Internal categories and anafunctors

Transcription

1 CHAPTER 1 Internal cateories and anafunctors In this chapter we consider anafunctors [Mak96, Bar6] as eneralised maps between internal cateories [Ehr63], and show they formally invert fully faithful, essentially surjective functors (this localisation was developed in [Pro96] without anafunctors). To do so we need our ambient cateory S to be a site, to furnish us with a class of arrows that replaces the class of surjections in the case S = Set. The site comes with collections called coverin families, or covers, and ive meanin to the phrase essentially surjective when workin internal to S. A useful analoy to consider is when S = Top, and the coverin families are open covers in the usual way. In that settin, surjective is replaced by admits local sections, and the same is true for an arbitrary site - surjections are replaced by maps admittin local sections with respect to the iven class of covers. The class of such maps does not determine the covers with which one started, and we use this to our advantae. A superextensive site 1 is a one where out of each coverin family {U i A i I} we can form a sinle map I U i A, and use these as our covers. A maps admits local sections over the oriinal coverin family if and only if it admits sections over the new cover, and it is with these we can define anafunctors. Finally we show that different collections of covers will ive equivalent results if they ive rise to the same collection of maps admittin local sections. Most of the definitions in this chapter are standard. We draw without reference on the backround to bicateories collected in Appendix A. The material on anafunctors and localisin bicateories, althouh not new, do not seem to be widely known. Theorem 6.6 is new, but note analoues have appeared in the literature for Lie roupoids [] and étale Lie roupoids [Pro96] (which are not covered by this chapter), and for étale topoloical roupoids and alebraic roupoids (étale roupoids internal to schemes) [Pro96]. Theorem 6.8 and its corollaries are also new. 1. Internal cateories and roupoids Internal cateories were introduced by Ehresmann [Ehr63], startin with differentiable and topoloical cateories (i.e. internal to Diff and Top respectively). We collect here the necessary definitions and terminoloy without burdenin the reader with paes of diarams. For a thorouh recent account, see [BL4] or [Bar6]. Familiarity with basic cateory theory [Mac71] is assumed. Let S be a cateory with binary products and pullbacks. It will be referred to as the ambient cateory. 1 This concept is due to Toby Bartels and Mike Shulman 5

2 6 1. INTERNAL CATEGORIES AND ANAFUNCTORS Definition 1.1. An internal cateory X in a cateory S is a diaram X 1 X X 1 m X 1 s,t X e X 1 in S such that the multiplication m is associative, the unit map e is a two-sided unit for m and s and t are the usual source and taret. The pullback in the diaram is X 1 X X 1 X 1 t X 1 s X. This, and pullbacks like this (where source is pulled back alon taret), will occur often. If confusion can arise, the maps in question will be noted down, as in X 1 s,x,tx 1. Also, since multiplication is associative, there is a well-defined map X 1 X X 1 X X 1 X 1, which will also be denoted by m. It follows from the definition 2 that there is a subobject X1 iso X 1 throuh which e factors and an involution ( ) 1 : X1 iso X1 iso sendin arrows to their inverses such that the restriction of the structure maps to X1 iso make X1 iso X an internal cateory, and that ( ) 1 e = e. Often an internal cateory will be denoted X 1 X, the arrows m, s, t, e will be referred to as structure maps and X 1 and X called the objects of arrows and objects respectively. Remark 1.2. A very often used class of internal cateories is that of Lie roupoids (e.. [Mac5]). Since Diff doesn t have all pullbacks, modifications need to be made to the above definition. Since submersions admit pullbacks and are stable, s and t are assumed to be surjective submersions. Various other constructions involvin pullbacks later on in this chapter also need care, and there is an established literature on the subject. More enerally, one can consider internal cateory theory for ambient cateories without pullbacks, iven a class of maps analoous to submersion, but we will not do this in the present work. Example 1.3. If M is a monoid object in S and a: M X X is an action, there is a cateory M X X, called the action cateory, where the source and taret are projection and the action respectively. The subobject of invertible arrows is M X. In particular, consider the case when X is the terminal object (assumed to exist so as to define the unit of the monoid). Then such a cateory is precisely a monoid. Example 1.4. If X Y is an arrow in S admittin iterated kernel pairs, there is a cateory Č(X) with Č(X) = X, Č(X) 1 = X Y X, source and taret are projection on first and second factor, and the multiplication is projectin out the middle factor in X Y X Y X. The subobject of invertible arrows is all of Č(X) 1. 2 [BP79], but see [EKvdL5] for some more details.

3 1. INTERNAL CATEGORIES AND GROUPOIDS 7 A lot of interest in internal cateories is for definin stacks over the ambient cateory (once it has the structure of a site, for which see below), and specifically, stacks of roupoids. These lead to considerin internal roupoids as local models for the stack over the site (e.. [BP79] in the case of a reular, finitely complete cateory). Definition 1.5. If an internal cateory X has X iso 1 X 1, then it is called an internal roupoid. A lot of the terminoloy and machinery will be described here for internal cateories, even thouh most of the examples of interest are internal roupoids. Example 1.6. Let S be a cateory. For each object A S there is an internal roupoid disc(a) which has disc(a) 1 = disc(a) = A and all structure maps equal to id A. Such a cateory is called discrete. It oes without sayin that disc(a B) = disc(a) disc(b). If S has binary products, there is an internal roupoid codisc(a) with codisc(a) = A, codisc(a) 1 = A A and where source and taret are projections on the first and second factor respectively. The unit map is the diaonal and composition is projectin out the middle factor in codisc(a) 1 codisc(a) codisc(a) 1 = A A A. Such a roupoid is called codiscrete. Aain, we have codisc(a B) = codisc(a) codisc(b). Example 1.7. The codiscrete roupoid is obviously a special case of example 1.4, which is called the Čech roupoid of the map X Y. The oriin of the name is that in Top, for maps of the form I U i Y, the Čech roupoid Č( I U i) appears in the definition of Čech cohomoloy. Example 1.8. If G is a roup object in a cateory S with finite products, the roupoid BG has BG =, BG 1 = G. Example 1.9. If C is a cateory with a set of objects enriched in Top, then let C int = Obj(C) and C1 int = Obj(C) C(a, b). Then C int is a cateory internal to 2 Top. This example can be eneralised to monoidal cateories other than Top in which sufficient coproducts of the unit exist. Example 1.1. If X is a topoloical space which has a universal coverin space (i.e. is path-connected, locally path-connected and semilocally simply connected), then the fundamental roupoid Π 1 (X) can be made into a roupoid internal to Top. Definition Given internal cateories X and Y in S, and internal functor f : X Y is a pair of maps f : X Y f 1 : X 1 Y 1 called the object and arrow component respectively. The map f 1 restricts to a map f 1 : X1 iso Y1 iso and both components commute with all the structure maps. Example Given a homomorphism φ between monoids or roups, there is a functor between the cateories/roupoids in example 1.3. More enerally, iven an equivariant map between objects with an M-action, it ives rise to a functor between the associated action cateories.

4 8 1. INTERNAL CATEGORIES AND ANAFUNCTORS Example If A B is a map in S, there are functors disc(a) disc(b) and codisc(a) codisc(b). Example If A C and B C are maps admittin iterated kernel pairs, and A B is a map over C, there is a functor Č(A) Č(B). Example A map X Y in Top induces a functor Π 1 (X) Π 1 (Y ) (when these exist). Definition Given internal cateories X, Y and internal functors f, : X Y, an internal natural transformation (or simply transformation) a: f is a map a: X Y 1 such that s a = f, t a = and the followin diaram commutes X 1 (a t,f 1 ) ( 1,a s) Y 1 Y Y 1 Y 1 Y Y 1 m Y 1 expressin the naturality of a. If a factors throuh Y1 iso, then it is called a natural isomorphism. Clearly there is no distinction between natural transformations and natural isomorphisms when Y is an internal roupoid. We can reformulate the naturality diaram above in the case that a is a natural isomorphism. Denote by a the composite arrow X a Y iso 1 m ( ) 1 Y iso 1 Y 1. Then the above diaram commutin is equivalent to this diaram commutin (1) X X X 1 X X a f a X 1 which we will use repeatedly. Y 1 Y Y 1 Y Y 1 Example Let V ρ,v ρ be the action roupoids associated to representations ρ, ρ of G on V. They are iven by functors from G to GL(V ) as described in example A natural transformation between these functors is precisely an intertwiner. Example If X is a roupoid in S, A is an object of S and f, : X codisc(a) are functors, there is a natural isomorphism f. Internal cateories (resp. roupoids), functors and transformations form a 2-cateory Cat(S) (resp. Gpd(S)) [Ehr63]. There is clearly a 2-functor Gpd(S) Cat(S). Also, disc and codisc, described in examples 1.6 and 1.13 are 2-functors S Gpd(S), whose underlyin functors are left and riht adjoint to the functor Y 1 ( ) : Gpd 1 (S) S, (X 1 X ) X. m

5 2. SITES AND COVERS 9 Here Gpd 1 (S) is the cateory underlyin the 2-cateory Gpd(S). Hence for an internal cateory X in S, there are functors disc(x ) X and X codisc(x ), the latter sendin an arrow to the pair (source,taret). Definition An internal or stron equivalence of internal cateories is an equivalence in this 2-cateory: an internal functor f : X Y such that there is a functor f : Y X and natural isomorphisms f f id Y, f f id X. In all that follows, cateory will mean internal cateory in S and similarly for functor and natural transformation/isomorphism. We will not be considerin here the effect a functor S S between ambient cateories has on internal cateory theory. 2. Sites and covers All the material in this section is standard. Even thouh we are assumin our ambient cateory has pullbacks, a lot of the definitions are made for more eneral cateories. Definition 2.1. A Grothendieck pretopoloy (or simply pretopoloy) on a cateory S is a collection J of families {(U i A) i I } for each object A S satisfyin the followin properties (1) (id: A A) is in J for every object A. (2) Given a map B A, for every (U i A) i I in J the pullbacks B A A i exist and (B A A i B) i I is in J. (3) For every (U i A) i I in J and for a collection (Vk i U i) k Ki from J for each i I, the composites (Vk i A) k Ki,i I are in J. Families in J are called coverin families. A cateory S equipped with a pretopoloy is called a site, denoted (S, J). Example 2.2. The basic example is the lattice of open sets of a topoloical space, seen as a cateory in the usual way, where a coverin family of an open U X is an open cover of U by opens in X. This is to be contrasted with the pretopoloy O on Top, where the coverin families of a space are just open covers of the whole space. Example 2.3. On Grp the class of surjective homomorphisms form a pretopoloy. Example 2.4. On Top the class of numerable open covers (i.e. those that admit a subordinate partition of unity [Dol63]) form a pretopoloy. Much of traditional bundle theory is carried out usin this site, for example, the Milnor classifyin space classifies bundles which are locally trivial over numerable covers??. Definition 2.5. Let (S, J) be a site. The pretopoloy J is called a sinleton pretopoloy if every coverin family consists of a sinle arrow (U A). In this case a coverin family is called a cover.

6 1 1. INTERNAL CATEGORIES AND ANAFUNCTORS Example 2.6. In Top, the classes of coverin maps, local section admittin maps, surjective étale maps and open surjections are all examples of sinleton pretopoloies. The results of [Pro96] pertainin to topoloical roupoids were carried out usin the site of open surjections. Definition 2.7. A coverin family (U i A) i I is called effective if A is the colimit of the followin diaram: the objects are the U i and the pullbacks U i A U j, and the arrows are the projections U i U i A U j U j. If the coverin family consists of a sinle arrow (U A), this is the same as sayin U A is a reular epimorphism. Definition 2.8. A site is called subcanonical if every coverin family is effective. Example 2.9. On Top, the usual pretopoloy of opens, the pretopoloy of numerable covers and that of open surjections are subcanonical. Example 2.1. In a reular cateory, the reular epimorphisms form a subcanonical sinleton pretopoloy. In fact, the (pullback stable) reular epimorphisms in any cateory form the larest subcanonical topoloy, so it has its own name 3 Definition The canonical sinleton pretopoloy R is the class of all reular epimorphisms which are pullback stable. It contains all the subcanonical sinleton pretopoloies. Remark If U A is an effective cover, a functor Č(U) disc(b) ives a unique arrow A B. This follows immediately from the fact A is the colimit of Č(U). Definition A finitary (resp. infinitary) extensive cateory is a cateory with finite (resp. small) coproducts such that the followin condition holds: let I be a a finite set (resp. any set), then, iven a collection of commutin diarams x i z a i i I a i, one for each i I, the squares are all pullbacks if and only if the collection {x i z} I forms a coproduct diaram. In such a cateory there is a strict initial object (i.e. iven a map A, A ). Example Top is infinitary extensive. Example Rin op is finitary extensive. In Top we can take an open cover {U i } I of a space X and replace it with the sinle map I U i X, and work just as before usin this new sort of cover, usin the fact Top is extensive. The sort of sites that mimic this behaviour are called superextensive. 3 of course, the nomenclature was decided the other way around - subcanonical meanin contained in the canonical pretopoloy.

7 2. SITES AND COVERS 11 Definition (Bartels-Shulman) A superextensive site is an extensive cateory S equipped with a pretopoloy J containin the families (U i I U i ) i I and such that all coverin families are bounded. This means that for a finitely extensive site, the families are finite, and for an infinitary site, the families are small. Example Given an extensive cateory S, the extensive pretopoloy has as coverin families the bounded collections (U i I U i) i I. The pretopoloy on any superextensive site contains the extensive pretopoloy. Example The cateory Top with its usual pretopoloy of open covers is a superextensive site. Given a superextensive site, one can form the class J of arrows I U i A. Proposition The class J is a sinleton pretopoloy, and is subcanonical if and only if J is. Proof. Since identity arrows are covers for J they are covers for J. The pullback of a J-cover I U i A alon B A is a J-cover as coproducts and pullbacks commute by definition of an extensive cateory. Now for the third condition. we use the fact that in an extensive cateory a map f : B I A i implies that B I B i and f = i f i. Given J-covers I U i A and J V j ( I U i), we see that J V j I W i. By the previous point, the pullback U k I U W i i I is a J-cover of U i, and hence (U k I U i W i U k ) i I is a J-coverin family for each k I. Thus (U k I U i W i A) i,k I is a J-coverin family, and so V j ( ) U k I U W i i A J k I I is a J-cover. The map I U i A is the coequaliser of I I U i A U j I U i if and only if A is the colimit of the diaram in definition 2.7. Hence ( I U i A) is effective if and only if (U i A) i I is effective Notice that the oriinal pretopoloy J is enerated by the union of J and the extensive pretopoloy.

8 12 1. INTERNAL CATEGORIES AND ANAFUNCTORS Definition 2.2. Let (S, J) be a site. An arrow P A in S is a J-epimorphism (or simply J-epi) if there is a coverin family (U i A) i I and a lift U i P A for every i I. The class of J-epimorphisms will be denoted (J-epi). This definition is equivalent to the definition in III.7.5 in [MM92]. The dotted maps in the above definition are called local sections, after the case of the usual open cover pretopoloy on Top. If the pretopoloy is left unnamed, we will refer to local epimorphisms. One reason we are interested in superextensive sites is the followin Lemma If (S, J) is a superextensive site, the class of J-epimorphisms is precisely the class of J-epimorphisms. If S has all pullbacks then the class of J-epimorphisms form a pretopoloy. In fact they form a pretopoloy with an additional condition it is saturated. The followin is adapted from [BW84]: 4 Definition A sinleton pretopoloy K is saturated if whenever the composite V U A is in K, then U A is in K. In fact only a slihtly weaker condition on S is necessary for (J-epi) to be a pretopoloy. Example Let (S, J) be a site. If pullbacks of J-epimorphisms exist then the collection (J-epi) of J-epimorphisms is a saturated pretopoloy. There is a definition of saturated for arbitrary pretopoloies, but we will use only this one. Another way to pass from an arbitrary pretopoloy to a sinleton one in a canonical way is this: Definition The sinleton saturation of a pretopoloy on an arbitrary cateory S is the larest class J sat (J-epi) of those J-epimorphisms which are pullback stable. If J is a sinleton pretopoloy, it is clear that J J sat. In fact J sat contains all the coverin families of J with only one element when J is any pretopoloy. From lemma 2.21 we have Corollary In a superextensive site (S, J), the saturations of J and J coincide. One class of extensive cateories which are of particular interest is those that also have finite/small limits. These are called lextensive. For example, Top is infinitary 4 Note that in [BW84] what we are callin a Grothendieck pretopoloy, is called a Grothendieck topoloy.

9 3. WEAK EQUIVALENCES 13 lextensive, as is a Grothendieck topos. In contrast, a eneral topos is finitary lextensive. In a lextensive cateory J sat =( J) sat =(J-epi). Sometimes a pretopoloy J contains a smaller pretopoloy that still has enouh covers to compute the same J-epis. Definition If J and K are two sinleton pretopoloies with J K, such that K J sat, then J is said to be cofinal in K, denoted J K. Clearly J J sat. Lemma If J K, then J sat = K sat. 3. Weak equivalences For cateories internal to Set, equivalences are precisely those fully faithful, essentially surjective functors. For internal cateories, however, this is not the case.in addition, we need to make use of a pretopoloy to make the surjective part of essentially surjective meaninful. Definition 3.1. [BP79, EKvdL5] An internal functor f : X Y in a site (S, J) is called (1) fully faithful if (s,t) f 1 X 1 Y 1 (s,t) X X f f Y Y is a pullback diaram (2) essentially J-surjective if the arrow labelled is in (J-epi) X f X Y Y1 iso Y iso 1 s t Y Y (3) a J-equivalence if it is fully faithful and essentially J-surjective. The class of J-equivalences will be denoted W J, and if mention of J is suppressed, they will be called weak equivalences. Example 3.2. If X Y is an internal equivalence, then it is a J-equivalence for all pretopoloies J [EKvdL5]. In fact, if T denotes the trivial pretopoloy (only isomorphisms are covers) the T -equivalences are precisely the internal equivalences. Example 3.3. If J is a sinleton pretopoloy, and U A is a J-cover (or more enerally, is in J sat ), Č(U) disc(a) is a J-equivalence.

10 14 1. INTERNAL CATEGORIES AND ANAFUNCTORS Example 3.4. If f : X Y is a functor such that f is in (J-epi), then f is essentially J-surjective. A very important example of a J-equivalence requires a little set up. The strict pullback of internal cateories X Y Z Z X is the cateory with objects X Y Z, arrows X 1 Y1 Z 1, and all structure maps iven componentwise by those of X and Z. Definition 3.5. Let S be a cateory with binary products, X a cateory internal to S and p : M X an arrow in S. Define the induced cateory X[M] to be the strict pullback (2) X[M] X Y codisc(m) codisc(x ) with objects M and arrows M 2 X 2 X 1. The canonical functor in the top row has as object component p and is fully faithful. It follows immediately from the definition that iven maps M X, N M, there is a canonical isomorphism (3) X[M][N] X[N]. If we aree to follow the convention that M N N = M is the pullback alon the identity arrow id N, then X[X ]=X. This also simplifies other results of this chapter, 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 bracketins of a eneral iterated fibred product they are only canonically isomorphic. Example 3.6. If Č(B) is the Čech roupoid associated to a map j : B A in S, then disc(a)[b] Č(B). Of special interest is the case when j is a cover for some pretopoloy on S. Lemma 3.7. If (S, J) is a site, X a cateory in S and (U X ) is a coverin family, the functor X[U] X is a J-equivalence. Proof. The object component of the canonical functor X[U] X is U X and since it is in J it is in J sat. Hence X[U] X is a J-equivalence.

11 4. ANAFUNCTORS 15 Lemma 3.8. Let X be an internal cateory in S, and M X, N X arrows in S. Then the followin square is a strict pullback X[M X N] X[N] X[M] Proof. Consider the followin cube X[M X N] X X[N] X[M] X codisc(m X N) codisc(n) codisc(m) codisc(x ) The bottom and sides are pullbacks, either by definition, or usin (3), and so the top is a pullback. Fully faithful functors are stable under pullback, much like monomorphisms are. Lemma 3.9. If f : X Y is fully faithful, and : Z Y is any functor, ˆf in is fully faithful. Z Y X Proof. The followin chain of isomorphisms establishes the claim ˆf Z X Y (Z Y X ) 2 Z 2 Z 1 X 2 Y 2 Z 1 f the last followin from the fact f is fully faithful. (X 2 Y 2 Y 1 ) Y1 Z 1 X 1 Y1 Z 1, 4. Anafunctors Definition 4.1. [Mak96, Bar6] Let (S, J) be a site. An anafunctor in (S, J) from a cateory X to a cateory Y consists of a cover (U X ) and an internal functor f : X[U] Y.

12 16 1. INTERNAL CATEGORIES AND ANAFUNCTORS The anafunctor is a span in Cat(S), and will be denoted (U, f): X Y. Example 4.2. For an internal functor f : X Y in the site (S, J), define the anafunctor (X,f): X Y as the followin span X = X[X ] f Y. We will blur the distinction between these two descriptions. If f = id: X X, then (X, id) will be denoted simply by id X. Example 4.3. If U A is a cover in (S, J) and G is a roup object in S, an anafunctor (U, ): disc(a) BG is a Čech cocycle. Definition 4.4. [Mak96, Bar6] Let (S, J) be a site, (U, f), (V, ): X Y anafunctors in S. A transformation α: (U, f) (V, ) from (U, f) to (V, ) is an internal natural transformation X[U] X[U X V ] f α X[V ] Y If α: U X V Y 1 factors throuh Y iso 1, then α is called an isotransformation. In that case we say (U, f) is isomorphic to (V, ). Clearly all transformations between anafunctors between internal roupoids are isotransformations. Example 4.5. Given functors f, : X Y between cateories in S, and a natural transformation a: f, there is a transformation a: (X,f) (X,) of anafunctors, iven by X X X = X a Y 1. Example 4.6. If (U, ), (V, h): disc(a) BG are two Čech cocycles, a transformation between them is a coboundary on the cover U A V A. Example 4.7. Let (U, f): X Y be an anafunctor in S. There is an isotransformation 1 (U,f) :(U, f) (U, f) called the identity transformation, iven by the natural transformation with component (4) U X U (U U) X 2 X id 2 U e X[U] 1 f 1 Y1 Example 4.8. [Mak96] Given anafunctors (U, f): X Y and (V, f k): X Y where k : V U is an isomorphism over X,arenamin transformation (U, f) (V, f k) is an isotransformation with component 1 (U,f) (k id) : V X U U X U Y 1

13 4. ANAFUNCTORS 17 k will be referred to as a renamin isomorphism. More enerally, we could let k : V U be any refinement, and this prescription also ives an isotransformation (U, f) (V, f k). Example 4.9. As a concrete and relevant example of a renamin transformation we can consider the triple composition of anafunctors (U, f): X Y, (V, ): Y Z, (W, h): Z A. The two possibilities of composin these are ( (U Y V ) Z W, h (f V ) ) ( W, U Y (V Z W ),h W f ) V Z W The unique isomorphism (U Y V ) Z W U Y (V Z W ) commutin with the various projections is then the required renamin isomorphism. The isotransformation arisin from this renamin transformation is the associator. We define the composition of anafunctors as follows. Let (U, f): X Y,(V, ): Y Z be anafunctors in the site (S, J). Their composite (V, ) (U, f) is the composite span defined in the usual way. X[U Y V ] f V X[U] Y [V ] f X Y Z The pullback is as shown by lemma 3.8, and the resultin span is an anafunctor because V Y, and hence U Y V X, is a cover, and usin (3). We will sometimes denote the composite by (U Y V, f V ). Remark 4.1. If one doesn t impose the existence of pullbacks on S (as in say Diff, see comment??), this composite span still exists, because V Y is a cover Consider the special case when V = Y, and hence (Y,) is just an ordinary functor. Then there is a renamin transformation (the identity transformation!) (Y,) (U, f) (U, f), usin the equality U Y Y = U. If we let = id Y, then we see that (Y, id Y ) is a strict unit on the left for anafunctor composition. Similarly, considerin (V, ) (Y, id), we see that (Y, id Y ) is a two-sided strict unit for anafunctor composition. In fact, we have also proved Lemma Given two functors f : X Y, : Y Z in S, their composition as anafunctors is equal to their composition as functors: (Y,) (X,f) = (X, f). A simple but useful criterion for describin isotransformations where either of the anafunctors is a functor is as follows.

14 18 1. INTERNAL CATEGORIES AND ANAFUNCTORS Lemma An anafunctor (V, ): X Y is isomorphic to a functor f : X Y if and only if there is a natural isomorphism X[V ] X Y In a site (S, J) where the axiom of choice holds (that is, every epimorphism has a section), one can prove that every J-equivalence between internal cateories is in fact an internal equivalence of cateories. It is precisely the lack of splittins that prevents this theorem from holdin in eneral sites. The best one can do in a eneral site is described in the the followin two lemmas. Lemma Let f : X Y be a J-equivalence in (S, J), and choose a cover U Y and a local section s: U X Y Y1 iso. Then there is a functor Y [U] X with object component s := pr 1 s: U X. Proof. The object component is iven, we just need the arrow component. Denote the local section by (s,ι) :U X Y Y iso 1. Consider the composite Y [U] 1 U Y Y 1 Y U (s,ι) id ( ι,s ) (X Y Y iso 1 ) Y Y 1 Y (Y iso 1 Y X ) X Y Y 3 Y X f id m id X Y Y 1 Y X X 1 It is clear that this commutes with source and taret, because these are projection on the first and last factor at each step. To see that it respects identities and composition, just use the fact that the ι component will cancel with the ι component. Hence there is an anafunctor Y X, and the next proposition tells us this is a pseudoinverse to f (in a sense to be made precise in proposition 4.19 below). Lemma Let f : X Y be a J-equivalence in S. There is an anafunctor and isotransformations (U, f): Y X ι: (X,f) (U, f) id Y ɛ: (U, f) (X,f) id X Proof. We have the anafunctor (U, f) from the previous lemma. Since the anafunctors id X, id Y are actually functors, we can use lemma Usin the special case of anafunctor composition when the second is a functor, this tells us that ι will be iven by a natural isomorphism X f f Y [U] Y

15 4. ANAFUNCTORS 19 This has component ι: U Y iso 1, usin the notation from the proof of the previous lemma. Notice that the composite f 1 f 1 is just Y [U] 1 U Y Y 1 Y U ι id ι Y iso 1 Y Y 1 Y Y iso 1 Y 3 m Y 1. Since the arrow component of Y [U] Y is U Y Y 1 Y U pr 2 Y 1, ι is indeed a natural isomorphism usin the diaram (1). The other isotransformation is between (X Y U, f pr 2 ) and (X, id X ), and is iven by the arrow ɛ: X X X Y U X Y U id (s,a) X Y (X Y Y 1 ) X 2 Y 2 Y 1 X 1 This has the correct source and taret, as the object component of f is s, and the source is iven by projection on the first factor of X Y U. This diaram (X Y 2 U) 2 X 2 X 1 U Y X 1 Y U pr 2 X 1 ι f ι (X Y Y iso 1 ) Y Y 1 Y (Y iso 1 Y X ) id m id X Y Y 1 Y X commutes, and usin (1) we see that ɛ is natural. Just as there is composition of natural transformations between internal functors, there is a composition of transformations between internal anafunctors [Bar6]. This is where the effectiveness of our covers will be used in order to construct a map locally over some cover. Consider the followin diaram X[U X V X W ] X[U X V ] X[V X W ] a X[U] X[V ] b X[W ] f h Y from which we can form a natural transformation between the leftmost and the rihtmost composites as functors in S. This will have as its component the arrow ba: U X V X W id id U X V X V X W a b Y 1 Y Y 1 m Y 1 in S. Notice that the Čech roupoid of the cover (5) U X V X W U X W

16 2 1. INTERNAL CATEGORIES AND ANAFUNCTORS is U X V X V X W U X V X W, usin the two projections V X V V. Denote this pair of parallel arrows by s, t: UV 2 W UV W for brevity. In [Bar6] we find this commutin diaram (6) UV 2 W t UV W s UV W eba Y 1 and so we have a functor Č(U X V X W ) disc(y 1 ). Our pretopoloy J is assumed to be subcanonical, and usin remark 2.12 this ives us a unique arrow ba: U X W Y 1, the composite of a and b. Remark In the special case that U X V X W U X W is an isomorphism (or is even just split), the composite transformation has U X W U X V X W eba e ba Y 1 as its component arrow. In particular, this is the case if one of a or b is a renamin transformation. Example Let (U, f) :X Y be an anafunctor and U j U j U successive refinements of U X (e. isomorphisms). Let (U,f U ), (U,f U ) denote the composites of f with X[U ] X[U] and X[U ] X[U] respectively. The arrow U X U id U j j U X U Y 1 is the component for the composition of the isotransformations (U, f) (U,f U ), (U,f U ) described in example 4.8. Thus we can see that the composite of renamin transformations associated to isomorphisms φ 1,φ 2 is simply the renamin transformation associated to their composite φ 1 φ 2. Example If a: f, b: h are natural transformations between functors f,, h: X Y in S, their composite as transformations between anafunctors (X,f), (X,), (X,h): X Y. is just their composite as natural transformations. This uses the equality X X X X X = X X X = X. Theorem [Bar6] For a site (S, J) where J is a subcanonical sinleton pretopoloy, internal cateories (resp. roupoids), anafunctors and transformations form a bicateory AnaCat(S, J) (resp. Ana(S, J)). There is a strict 2-functor Ana(S, J) AnaCat(S, J) which is the identity on -cells and induces isomorphisms on hom-cateories. The followin is the main result of this section, and allows us to relate anafunctors to the localisations considered in the next section.

17 5. LOCALISING BICATEGORIES AT A CLASS OF 1-CELLS 21 Proposition There are strict 2-functors α J : Cat(S) AnaCat(S, J), β J = α Gpd(S) J : Gpd(S) Ana(S, J) sendin J-equivalences to equivalences such that Gpd(S) Cat(S) β J Ana(S, J) α J AnaCat(S, J) commutes. Proof. We define α J and β J to be the identity on objects, and as described in examples 4.2, 4.5 on 1-cells and 2-cells (i.e. functors and transformations). We need first to show that this ives a functor Cat(S)(X, Y ) AnaCat(S, J)(X, Y ). This is precisely the content of example Since the identity 1-cell on a cateory X in AnaCat(S, J) is the imae of the identity functor on S in Cat(S), α J and β J respect identity 1-cells. Also, lemma 4.11 tells us that α J and β J respect composition. That α J and β J send J-equivalences to equivalences is the content of lemma Localisin bicateories at a class of 1-cells Ultimately we are interestin in invertin all weak equivalences in Gpd(S), and so need to discuss what it means to add the formal pseudoinverses to a class of 1-cells in 2-cateory - a process known as localisation. This was done in [Pro96] for the more eneral case of a class of 1-cells in a bicateory, where the resultin bicateory is constructed and its universal properties (analoous to those of a quotient) examined. The application in loc. cit. is to showin the equivalence of various bicateories of stacks to localisations of 2-cateories of roupoids. The results of this chapter can be seen as one-half of a eneralisation of these results to an arbitrary site with pullbacks. Definition 5.1. Let E be a class of arrows in the ambient cateory S. E is called a class of admissible maps for J if it is a sinleton pretopoloy in which a iven sinleton pretopoloy J is cofinal, and satisfyin the followin condition: (S) E contains the split epimorphisms, and if e: A B is a split epimorphism, and A e B p C is in M, then p M. Example 5.2. If E is a saturated sinleton pretopoloy, it is a class of admissible maps for itself, and (J-epi) is a class of admissible maps for J (they satisfy condition (S) because they are saturated). A sinleton pretopoloy satisfyin condition (S) is a class of admissible maps for itself, and will just be referred to as a class of admissible maps. In particular, E could be the class of J-epimorphisms for a non-sinleton pretopoloy J.

18 22 1. INTERNAL CATEGORIES AND ANAFUNCTORS Definition 5.3. [EKvdL5] Let E be some class of admissible maps in a cateory S. A functor X Y in S is called an E-equivalence if it is fully faithful, and X Y Y iso 1 t pr 2 Y is in E. If this last condition holds we will say the functor is essentially E-surjective. If E =(J-epi) for some pretopoloy J, we will still refer to J-equivalences. The class of E-equivalences will be denoted W E. Definition 5.4. [Pro96] Let B be a bicateory and W B 1 a class of 1-cells. A localisation of B with respect to W is a bicateory 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 ives an equivalence of bicateories U : Hom(B[W 1 ],D) Hom W (B, D), where Hom W denotes the sub-bicateory of weak 2-functors that send elements of W to equivalences (call these W -invertin, abusin notation slihtly). The universal property means that W -invertin weak 2-functors F : B D factor, up to a transformation, throuh B[W 1 ], inducin an essentially unique weak 2-functor F : B[W 1 ] D. Definition 5.5. [Pro96] Let B be a bicateory B with a class W of 1-cells. W is said to admit a riht calculus of fractions if it satisfies the followin conditions 2CF1. W contains all equivalences 2CF2. a) W is closed under composition b) If a W and a iso-2-cell a b then b W 2CF3. For all w : A A, f : C A with w W there exists a 2-commutative square P A C v f with v W. 2CF4. If α: w f w is a 2-cell and w W there is a 1-cell v W and a 2-cell β : f v v such that α v = w β. Moreover: when α is an iso-2-cell, 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 iso-2-cell A w

19 5. LOCALISING BICATEGORIES AT A CLASS OF 1-CELLS 23 ɛ: v u v u such that the followin diaram commutes: (7) f v u β u v u f ɛ ɛ f v u β u v u Remark 5.6. In particularly nice cases (as in the next section), the first half of 2CF4 holds due to left-cancellability of elements of W, ivin us the canonical choice v = I. Theorem 5.7. [Pro96] A bicateory B with a class W that admits a calculus of riht fractions has a localisation with respect to W. From now on we shall refer to a calculus of riht fractions as simply a calculus of fractions, and the resultin localisation as a bicateory of fractions. Since B[W 1 ] is defined only up to equivalence, it is of reat interest to know when a bicateory D in which elements of W are converted to equivalences is itself equivalent to B[W 1 ]. In particular, one would be interested in findin such an equivalent bicateory with a simpler description than that which appears in [Pro96]. Thanks are due to Matthieu Dupont for pointin out (in personal communication) that the statement in loc. cit. actually only holds in one direction, as stated below, not in both. Proposition 5.8. [Pro96] A weak 2-functor F : B D which sends elements of W to equivalences induces an equivalence of bicateories F : B[W 1 ] D if the followin conditions hold EF1. F is essentially surjective, EF2. For every 1-cell f D 1 there is a w W and a B 1 such that F f Fw, EF3. F is locally fully faithful. The followin is useful in showin a weak 2-functor sends weak equivalences to equivalences, because this condition only needs to be checked on a class that is in some sense cofinal in the weak equivalences. Theorem 5.9. Let the bicateory B admit a calculus of fractions for W, and let V W be a class of 1-cells such that for all w W, there exists v V and s W such that there is an invertible 2-cell a s b c. v w

20 24 1. INTERNAL CATEGORIES AND ANAFUNCTORS Then a weak 2-functor F : B D that sends elements of V to equivalences also sends elements of W to equivalences. Proof. In the followin the coherence cells will be implicit. First we show that Fw has a pseudosection in C for any w W. Let v, s be as above. Let Fv be a pseudoinverse of Fv, and let j = Fs Fv. Then there is the followin invertible 2-cell Fw j F (w s) Fv Fv Fv I. We now show that j is in fact a pseudoinverse for Fw. Since s W, there is a v V and s W and a 2-cell ivin the followin diaram s v d a s w b c. v Apply the functor F, and denote pseudoinverses of F v, F v by F v, Fv. Usin the 2-cell I Fv Fv we et the followin 2-cell Fd Fv Fa Fs Fw Fb Then there is this composite invertible 2-cell j Fw (Fs Fv) (Fv (Fs Fv )) (Fs Fs ) Fv Fv Fv I, makin Fwis an equivalence. Hence F sends all elements of W to equivalences. Fv Fc 6. Anafunctors are a localisation In this section we present the new result that Cat(S) and Gpd(S) admit calculi of fractions for the weak equivalences, and the bicateory of anafunctors is an equivalent localisation. Definition 6.1. (see, e.. [EKvdL5]) The isomorphism cateory of an internal cateory X is the internal cateory denoted X I, with X I = X iso 1, X I 1 =(X 1 s,x,t X iso 1 ) X1 (X iso 1 s,x,t X 1 ). where the fibred product over X 1 arises by considerin the composition maps X 1 s,x,t X iso 1 X 1 X iso 1 s,x,t X 1 X 1.

21 6. ANAFUNCTORS ARE A LOCALISATION 25 Composition in X I is the same as commutative squares in the case of ordinary cateories. There are two functors s, t: X I X which have the usual source and taret maps of X as their respective object components. This construction is internal version of the functor cateory Cat(I,C), since the roupoid I =( ) doesn t always exist internal to S. Remark 6.2. There is an isomorphism X I 1 X iso 1 t,x,t X 1 s,x,t X iso 1 iven by projectin out the last factor in (X 1 s,x,t X iso 1 ) X1 (X iso 1 s,x,t X 1 ). The astute reader will reconise the followin as an internalisation of the usual notion of weak pullback Definition 6.3. The weak pullback X Y Z of a diaram of internal cateories X is iven by the pullback X Y,s Y I t,y Z. There is a 2-commutative square Z Y X Y Z Z X Y The followin terminoloy is adapted from [EKvdL5], althouh strictly speakin this map is only a fibration when model structure from loc. cit. exists. Definition 6.4. An internal functor f : X Y is called a trivial E-fibration if it is fully faithful and f E. Lemma 6.5. If a functor f : X Y is an E-equivalence, is a trivial E-fibration. X Y Y I t pr 2 Y Proof. The object component of t pr 2 is t pr 2, which is in E by definition if f is essentially E-surjective. Consider now the pullback (X Y Y iso 1 ) 2 Y 2 Y 1 Y 1 (X Y Y iso 1 ) 2 Y Y

22 26 1. INTERNAL CATEGORIES AND ANAFUNCTORS Remark 6.2 tells is that the pullback is isomorphic to X 2 Y 2 Y I 1 in the pullback X 2 Y 2 Y1 I pr 2 Y I 1 pr 1 Y 1 X 2 Y Y but if f is fully faithful, hence t pr 2 is fully faithful. X 2 Y 2 Y1 I X 2 Y 2 Y 1 Y1 Y1 I X 1 Y1 Y1 I, The internal cateory X Y Y I is called the mappin path space construction in [EKvdL5]. If the model structure in loc. cit. exists, the above follows from cofibration-acyclic fibration factorisation. Theorem 6.6. Let S be a cateory with pullbacks and a class E of admissible maps. The 2-cateories Cat(S) and Gpd(S) admit riht calculi of fractions for the class W E in each. Before we prove the theorem, we introduce a lemma Lemma 6.7. Let f, : X Y be functors and a: f a natural isomorphism. There is an isomorphism commutin with the projections to X 2. X 2 f 2,Y 2 Y 1 X 2 2,Y 2 Y 1 Proof. Supressin the canonical isomorphisms X 2 Y 2 Y 1 X Y Y 1 Y X, the required isomorphism is X f,y Y 1 Y,fX (id, a) id (a,id) id m id X,Y Y 1 Y Y 1 Y Y 1 Y,X X,Y Y 1 Y,X. which is the identity map when restricted to the X factors, from which the claim follows. Now the proof of theorem 6.6. Proof. We show the conditions of definition 5.5 hold. 2CF1. Since E contains all the split epis, an internal equivalence is essentially E- surjective. Let f : X Y be an internal equivalence, and : Y X a pseudoinverse. By definition there are natural isomorphisms a: f id X and b: f id Y. To show that f is fully faithful, we first show that the map q : X 1 X 2 Y 2 Y 1

23 6. ANAFUNCTORS ARE A LOCALISATION 27 is a split monomorphism over X 2. This diaram commutes X 1 X 2 Y 2 Y 1 X 1 X 2 f,x 2 X 1, by the naturality of a, the marked isomorphism comin from lemma 6.7. The splittin commutes with projection to X 2 because the isomorphism does. Call the splittin s. The same arument implies that Y 1 Y 2 X 2 X 1 is a split monomorphism over Y 2, and this implies the arrow l : X 2 Y 2 Y 1 X 2 Y 2 Y 2 X 2 X 1 X 2 f,x 2 X 1 is a split monomorphism. This diaram commutes X 2 Y 2 Y 1 l s X 1 q X 2 Y 2 Y 1 l X 2 f,x 2 X 1 X 1 X 2 f,x 2 X 1 X 1 usin naturality aain, and so q s = id. Thus q is an isomorphism, and f is fully faithful. 2CF2 a). That the composition of fully faithful functors is aain fully faithful is trivial. To show that the composition of essentially E-surjective functors f : X Y, : Y Z is aain so, consider the followin diaram Y Z Z 1 Z 1 t Z X Y Y 1 Y 1 t Y Z s X f Y s where the curved arrows are in E by assumption. The lower such arrow pulls back to an arrow X Y Y 1 Z Z 1 Y Z Z 1 (aain in E). Hence the composite is in E, and is equal to the composite The map X Y Y 1 Z Z 1 t pr X Y Y 1 Z Z 1 Y Z Z 2 1 Z id id id m t pr X Z Z 1 Z Z 1 X Z Z 2 1 Z. X Z Z 1 X Y Y Z Z 1 id e id X Y Y 1 Z Z 1

24 28 1. INTERNAL CATEGORIES AND ANAFUNCTORS is a section of X Y Y 1 Z Z 1 id id id m X Z Z 1 Z Z 1 X Z Z 1. t pr Now condition (S) tells us that X Z Z 2 1 Z is in E, and f is essentially E-surjective. 2CF2 b). We will show this in two parts: fully faithful functors are closed under isomorphism, and essentially E-surjective functors are closed under isomorphism. Let w, f : X Y be functors and a: w f be a natural isomorphism. First, let w be essentially E-surjective. That is, t pr (8) X w,y,s Y 2 1 Y is in E. Now note that the map (id, a) id id m (9) X f,y,s Y 1 X w,y,sy 1 t,y,s Y 1 X w,y,s Y 1 is an isomorphism, and so the composite of 9 and 8 is in E. Thus f is essentially E-surjective. Now let w be fully faithful. Thus w X 1 Y 1 X X Y Y is a pullback square. Usin lemma 6.7 there is an isomorphism X 1 X w,y Y 1 Y,w X X f,y Y 1 Y,f X. The composite of this with projection on X 2 is (s, t): X 1 X, 2 and the composite with pr 2 : X f,y Y 1 Y,f X Y 1 is just f 1 by the diaram 1, and so this diaram commutes X 1 X 2 f 2,Y 2 Y 1 Y 1 X 2 i.e. f is fully faithful. 2CF3. The existence of a 2-commutin square is easy: take the weak pullback (definition 6.3). Since the weak pullback of an E-equivalence is the strict pullback of a trivial E-fibration (usin lemma 6.5), we only need to show that the strict pullback of a trivial E-fibration is an E-equivalence. By lemma 3.9, the pullback of a trivial E-fibration is fully faithful. Since the object component of pulled back map is the pullback of the object component, which is in E, the pullback of the trivial E-fibration is aain a trivial E-fibration. f 2 Y 2

25 6. ANAFUNCTORS ARE A LOCALISATION 29 2CF4. It is proved in [Pro96] that iven a natural transformation X f Y w a Z Y w where w is fully faithful (e.. w is in W E ), there is a unique a : f such that X f Y w a Z Y w = X f a Y w Z. This is the first half of 2CF4, where v = id X. If v : W X W E such that there is a transformation satisfyin X v f W b Y v X X v f W b Y v X w Z = W v X f Y w a Z Y w f = W v X a Y w Z,

26 3 1. INTERNAL CATEGORIES AND ANAFUNCTORS we can choose a J-cover U X, a functor u : X[U] W and a natural isomorphism W X[U] u u ɛ X where, since J E, u W E, and since v u u, v u W E by 2CF2 a) above. We can apply the first step aain, usin uniquess to et We paste this with ɛ, X v f W b Y v X X[U] ɛ X v u W u f b Y v X v = W v X which is precisely the diaram (7). Hence 2CF4 holds. = X[U] f a Y, W v X If E is a class of admissible maps for J, E-equivalences are J-equivalences and so W E W J. This means that the 2-functors α J,β J in proposition 4.19 send E- equivalences to equivalences. We use this fact and proposition 5.8 to show the followin. Theorem 6.8. Let (S, J) be a site with a subcanonical sinleton pretopoloy J and let E be a class of admissible maps for J. Then there are equivalences of bicateories AnaCat(S, J) Cat(S)[W 1 E ] Ana(S, J) Gpd(S)[W 1 E ] Proof. Let us show the conditions in proposition 5.8 hold. We will only supply the details for α J, the same aruments clearly apply to β J. EF1. α J (and β J ) are the identity on -cells, and hence surjective. EF2. This is equivalent to showin that for any anafunctor (U, f): X Y there are functors w, such that w is in W E and (U, f) α J () α J (w) 1 ɛ f a Y,

27 6. ANAFUNCTORS ARE A LOCALISATION 31 where α J (w) 1 is some pseudoinverse for α J (w). Let w be the functor X[U] X this has object component in J E, hence an E-equivalence and let = f : X[U] Y. First, note that is a pseudoinverse for α J (w) = Then the composition α J (f) α J (w) 1 is X[U] = X X[U] X[U][U]. X[U] X X[U U U U U] X Y which is isomorphic to (U, f) by the renamin transformation arisin from the isomorphism U U U U U U. EF3. If a: (X,f) (X,) is a transformation of anafunctors for functors f, : X Y, it is iven by a natural transformation with component X X X Y 1. Simply precompose with the isomorphism X X X X to et a unique natural transformation a: f such that a is the imae of a under α J. We now finish on a series of results followin from this theorem, usin basic properties of pretopoloies from section 2. Corollary 6.9. When J and K are two subcanonical sinleton pretopoloies on S such that J sat = K sat, there is an equivalence of bicateories Ana(S, J) Ana(S, K) Usin corollary 6.9 we see that usin a cofinal pretopoloy ives an equivalent bicateory of anafunctors. If E is any class of admissible maps for subcanonical J, the bicateory of fractions invertin W E is equivalent to that of J-anafunctors. Hence Corollary 6.1. Let E be a class of admissible maps for the subcanonical pretopoloy J. There is an equivalence of bicateories Cat(S)[W 1 1 ] Cat(S)[W E where of course W J = W Jsat. The same result holds with Cat replaced by Gpd. J ]

28 32 1. INTERNAL CATEGORIES AND ANAFUNCTORS Finally, if (S, J) is a superextensive site (like Top with its usual pretopoloy of open covers), we have the followin result which is useful when J is not a sinleton pretopoloy. Corollary Let (S, J) be a superextensive site where J is a subcanonical pretopoloy. Then Gpd(S)[W 1 J sat ] Ana(S, J) Proof. This essentially follows from the corollary to lemma Obviously this can be combined with previous results, for example if K J, for J a non-sinleton pretopoloy, K-anafunctors localise Gpd(S) at the class of J- equivalences.