

Proof. This follows using proposition 5.4.

We would also like to know when the class of Bunge-Pare weak equivalences is exactlythe class of weak equivalences. For this to be the case, we need to have a condition onthe pretopology under consideration.

7.4. Definition. A singleton pretopology E is called saturated if whenever the composite

Ah // B

g// C is in E, then g ∈ E.

7.5. Proposition. If (S,E) is a finitely complete unary site with E saturated then everyE-equivalence is a Bunge-Pare E-equivalence.

Proof.

7.6. Corollary. The canonical singleton pretopology on a finitely complete category issaturated. Hence WBP

E = WE for this site.

However, this result is not restricted to finitely complete categories.

We thus need to furnish examples of 2-categories Cat′(S) that admit base changealong some class of arrows. The following lemma give a sufficiency condition for this tobe so.

7.7. Lemma. Let CatM(S) be defined as the full sub-2-category of Cat(S) with objectsthose categories such that the source and target maps belong to a singleton pretopologyM. Then CatM(S) admits base change along arrows in M, as does the corresponding2-category of groupoids.

Proof. We shall denote arrows in M by for the purposes of the current proof.Let X be an object of CatM(S) and f : M X0 . In the following two diagrams, allthe squares are pullbacks

X[M ]1 //

s′

M

��

X[M ]1[rr]t′

��

X1 ×X0 M

��

M

��M ×X0 X1

// X1// X0 X1

��

X0

M // X0 M X0

The maps marked s′, t′ are the source and target maps for the base change along f , soX[M ] is in CatM(S). The same argument holds for groupoids verbatim.

27

In practice one often only wants base change along a subclass of M, such as (opencovers) ⊂ (open maps) as classes of maps in Top.

7.8. Example. In Top, the classes of covering maps, maps admitting local sections,surjective local homeomorphisms and open surjections are all examples of singleton pre-topologies. The results of [Pronk, 1996] pertaining to topological groupoids were carriedout using the site of open surjections.

7.9. Example. Consider the example of the 2-category of groupoid schemes X withsource and target maps etale (and satisfying some condition on (s, t) : X1

// X0 × X0,assumed to be a pullback-stable property). These correspond (up to details on (s, t)) toDeligne-Mumford stacks. By lemma 7.7 the base change along etale maps exists in the2-category of such groupoids.

7.10. Example. The prototypical example is that of Lie groupoids, where the sourceand target maps are required to be submersions. Restricting further we could consideretale Lie groupoids, where the source and target maps are local diffeomorphisms.

7.11. Example. If LieGpd denotes the 2-category of Lie groupoids, which are definedsuch that the source and target maps are submersions, then consider the following sub-2-categories:

• Proper Lie groupoids: those X where the map (s, t) : X1//X2

0 is proper,

• etale Lie groupoids: those X where the maps s, t : X1// X0 are local diffeomor-

phisms (i.e. etale),

• Proper etale Lie groupoids – also known as orbifolds [Moerdijk, 2002].

The third item in example 7.11 illustrates an important point: the intersection ofany two sub-2-categories of Cat(S) that satisfy our assumptions will also satisfy theassumptions.

7.12. Example. In [Noohi, 2005a] we find the definition of a class LF of local fibrations,which are maps in Top satisfying the properties:

1. Open embeddings are in LF,

2. LF is closed under composition,

3. LF is stable under pullback and local on the target,2

4. LF is stable under coproducts when considered as elements of a slice categoryTop/Y .

NEW VERSION OF AXIOMS

2A class of maps is local on the target if given an open cover U //Y , and if the projection U×Y X //Uis in that class, then X // Y is in that class.

28

1. LF is a singleton pretopology

2. closed under coproducts

3. containing the open embeddings

4. local on the target for open covers

COMMENTS

1. LF is very much like a superextensive pretopology

2. what’s the interplay between items 2 and 3?

We then consider the 2-categories CatLF (Top), GpdLF (Top) of topological categoriesand groupoids where the source and target maps belong to LF . Examples of classesLF include: open maps, local homeomorphisms, locally split maps, local Serre/Hurewiczfibrations.

Groupoids in GpdLF (Top) model topological stacks with varying properties, and alladmit base change along arrows of the form

∐Ui //X where {Ui} is an open cover of

X (see lemma 7.7).

These two examples have a ‘geometric’ flavour, but we are not restricted to examplesthat look like stacks. Since Grp is a Mal’cev category, categories in Grp are all groupoids,so we only consider groupoids in Grp. It is well known that groupoids in Grp modelpointed, connected 2-types, and π1(X) := Aut(1X0) and π0(X) := X0/X1 correspondto the second and first homotopy groups of the homotopy type corresponding to thegroupoid X. It is therefore an interesting question to consider models of localisations of2-types, for example at a set P of primes, or models of nilpotent 2-types. More generally,consider a class of groups C satisfying: if given an epimorphism of groups, p : G //H, thenG ∈ C if and only if H, ker(p) ∈ C. For example, we could take the classes of p-groups,finitely generated p-groups, nilpotent groups. We can then consider sub-2- categories ofGpd(Grp) where π1 and π0 are either elements of C, or are localised at C. We shall notpursue this further here.

7.13. Example. Let C be a Serre class in an abelian category A. Then consider the 2-category of groupoids X in A such that the kernel and cokernel of the associated crossedmodule s−1(0) //X0 are in C. One could denote the resulting 2-category by GpdC(S).When S = Ab, then these groupoids represent pointed connected 2-types with abelianfundamental group and homotopy groups in C.

One last example we consider is groupoid schemes, which correspond to algebraicstacks. Without going into details, algebraic stacks are stacks of groupoids

X : Schop //Gpd

on the category of schemes, satisfying two generic conditions:

29

1. There is a representable cover X // X , from a representable stack X, defined viasome pretopology J on Sch (in particular, it is a map belonging to a pullback stableclass) for example surjective etale or smooth maps;

2. The diagonal X //X ×X is representable and belongs to a class of maps which ispullback stable.

We gloss over the details about what it means for these various classes of maps to workand so on. The important point for us is that these two conditions imply that there is agroupoid X [2] := X ×X X ⇒ X internal to Sch where the source and target maps belongto J , and where (s, t) : X [2] //X ×X belongs to a pullback-stable class of maps. SinceSch is finitely complete, we do not need to worry about pullbacks existing, only aboutthe base change existing in the 2-category Cat′(Sch) we are interested in. (Note that thissort of base change is not the same as base change as considered in algebraic geometry –caveat lector.) Since (s, t) belongs to a pullback-stable class, this obviously presents noobstruction to the existence of base change along any map. The only thing we need tocheck is that source and target maps are what they need to be. See example 7.9.

A. Superextensive sites

The usual sites of topological spaces, manifolds and schemes all share a common property:one can (generally) take coproducts of covering families and end up with a coveringfamily. In this appendix we gather some results that generalise this fact, none of whichare especially deep, but help provide examples of bicategories of anafunctors.

A.1. Definition. [Carboni-Lack-Walters, 1993] A finitary (resp. infinitary) extensivecategory is a category with finite (resp. small) coproducts such that the following condi-tion holds: let I be a a finite set (resp. any set), then, given a collection of commutingdiagrams

Xi//

��

Z

��Ai //

∐i∈I Ai ,

one for each i ∈ I, the squares are all pullbacks if and only if the collection {Xi//Z}i∈I

forms a coproduct diagram.

In such a category there is a strict initial object (i.e. given a map A //0 with 0 initial,we have A ' 0).

A.2. Example. Top is infinitary extensive. Ringop, the category of affine schemes, isfinitary extensive.

30

In Top we can take an open cover {Ui}I of a space X and replace it with the singlemap

∐I Ui

//X, and work just as before using this new sort of cover, using the fact Topis extensive. The sort of sites that mimic this behaviour are called superextensive.

A.4. Definition. (Bartels-Shulman) A superextensive site is an extensive category Sequipped with a pretopology J containing the families

(Ui //∐I

Ui)i∈I

and such that all covering families are bounded; this means that for a finitely extensive site,the families are finite, and for an infinitary site, the families are small. The pretopologyin this instance will also be called superextensive.

A.5. Example. Given an extensive category S, the extensive pretopology has as coveringfamilies the bounded collections (Ui //

∐I Ui)i∈I . The pretopology on any superextensive

site contains the extensive pretopology.

A.6. Example. The category Top with its usual pretopology of open covers is a su-perextensive site.

A.7. Example. An elemantary topos with the coherent pretopology is finitary superex-tensive, and a Grothendieck topos with the canonical pretopology is infinitary superex-tensive.

Given a superextensive site (S, J), one can form the class qJ of arrows of the form∐I Ui

//A for covering families {Ui //A}i∈I in J (more precisely, all arrows isomorphicin S/A to such arrows).

A.8. Proposition. The class qJ is a singleton pretopology, and is subcanonical if andonly if J is.

Proof. Since identity arrows are covers for J they are covers for qJ . The pullback of aqJ-cover

∐I Ui

//A along B //A is a qJ-cover as coproducts and pullbacks commuteby definition of an extensive category. Now for the third condition we use the fact thatin an extensive category a map

f : B //∐I

Ai

implies thatB '∐

I Bi and f =∐

i fi. GivenqJ-covers∐

I Ui//A and

∐J Vj

//(∐

I Ui),we see that

∐J Vj '

∐IWi. By the previous point, the pullback∐

I

Uk ×∐I Ui′

Wi

is a qJ-cover of Ui, and hence (Uk ×∐I Ui′

Wi// Uk)i∈I is a J-covering family for each

k ∈ I. Thus(Uk ×∐

I Ui′Wi

// A)i,k∈I

31

is a J-covering family, and so

∐J

Vj '∐k∈I

(∐I

Uk ×∐I Ui′

Wi

)// A

is a qJ-cover.The map

∐I Ui

//A is the coequaliser of∐

I×I Ui ×A Uj ⇒∐

I Ui if and only if A is thecolimit of the diagram in definition 2.3. Hence (

∐I Ui

// A) is effective if and only if(Ui // A)i∈I is effective

Notice that the original superextensive pretopology J is generated by the union of qJand the extensive pretopology.

One reason we are interested in superextensive sites is the following (WHY??)

A.9. Lemma. In a superextensive site (S, J), we have Jun = (qJ)un.

One class of extensive categories which are of particular interest is those that also havefinite/small limits. These are called lextensive. For example, Top is infinitary lextensive,as is a Grothendieck topos. In contrast, an elementary topos is in general only finitarylextensive. We end with a lemma about WISC.

A.10. Lemma. If (S, J) is a superextensive site, (S, J) satisfies WISC if and only if(S,qJ) does.

References

O. Abbad, S. Mantovani, G. Metere, and E.M. Vitale, Butterflies are fractions of weakequivalences, Preprint, 2010. Available from http://users.mat.unimi.it/users/

metere/

P. Aczel, The type theoretic interpretation of constructive set theory, Logic Colloquium’77, Stud. Logic Foundations Math., vol. 96, North-Holland, 1978, pp. 55–66.

M. Barr and C. Wells, Toposes, triples and theories, Grundlehren der mathematischenwissenschaften, no. 278, Springer-Verlag, 1984, Version 1.1 available from http://

www.cwru.edu/artsci/math/wells/pub/ttt.html.

T. Bartels, Higher gauge theory I: 2-Bundles, Ph.D. thesis, University of California River-side, 2006, arXiv:math.CT/0410328.

J. Baez and A. Lauda, Higher dimensional algebra V: 2-groups, Theory and Appl. Categ.12 (2004), no. 14, 423–491.

J. Benabou, Les distributeurs, rapport 33, Universite Catholique de Louvain, Institut deMathematique Pure et Appliquee, 1973.

32

J. Benabou, Theories relatives a un corpus, C. R. Acad. Sci. Paris Ser. A-B 281 (1975),no. 20, Ai, A831–A834.

B. van den Berg, WISC is independent of ZF, Unpublished note, 2012. Available fromhttp://www.staff.science.uu.nl/∼berg0002/

B. van den Berg and I. Moerdijk, The Axiom of Mulitple Choice and models for construc-tive set theory, preprint (2012), [arXiv:1204.4045]

M. Breckes, Abelian metamorphosis of anafunctors in butterflies, Preprint, 2009.

M. Bunge and R. Pare, Stacks and equivalence of indexed categories, Cahiers TopologieGeom. Differentielle 20 (1979), no. 4, 373–399.

A. Carboni, S. Lack, and R. F. C. Walters, Introduction to extensive and distributivecategories, J. Pure Appl. Algebra 84 (1993), 145–158.

D. Carchedi, Compactly generated stacks: a cartesian-closed theory of topological stacks,Adv. Math. 229 (2009), 3339–33397. [arXiv:0907.3925]

H. Colman and C. Costoya, A Quillen model structure for orbifolds, preprint, 2009, Avail-able from http://faculty.ccc.edu/hcolman/.

T. tom Dieck, Klassifikation numerierbarer bundel, Arch. Math. 17 (1966), 395–399.

W. G. Dwyer and D. M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra17 (1980), no. 3, 267–284.

W. G. Dwyer and D. M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra18 (1980), no. 1, 17–35.

W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19(1980), no. 4, 427–440.

A. Dold, Partitions of unity in the theory of fibrations, Ann. Math. 78 (1963), no. 2,223–255.

C. Ehresmann, Categories topologiques et categories differentiables, Colloque Geom. Diff.Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, 1959, pp. 137–150.

C. Ehresmann, Categories structurees, Annales de l’Ecole Normale et Superieure 80(1963), 349–426.

T. Everaert, R.W. Kieboom, and T. van der Linden, Model structures for homotopy ofinternal categories, TAC 15 (2005), no. 3, 66–94.

P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag,1967.

33

M. Hilsum and G. Skandalis, Morphismes-orientes d‘epsaces de feuilles et fonctorialiteen theorie de Kasparov (d’apres une conjecture d’A. Connes), Ann. Sci. Ecole Norm.Sup. 20 (1987), 325–390.

P. Johnstone, Sketches of an elephant, a topos theory compendium, Oxford Logic Guides,vol. 43 and 44, The Clarendon Press Oxford University Press, 2002.

A. Joyal and M. Tierney, Strong stacks and classifying spaces, Category theory (Como,1990), Lecture Notes in Math., vol. 1488, Springer, 1991, pp. 213–236.

G.M. Kelly, Basic concepts of enriched category theory, Reprints in Theory and Applica-tion of Categories 10 (2005), vi+137, Reprint of the 1982 original [Cambridge Univ.Press].

N.P. Landsman, Bicategories of operator algebras and Poisson manifolds, Fields InstituteCommunications 30 (2001), 271–286.

E. Lerman, Orbifolds as stacks?, L’Enseign Math. (2) 56 (2010), no. 3-4, 315–363.[arXiv:0806.4160].

J. Lurie, (∞, 2)-categories and the Goodwillie Calculus I, Preprint, 2009. Available fromhttp://www.math.harvard.edu/∼lurie

S. MacLane and I. Moerdijk, Sheaves in geometry and logic, Springer-Verlag, 1992.

M. Makkai, Avoiding the axiom of choice in general category theory, JPAA 108 (1996),109–173.

J. Milnor, Construction of universal bundles II, Ann. Math. 63 (1956), 430–436.

I. Moerdijk, The classifying topos of a continuous groupoid I, Trans. Amer. Math. Soc.310 (1988), 629–668.

I. Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics(Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI,2002, pp. 205–222. [arXiv:math/0203100]

I. Moerdijk and J. Mrcun, Introduction to foliations and lie groupoids, Cambridge studiesin advanced mathematics, vol. 91, Cambridge University Press, 2003.

I. Moerdijk and J. Mrcun, Lie groupoids, sheaves and cohomology, Poisson geometry,deformation quantisation and group representations, London Math. Soc. Lecture NoteSeries, no. 323, Cambridge University Press, 2005, pp. 145–272.

B. Noohi, Foundations of topological stacks I, 2005, [arXiv:math.AG/0503247].

B. Noohi, On weak maps between 2-groups, 2005, [arXiv:math/0506313v2].

34

J. Pradines, Morphisms between spaces of leaves viewed as fractions, Cahiers TopologieGeom. Differentielle Categ. 30 (1989), no. 3, 229–246.

D. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996),no. 3, 243–303.

E. M. Vitale, Bipullbacks and calculus of fractions, Cah. Topol. Geom. Differ. Categ.51 (2010), no. 2, 83–113. Available from http://perso.uclouvain.be/enrico.

vitale/

School of Mathematical Sciences,University of AdelaideAdelaide, SA 5005Australia

Email: david.roberts@adelaide.edu.au