Internal categories, anafunctors and localisations

INTERNAL CA TEGORIES, ANAFUNCTORS AND LOCALISA TIONS In memory of Luanne Palmer (1965-2011) D A VID MICHAEL ROBER TS Abstract. In this article w e review the theory of anafunctors introduced by Makk ai and Bartels, and show that giv en a sub canonical site S , one can form a bicategorical lo calisation of v arious 2-categories of in ternal categories or groupoids at w eak equiv alences using anafunctors as 1-arro ws. This uniﬁes a n um ber of pro ofs throughout the literature, using the fewest assumptions p ossible on S . 1. In tro duction It is a w ell-kno wn classical result of category theory that a functor is an equiv alence (that is, in the 2-category of categories) if and only if it is fully faithful and essentially surjective. This fact is equiv alent to the axiom of choice. It is therefore not true if one is working with categories internal to a category S whic h do esn’t satisfy the (external) axiom of c hoice. This is may fail ev en in a category very m uc h lik e the category of sets, suc h as a w ell-p oin ted b o olean top os, or even the category of sets in constructive foundations. As internal categories are the ob jects of a 2-category Cat ( S ) we can talk ab out in ternal equiv alences, and ev en fully faithful functors. In the case S has a singleton pretop ology J (i.e. cov ering families consist of single maps) we can deﬁne an analogue of essen tially surjectiv e functors. In ternal functors whic h are fully faithful and essen tially surjective are called we ak e quivalenc es in the literature, going back to [ Bunge-P ar ´ e 1979 ]. W e shall call them J -equiv alences for clarity . W e can recov er the classical result men tioned ab o v e if we lo calise the 2-category Cat ( S ) at the class W J of J -equiv alences. W e are not just in terested in lo calising Cat ( S ), but v arious full sub-2-categories C  → Cat ( S ) which arise in the study of presentable stac ks, for example algebraic, top ological, diﬀeren tiable, etc. stac ks. As such it is necessary to ask for a compatibilit y condition b et w een the pretop ology on S and the sub-2-category we are interested in. W e call this condition existence of b ase change for co v ers of the pretoplogy , and demand that for any co v er p : U / / X 0 (in S ) of the ob ject of ob jects of X ∈ C , there is a fully faithful functor in C with ob ject comp onen t p . An Australian Postgraduate Award pro vided ﬁnancial supp ort during part of the time the material for this pap er was written. 2000 Mathematics Sub ject Classiﬁcation: Primary 18D99;Secondary 18F10, 18D05, 22A22. Key words and phrases: internal categories, anafunctors, lo calization, bicategory of fractions. c  Da vid Michael Rob erts, 2011. Permission to cop y for priv ate use granted. 1 2 1.1. Theorem . L et S b e a c ate gory with singleton pr etop olo gy J and let C b e a ful l sub-2-c ate gory of Cat ( S ) which admits b ase change along arr ows in J . Then C admits a c alculus of fr actions for the J -e quivalenc es. Pronk giv es us the appropriate notion of a calculus of fractions for a 2-category in [ Pronk 1996 ] as a generalisation of the usual construction for categories [ Gabriel-Zisman 1967 ]. In her construction, 1-arrows are spans and 2-arro ws are equiv alence classes of bicategorical spans of spans. This construction, while canonical, can b e a little un wieldy so w e lo ok for a simpler construction of the lo calisation. W e ﬁnd this in the notion of anafunctor , in tro duced b y Makk ai for plain small categories [ Makk ai 1996 ] (Kelly described them brieﬂy in [ Kelly 1964 ] but did not dev elop the concept further). In his setting an anafunctor is a span of functors suc h that the left (or source) leg is a surjective-on-ob jects, fully faithful functor. 1 F or a general category S with a sub c anonic al singleton pretop ology J [ Bartels 2006 ], the analogon is a span with left leg a fully faithful functor with ob ject component a co v er. Comp osition of anafunctors is giv en b y comp osition of spans in the usual wa y , and there are 2-arrows b et ween anafunctors (a certain sort of span of spans) that give us a bicategory Cat ana ( S, J ) with ob jects internal categories and 1-arro ws anafunctors. W e can also deﬁne the full sub-bicategory C ana ( J )  → Cat ana ( S, J ) analogous to C , and there is a strict inclusion 2-functor C  → C ana ( J ). This giv es us our second main theorem. 1.2. Theorem . L et S b e a c ate gory with sub c anonic al singleton pr etop olo gy J and let C b e a ful l sub-2-c ate gory of Cat ( S ) which admits b ase change along arr ows in J , Then C  → C ana ( J ) is a lo c alisation of C at the class of J -e quivalenc es. So far w e ha v en’t men tioned the issue of size, which usually is imp ortan t when constructing lo calisations. If the site ( S, J ) is lo cally small, then C is lo cally small, in the sense that the hom-categories are small. This also implies that C ana ( J ) and hence any C [ W − 1 J ] has lo c al ly small hom-categories i.e. has only a set of 2-arrows b et w een an y pair of 1-arro ws. T o pro v e that the lo cali sation is lo cally essentially small (that is, hom-categories are equiv alent to small categories), we need to assume a size restriction axiom on the pretop ology J , called WISC (W eakly Initial Sets of Co v ers). WISC can b e seen as an extremely weak c hoice principle, weak er than the existence of enough pro jectives, and states that for ev ery ob ject A of S , there is a set of J -co v ers of A whic h is coﬁnal in all J -co v ers of A . It is automatically satisﬁed if the pretop ology is sp eciﬁed as an assignmen t of a set of cov ers to each ob ject. 1.3. Theorem . L et S b e a c ate gory with sub c anonic al singleton pr etop olo gy J satisfying WISC, and let C b e a ful l sub-2-c ate gory of Cat ( S ) which admits b ase change along arr ows in J . Then any lo c alisation of C at the class of J -e quivalenc es is lo c al ly essential ly smal l. 1 Anafunctors were so named by Makk ai, on the suggestion of Pa vlo vic, after profunctors, in analogy with the pair of terms anaphase/prophase from biology . F or more on the relationship b et w een anafunctors and profunctors, see b elo w. 3 Since a singleton pretopology can be conv enien tly deﬁned as a certain wide sub category , this is not a v acuous statemen t for large sites, such as T op or Grp ( E ) (group ob jects in a top os E ). In fact WISC is indep enden t of the Zermelo-F raenk el axioms (without Choice) [ v an den Berg 2012 , Rob erts 2013 ]. It is thus p ossible to hav e the theorem fail for the top os S = Set ¬ AC with surjections as cov ers. Since there ha v e b een man y v ery closely related approac hes to lo calisation of 2- categories of internal categories and group oids, we give a brief sketc h in the following section. Sections 3 to 6 of this article then give necessary background and notation on sites, in ternal categories, anafunctors and bicategories of fractions resp ectiv ely . Section 7 contains our main results, while section 8 sho ws examples from the literature that are co v ered by the theorems from section 7. A short app endix detailing sup erextensiv e sites is included, as this material do es not app ear to b e well-kno wn (they w ere discussed in the recen t [Shulman 2012], Example 11.12). This article started out based on the ﬁrst c hapter of the author’s PhD thesis, which only dealt with group oids in the site of top ological spaces and op en cov ers. Man y thanks are due to Mic hael Murray , Mathai V arghese and Jim Stasheﬀ, sup ervisors to the author. The patrons of the n -Category Caf ´ e and n Lab, esp ecially Mik e Sh ulman and T ob y Bartels, pro vided helpful input and feedback. Stev e Lac k suggested a n um b er of improv emen ts, and the referee ask ed for a complete rewrite of this article, whic h has greatly impro v ed the theorems, pro ofs, and hop efully also the exp osition. Any dela ys in publication are due en tirely to the author. 2. Anafunctors in con text The theme of giving 2-categories of in ternal categories or group oids more equiv alences has b een approached in sev eral diﬀerent w a ys ov er the decades. W e sk etc h a few of them, without necessarily ﬁnding the original references, to giv e an idea of how widely the results of this pap er apply . W e giv e some more detailed examples of this applicabilit y in section 8. P erhaps the oldest related construction is the distributors of B´ enab ou, also known as mo dules or profunctors [ B ´ enab ou 1973 ] (see [ Johnstone 2002 ] for a detailed treatmen t of internal profunctors, as the original article is diﬃcult to source). B ´ enab ou p oin ted out [ B ´ enab ou 2011 ], after a preprin t of this article w as released, that in the case of the category Set (and more generally in a ﬁnitely complete site with reﬂexive co equalisers that are stable under pullback, see [ MMV 2012 ]), the bicategory of small (resp. internal) categories with representable profunctors as 1-arrows is equiv alen t to the bicategory of small categories with anafunctors as 1-arro ws. In fact this was discussed by Baez and Makk ai [ Baez-Makk ai 1997 ], where the latter p oin ted out that represen table profunctors corresp ond to satur ate d anafunctors in his setting. The author’s preference for anafunctors lies in the fact they can b e deﬁned with w eak er assumptions on the site ( S, J ), and in fact in the sequel [ Rob erts B ], do not require the 2-category to ha v e ob jects whic h are in ternal categories. In a sense this is analogous to [ Street 1980 ], where the formal bicategorical approac h to profunctors b et w een ob jects of a bicategory is giv en, alb eit still requiring 4 more colimits to exist than anafunctors do. B ´ enab ou has p oin ted out in priv ate communication that he has an unpublished distributor-lik e construction that do es not rely on existence of reﬂexiv e co equalisers; the author has not seen any details of this and is curious to see ho w it compares to anafunctors. Related to this is the original work of Bunge and Par ´ e [ Bunge-P ar ´ e 1979 ], where they consider functors b et w een indexed categories asso ciated to internal categories, that is, the externalisation of an in ternal category and stack completions thereof. This was one motiv ation for considering w eak equiv alences in the ﬁrst place, in that a pair of internal categories ha v e equiv alent stac k completed externalisations if and only if they are connected b y a span of internal functors which are weak equiv alences. Another approac h is constructing bicategories of fractions ` a la Pronk [ Pronk 1996 ]. This has b een follow ed by a num ber of authors, usually follow ed up by an explicit construction of a lo calisation simplifying the canonical one. Our work here sits at the more general end of this sp ectrum, as others hav e tailored their constructions to take adv antage of the structure of the site they are interested in. F or example, butterﬂies (originally called papillons) hav e been used for the category of groups [ No ohi 2005b , Aldro v andi-No ohi 2009 , Aldro v andi-No ohi 2010 ], ab elian categories [ Brec k es 2009 ] and semiab elian categories [ AMMV 2010 , MMV 2012 ]. These are similar to the meromorphisms of [ Pradines 1989 ], in tro duced in the con text of the site of smooth manifolds; though these only use a 1-categorical approach to lo calisation. Vitale [ Vitale 2010 ], after ﬁrst sho wing that the 2-category of group oids in a regular category has a bicategory of fractions, then sho ws that for protomo dular regular categories one can generalise the pullbac k congruences of B´ enab ou in [ B ´ enab ou 1989 ] to discuss bicategorical lo calisation. This approac h can b e applied to internal categories, as long as one restricts to in v ertible 2-arro ws. Similarly , [ MMV 2012 ] giv e a construction of what they call fr actors b et w een in ternal group oids in a Mal’tsev category , and show that in an eﬃciently regular category (e.g. a Barr-exact category) fractors are 1-arrows in a lo calisation of the 2-category of internal group oids. The pro of also works for internal categories if one considers only in v ertible 2-arrows. Other authors, in dealing with in ternal group oids, hav e adopted the approach pi- oneered by Hilsum and Sk andalis [ Hilsum-Sk andalis 1987 ], which has gone by v arious names including Hilsum-Sk andalis morphisms, Morita morphisms, bimo dules, bibundles, righ t principal bibundles and so on. All of these are v ery closely related to saturated anafunctors, but in fact no published deﬁnition of a saturated anafunctor in a site other than Set ([ Makk ai 1996 ]) has app eared, except in the guise of in ternal profunctors (e.g. [ Johnstone 2002 ], section B2.7). Note also that this approach has only b een applied to internal group oids. The review [ Lerman 2010 ] cov ers the case of Lie group oids, and in particular orbifolds, while [ Mr ˇ cun 2001 ] treats bimo dules b et w een group oids in the category of aﬃne schemes, but from the p oin t of view of Hopf algebroids. The link b et w een localisation at w eak equiv alences and presentable stacks is considered in (of course) [ Pronk 1996 ], as w ell as more recen tly in [ Carc hedi 2012 ], [ Sc h¨ appi 2012 ], in the cases of top ological and algebraic stac ks resp ectiv ely , and for example [ TXL-G 2004 ] 5 in the case of diﬀerentiable stac ks. A third approach is b y considering a mo del category structure on the 1-category of in ternal categories. This is considered in [ Jo y al-Tierney 1991 ] for categories in a top os, and in [ EKvdL 2005 ] for categories in a ﬁnitely complete sub canonical site ( S, J ). In the latter case the authors show when it is p ossible to construct a Quillen mo del category structure on Cat ( S ) where the weak equiv alences are the weak equiv alences from this pap er. Suﬃcien t conditions on S include b eing a top os with nno, b eing lo cally ﬁnitely presen table or b eing ﬁnitely complete regular Mal’tsev – and additionally having enough J -pro jective ob jects. If one is willing to consider other mo del-category-lik e structures, then these assumptions can b e dropp ed. The pro of from [ EKvdL 2005 ] can b e adapted to show that for a ﬁnitely complete site ( S, J ), the category of group oids with source and target maps restricted to b e J -co v ers has the structure of a category of ﬁbran t ob jects, with the same w eak equiv alences. W e note that [Colman-Costo y a 2009] giv es a Quillen mo del structure for the category of orbifolds, whic h are there deﬁned to b e proper top ological group oids with discrete hom-spaces. In a similar v ein, one could consider a localisation using hammo ck lo calisation [ Dwy er-Kan 1980a ] of a category of in ternal categories, whic h puts one squarely in the realm of ( ∞ , 1)-categories. Alternativ ely , one could work with the ( ∞ , 1)-category arising from a 2-category of internal categories, functors and natural isomorphisms and consider a lo calisation of this as giv en in, say [ Lurie 2009a ]. Ho w ev er, to deal with general 2-categories of in ternal categories in this wa y , one needs to pass to ( ∞ , 2)-categories to handle the non-inv ertible 2-arro ws. The theory here is not so w ell-dev elop ed, how ever, and one could see the results of the curren t pap er as giving to y examples with which one could work. This is one motiv ation for making sure the results shown in this pap er apply to not just 2-categories of group oids. Another is extending the theory of presen table stac ks from stac ks of group oids to stac ks of categories [Rob erts A]. 3. Sites The idea of surje ctivity is a necessary ingredien t when talking ab out equiv alences of categories—in the guise of just essential surjectivit y—but it do esn’t generalise in a straigh t- forw ard w a y from the category Set . The necessary prop erties of the class of surjective maps are enco ded in the deﬁnition of a Grothendiec k pretop ology , in particular a singleton pretop ology . This section gathers deﬁnitions and notations for later use. 3.1. Definition . A Gr othendie ck pr etop olo gy (or simply pr etop olo gy ) on a category S is a collection J of families { ( U i / / A ) i ∈ I } A ∈ Ob j( S ) of morphisms for each ob ject A ∈ S satisfying the follo wing prop erties 1. ( A 0 ∼ / / A ) is in J for every isomorphism A 0 ' A . 6 2. Giv en a map B / / A , for ev ery ( 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. F or ev ery ( U i / / A ) i ∈ I in J and for a collection ( V i k / / U i ) k ∈ K i from J for each i ∈ I , the family of comp osites ( V i k / / A ) k ∈ K i ,i ∈ I are in J . F amilies in J are called c overing families . W e call a category S equipp ed with a pretopology J a site , denoted ( S, J ) (note that often one sees a site deﬁned as a category equipp ed with a Grothendiec k top olo gy ). The pretop ology J is called a singleton pretop ology if ev ery cov ering family consists of a single arrow ( U / / A ). In this case a co v ering family is called a c over and we call ( S, J ) a unary site. V ery often, one sees the deﬁnition of a pretop ology as b eing an assignment of a set co v ering families to eac h ob ject. W e do not require this, as one can deﬁne a singleton pretop ology as a sub category with certain prop erties, and there is not necessarily then a set of co v ers for each ob ject. One example is the category of groups with surjectiv e homomorphisms as co v ers. This distinction will b e imp ortan t later. One thing w e will require is that sites come with sp e ciﬁe d pullbac ks of co v ering families. If one do es not mind applying the axiom of c hoice (resp. axiom of choice for classes) then an y small site (resp. large site) can b e so equipp ed. But often sites that arise in practice ha v e more or less canonical c hoices for pullbac ks, such as the category of ZF-sets. 3.2. Example . The protot ypical example is the pretopology O on T op , where a co v ering family is an op en co v er. The class of n umerable op en cov ers (i.e. those that admit a sub ordinate partition of unit y [ Dold 1963 ]) also forms a pretop ology on T op . Much of traditional bundle theory is carried out using this site; for example the Milnor classifying space classiﬁes bundles which are lo cally trivial ov er n umerable cov ers. 3.3. Definition . A co v ering family ( U i / / A ) i ∈ I is called eﬀe ctive if A is the colimit of the following diagram: the ob jects are the U i and the pullbac ks U i × A U j , and the arrows are the pro jections U i ← U i × A U j / / U j . If the co v ering family consists of a single arro w ( U / / A ), this is the same as saying U / / A is a regular epimorphism. 3.4. Definition . A site is called sub c anonic al if ev ery cov ering family is eﬀectiv e. 3.5. Example . On T op , the usual pretop ology O of opens, the pretop ology of n umerable co v ers and that of op en surjections are sub canonical. 7 3.6. Example . In a regular category , the class of regular epimorphisms forms a sub- canonical singleton pretop ology . In fact w e can make the following deﬁnition. 3.7. Definition . F or a category S , the largest class of regular epimorphisms of whic h all pullbacks exist, and whic h is stable under pullbac k, is called the c anonic al singleton pr etop olo gy and denoted c . This is a to b e con trasted to the canonical top olo gy on a category , whic h consists of co v ering siev es rather than cov ers. The canonical singleton pretop ology is the largest sub canonical singleton pretop ology on a category . 3.8. Definition . Let ( S, J ) b e a site. An arro w P / / A in S is called a J -epimorphism if there is a cov ering family ( U i / / A ) i ∈ I and a lift P   U i ? ? / / A for ev ery i ∈ I . A J -epimorphism is called universal if its pullbac k along an arbitrary map exists. W e denote the singleton pretop ology of univ ersal J -epimorphisms by J un . This deﬁnition of J -epimorphism is equiv alen t to the deﬁnition in I II.7.5 in [ Mac Lane-Mo erdijk 1992 ]. The dotted maps in the abov e deﬁnition are called lo cal sections, after the case of the usual op en co v er pretop ology on T op . If J is a singleton pretop ology , it is clear that J ⊂ J un . 3.9. Example . The univ ersal O -epimorphisms for the pretop ology O of op en cov ers on Diﬀ form S ubm , the pretop ology of surjective submersions. 3.10. Example . In a ﬁnitely complete category the universal tr iv -epimorphisms are the split epimorphisms, where tr iv is the trivial pr etop olo gy where all co v ering families consist of a single isomorphism. In Set with the axiom of choice there are all the epimorphisms. Note that for a ﬁnitely complete site ( S, J ), J un con tains tr iv un , hence all the split epimirphisms. Although we will not assume that all sites w e consider are ﬁnitely complete, results similar to ours ha v e, and so in that case we can sa y a little more, given stronger prop erties on the pretop ology . 3.11. Definition . A singleton pretop ology J is called satur ate d if whenev er the com- p osite A h / / B g / / C is in J , then g ∈ J . The concept of a saturated pretop ology was in troduced b y B´ enab ou under the name c alibr ation [ B ´ enab ou 1975 ]. It follo ws from the deﬁnition that a saturated singleton pretop ology con tains the split epimorphisms (take h to b e a section of the epimorphism g ). 8 3.12. Example . The canonical singleton pretop ology c in a regular category (e.g. a top os) is saturated. 3.13. Example . Given a pretop ology J on a ﬁnitely complete category , J un is saturated. Sometimes a pretop ology J con tains a smaller pretop ology that still has enough cov ers to compute the same J -epimorphisms. 3.14. Definition . If J and K are tw o singleton pretop ologies with J ⊂ K , such that K ⊂ J un , then J is said to b e c oﬁnal in K . Clearly J is coﬁnal in J un for any singleton pretop ology J . 3.15. Lemma . If J is c oﬁnal in K , then J un = K un . W e ha v e the follo wing lemma, which is essentially pro v ed in [ Johnstone 2002 ], C2.1.6. 3.16. Lemma . If a pr etop olo gy J is sub c anonic al, then so any pr etop olo gy in which it is c oﬁnal. In p articular, J sub c anonic al implies J un sub c anonic al. As mentioned earlier, one ma y b e giv en a singleton pretop ology such that each ob ject has more than a set’s worth of co v ers. If such a pretopology contains a coﬁnal pretop ology with set-many cov ers for eac h ob ject, then w e can pass to the smaller pretop ology and reco v er the same results (in a wa y that will b e made precise later). In fact, we can get a w a y with something w eak er: one could ask only that the category of all cov ers of an ob ject (see deﬁnition 3.18 b elo w) has a set of w eakly initial ob jects, and suc h set ma y not form a pretop ology . This is the conten t of the axiom WISC b elo w. W e ﬁrst giv e some more precise deﬁnitions. 3.17. Definition . A category C has a we akly initial set I of ob jects if for ev ery ob ject A of C there is an arrow O / / A from some ob ject O ∈ I . F or example the large category Fields of ﬁelds has a weakly initial set, consisting of the prime ﬁelds { Q , F p | p prime } . T o contrast, the category of sets with surjections for arro ws do esn’t ha v e a weakly initial set of ob jects. Every small category has a w eakly initial set, namely its set of ob jects. W e pause only to remark that the statemen t of the adjoint functor theorem can b e expressed in terms of weakly initial sets. 3.18. Definition . Let ( S, J ) b e a site. F or any ob ject A , the c ate gory of c overs of A , denoted J / A has as ob jects the cov ering families ( U i / / A ) i ∈ I and as morphisms ( U i / / A ) i ∈ I / / ( V j / / A ) j ∈ J tuples consisting of a function r : I / / J and arro ws U i / / V r ( i ) in S/ A . When J is a singleton pretop ology this is simply a full sub category of S/ A . W e no w deﬁne the axiom WISC (W eakly Initial Set of Co v ers), due indep enden tly to Mike Sh ulman and Thomas Streic her. 9 3.19. Definition . A site ( S, J ) is said to satisfy WISC if for every ob ject A of S , the category J / A has a w eakly initial set of ob jects. A site satisfying WISC is in some sense constrained b y a small amount of data for eac h ob ject. Any small site satisﬁes WISC, for example, the usual site of ﬁnite-dimensional smo oth manifolds and op en cov ers. An y pretop ology J con taining a coﬁnal pretop ology K such that K / A is small for every ob ject A satisﬁes WISC. 3.20. Example . An y regular category (for example a top os) with enough pro jectives, equipp ed with the canonical singleton pretop ology , satisﬁes WISC. In the case of Set ‘enough pro jectives’ is the Presen tation Axiom (P Ax), studied, for instance, b y Aczel [Aczel 1978] in the context of constructive set theory . 3.21. Example . [Sh ulman] ( T op , O ) satisﬁes WISC, using A C in Set . Choice may b e more than is necessary here; it would b e interesting to see if w eak er c hoice principles in the site ( Set , sur j ections ) are enough to prov e WISC for ( T op , O ) or other concrete sites. 3.22. Lemma . If ( S, J ) satisﬁes WISC, then so do es ( S , J un ) . It is instructiv e to consider an example where WISC fails in a non-artiﬁcial wa y . The category of sets and surjections with all arrows co v ers clearly do esn’t satisfy WISC, but is con triv ed and not a ‘useful’ sort of category . F or the moment, assume the existence of a Grothendiec k universe U with cardinality λ , and let Set U refer to the category of U -small sets. Clearly we can deﬁne WISC relative to U , call it WISC U . Let G b e a U -large group and B G the U -large group oid with one ob ject asso ciated to G . The b oolean top os Set B G U of U -small G -sets is a unary site with the class epi of epimorphisms for co v ers. One could consider this top os as b eing an exotic sort of forcing construction. 3.23. Pr oposition . If G has at le ast λ -many c onjugacy classes of sub gr oups, then (Set B G U , epi ) do es not satisfy WISC U . Alternativ ely , one could w ork in foundations where it is legitimate to discuss a prop er class-sized group, and then consider the top os of sets with an action by this group. If there is a prop er class of conjugacy classes of subgroups, then this top os with its canonical singleton pretop ology will fail to satisfy WISC. Simple examples of such groups are Z U (giv en a univ erse U ) and Z K (for some prop er class K ). Recen tly , [v an den Berg 2012] (relativ e to a large cardinal axiom) and [Rob erts 2013] (with no large cardinals) ha v e shown that the category of sets may fail to satisfy WISC. The mo dels constructed in [Karaglia 2012] are also conjectured to not satisfy WISC. P erhaps of indep enden t interest is a form of WISC with a b ound: the weakly initial set for eac h category J / A has cardinalit y less than some cardinal κ (call this WISC κ ). Then one could consider, for example, sites where each ob ject has a w eakly initial ﬁnite or coun table set of cov ers. Note that the condition ‘enough pro jectives’ is the case κ = 2. 10 4. In ternal categories In ternal categories w ere introduced in [ Ehresmann 1963 ], starting with diﬀerentiable and top ological categories (i.e. internal to Diﬀ and T op resp ectiv ely). W e collect here the necessary deﬁnitions, terminology and notation. F or a thorough recent accoun t, see [Baez-Lauda 2004] or the encyclop edic [Johnstone 2002]. Fix a category S , referred to as the ambient c ate gory . 4.1. Definition . An internal c ate gory X in a category S is a diagram X 1 × X 0 X 1 × X 0 X 1 ⇒ X 1 × X 0 X 1 m − → X 1 s,t ⇒ X 0 e − → X 1 in S suc h that the multiplic ation m is asso ciativ e (w e demand the limits in the diagram exist), the unit map e is a tw o-sided unit for m and s and t are the usual sour c e and tar get . An internal gr oup oid is an internal category with an inv olution ( − ) − 1 : X 1 / / X 1 satisfying the usual diagrams for an inv erse. Since m ultiplication is asso ciativ e, there is a w ell-deﬁned map X 1 × X 0 X 1 × X 0 X 1 / / X 1 , whic h will also b e denoted by m . The pullbac k in the diagram in deﬁnition 4.1 is X 1 × X 0 X 1 / /   X 1 s   X 1 t / / X 0 . and the double pullbac k is the limit of X 1 t → X 0 s ← X 1 t → X 0 s ← X 0 . These, and pullbac ks lik e these (where source is pulled back along target), will o ccur often. If confusion can arise, the maps in question will b e explicity written, as in X 1 × s,X 0 ,t X 1 . One usually sees the requirement that S is ﬁnitely complete in order to deﬁne internal categories. This is not strictly necessary , and not true in the well-studied case of S = Diﬀ , the category of smo oth manifolds. Often an in ternal category will b e denoted X 1 ⇒ X 0 , the arro ws m, s, t, e (and ( − ) − 1 ) will b e referred to as structur e maps and X 1 and X 0 called the ob ject of arrows and the ob ject of ob jects resp ectiv ely . F or example, if S = T op , we ha v e the space of arro ws and the space of ob jects, for S = Grp w e hav e the group of arrows and so on. 4.2. Example . If X / / Y is an arro w in S admitting iterated k ernel pairs, there is an in ternal groupoid ˇ C ( X ) with ˇ C ( X ) 0 = X , ˇ C ( X ) 1 = X × Y X , source and target are pro jection on ﬁrst and second factor, and the m ultiplication is pro jecting out the middle factor in X × Y X × Y X . This group oid is called the ˇ Ce ch gr oup oid of the map X / / Y . The origin of the name is that in T op , for maps of the form ` I U i / / Y (arising from an op en co v er), the ˇ Cec h group oid ˇ C ( ` I U i ) app ears in the deﬁnition of ˇ Cec h cohomology . 11 4.3. Example . Let S b e a category with binary pro ducts. F or eac h ob ject A ∈ S there is an in ternal group oid disc ( A ) which has disc ( A ) 1 = disc ( A ) 0 = A and all structure maps equal to id A . Such a category is called discr ete . There is also an in ternal group oid co disc( A ) with co disc( A ) 0 = A, co disc( A ) 1 = A × A and where source and target are pro jections on the ﬁrst and second factor resp ectiv ely . Suc h a group oid is called c o discr ete . 4.4. Definition . Giv en internal categories X and Y in S , an internal functor f : X / / Y is a pair of maps f 0 : X 0 / / Y 0 and f 1 : X 1 / / Y 1 called the ob ject and arrow component resp ectiv ely . Both comp onen ts are required to comm ute with all the structure maps. 4.5. Example . If A / / C and B / / C are maps admitting iterated kernel pairs, and A / / B is a map ov er C , there is a functor ˇ C ( A ) / / ˇ C ( B ). 4.6. Example . If ( S, J ) is a sub canonical unary site, and U / / A is a co v er, a functor ˇ C ( U ) / / disc ( B ) gives a unique arrow A / / B . This follo ws immediately from the fact A is the colimit of the diagram underlying ˇ C ( U ). 4.7. Definition . Giv en internal categories X , Y and internal functors f , g : X / / Y , an internal natur al tr ansformation (or simply tr ansformation ) a : f ⇒ g is a map a : X 0 / / Y 1 suc h that s ◦ a = f 0 , t ◦ a = g 0 and the follo wing diagram commutes X 1 ( g 1 ,a ◦ s ) / / ( a ◦ t,f 1 )   Y 1 × Y 0 Y 1 m   Y 1 × Y 0 Y 1 m / / Y 1 (1) expressing the naturalit y of a . In ternal categories (resp. group oids), functors and transformations in a lo cal ly small category S form a lo cally small 2-category Cat ( S ) (resp. Gp d ( S )) [ Ehresmann 1963 ]. There is clearly an inclusion 2-functor Gp d ( S ) / / Cat ( S ). Also, disc and co disc , describ ed in example 4.3 , are 2-functors S / / Gp d ( S ), whose underlying functors are left and righ t adjoin t to the functor Ob j : Cat ( S ) ≤ 1 / / S, ( X 1 ⇒ X 0 ) 7→ X 0 . Here Cat ( S ) ≤ 1 is the 1-category underlying the 2-category Cat ( S ). Hence for an internal category X in S , there are functors disc ( X 0 ) / / X and X / / co disc ( X 0 ), the arrow comp onen t of the latter b eing ( s, t ) : X 1 / / X 2 0 . 12 W e say a natural transformation is a natur al isomorphism if it has an in v erse with resp ect to vertical composition. Clearly there is no distinction b et w een natural transformations and natural isomorphisms when the co domain of the functors is an in ternal group oid. W e can reformulate the naturality diagram ( 1 ) in the case that a is a natural isomorphism. Denote b y − a the in v erse of a . Then the diagram ( 1 ) comm utes if and only if the diagram X 0 × X 0 X 1 × X 0 X 0 − a × f 1 × a / / '   Y 1 × Y 0 Y 1 × Y 0 Y 1 m   X 1 g 1 / / Y 1 (2) comm utes, a fact we will use several times. 4.8. Example . If X is a category in S , A is an ob ject of S and f , g : X / / co disc ( A ) are functors, there is a unique natural isomorphism f ∼ ⇒ g . 4.9. Definition . An internal or str ong e quivalenc e of in ternal categories is an equiv alence in the 2-category of internal categories. That is, an in ternal functor f : X / / Y suc h that there is a functor f 0 : Y / / X and natural isomorphisms f ◦ f 0 ⇒ id Y , f 0 ◦ f ⇒ id X . 4.10. Definition . F or an internal category X and a map p : M / / X 0 in S the b ase change of X along p is any category X [ M ] with ob ject of ob jects M and ob ject of arrows giv en by the pullback M 2 × X 2 0 X 1 / /   X 1 ( s,t )   M 2 p 2 / / X 2 0 If C ⊂ Cat ( S ) denotes a full sub-2-category and if the base change along any map in a giv en class K of maps exists in C for all ob jects of C , then we say C admits b ase change along maps in K , or simply admits b ase change for K . 4.11. Remark . In all that follows, ‘category’ will mean ob ject of C and similarly for ‘functor’ and ‘natural transformation/isomorphism’. The strict pullbac k of internal categories X × Y Z / /   Z   X / / Y when it exists, is the internal category with ob jects X 0 × Y 0 Z 0 , arrows X 1 × Y 1 Z 1 , and all structure maps giv en comp onen t wise by those of X and Z . Often w e will b e able to prov e that certain pullbacks exist b ecause of conditions on v arious component maps in S . W e do not assume that all strict pullbacks of internal categories exists in our c hosen C . 13 It follows immediately from deﬁnition 4.10 that given maps N / / M and M / / X 0 , there is a canonical isomorphism X [ M ][ N ] ' X [ N ] . (3) with ob ject comp onen t the identit y map, when these base changes exist. 4.12. Remark . If w e agree to follo w the con v en tion that M × N N = M is the pullback along the identit y arro w id N , then X [ X 0 ] = X . This also simpliﬁes other results of this pap er, so will b e adopted from now on. One consequence of this assumption is that the iterated ﬁbre pro duct M × M M × M . . . × M M , brac k eted in any order, is e qual to M . W e cannot, how ever, equate tw o brac k etings of a general iterated ﬁbred pro duct; they are only canonically isomorphic. 4.13. Lemma . L et Y / / X b e a functor in S and j 0 : U / / X 0 a map. If the b ase change along j 0 exists, the fol lowing squar e is a strict pul lb ack Y [ Y 0 × X 0 U ] / /   X [ U ] j   Y / / X assuming it exists. Pr oof . Since base change along j 0 exists, we kno w that w e hav e the functor Y [ Y 0 × X 0 U ] / / Y , we just need to sho w it is a strict pullback of j . On the lev el of ob jects this is clear, and on the level of arrows, w e hav e ( Y 0 × X 0 U ) 2 × Y 2 0 Y 1 ' U 2 × X 2 0 Y 1 ' ( U 2 × X 2 0 X 1 ) × X 1 Y 1 ' X [ U ] 1 × X 1 Y 1 so the square is a pullbac k. W e are interested in 2-categories C whic h admits base change for a given pretop ology J on S , which w e shall cov er in more detail in section 8. Equiv alences in Cat —assuming the axiom of c hoice—are precisely the fully faithful, essen tially surjective functors. F or internal categories, how ev er, this is not the case. In addition, we need to mak e use of a pretop ology to make the ‘surjectiv e’ part of ‘essen tially surjectiv e’ meaningful. 14 4.14. Definition . Let ( S, J ) b e a unary site. An in ternal functor f : X / / Y in S is called 1. ful ly faithful if X 1 f 1 / / ( s,t )   Y 1 ( s,t )   X 0 × X 0 f 0 × f 0 / / Y 0 × Y 0 is a pullbac k diagram; 2. J -lo c al ly split if there is a J -co v er U / / Y 0 and a diagram Y [ U ] ¯ f   u   X f / / Y   comm uting up to a natural isomorphism; 3. a J -e quivalenc e if it is fully faithful and J -lo cally split. The class of J -equiv alences will b e denoted W J . If mention of J is suppressed, they will b e called we ak e quivalenc es . 4.15. Remark . There is another deﬁn tion of full faithfulness for internal categories, namely that of a functor f : Z / / Y b eing r epr esentably ful ly faithful . This means that for all categories Z , the functor f ∗ : Cat ( S )( Z, X ) / / Cat ( S )( Z , Y ) is fully faithful. It is a w ell-kno wn result that these t w o notions coincide, so we shall use either characterisation as needed. 4.16. Lemma . If f : X / / Y is a ful ly faithful functor such that f 0 is in J , then f is J -lo c al ly split. That is, the canonical functor X [ U ] / / X is a J -equiv alence whenev er the base c hange exists. Also, w e do not require that J is sub canonical. W e record here a useful lemma. 4.17. Lemma . Given a ful ly faithful functor f : X / / Y in C and a natur al isomorphism f ⇒ g , the functor g is also ful ly faithful. In p articular, an internal e quivalenc e is ful ly faithful. Pr oof . This is a simple application of the deﬁnition of represen table full faithfulness and the fact that the result is true in Cat . 15 The ﬁrst deﬁnition of w eak equiv alence of internal categories along the lines w e are considering app eared in [ Bunge-P ar ´ e 1979 ] for S a regular category , and J the class of regular epimorphisms (i.e. c ), in the context of stacks and indexed categories. This was later generalised in [ EKvdL 2005 ] to more general ﬁnitely complete sites to discuss mo del structures on the category of internal categories. Both w ork only with saturated singleton pretop ologies. Note that when S is ﬁnitely complete, the ob ject X iso 1  → X 1 of isomorphisms of a category X can b e constructed as a ﬁnite limit [ Bunge-P ar ´ e 1979 ], and in the case when X is a group oid we ha v e X iso 1 ' X 1 . 4.18. Definition . [ Bunge-P ar ´ e 1979 , EKvdL 2005 ] F or a ﬁnitely complete unary site ( S, J ) with J saturated, a functor f is called essential ly J -surje ctive if the arro w lab elled ~ b elo w is in J . X 0 × Y 0 Y iso 1 y y ~     X 0 f 0   Y iso 1 s y y t % % Y 0 Y 0 A functor is called a Bunge-Par´ e J -e quivalenc e if it is fully faithful and essentially J - surjectiv e. Denote the class of such maps b y W B P J . Deﬁnition 4.14 is equiv alen t to the one in [ Bunge-P ar ´ e 1979 , EKvdL 2005 ] in the sites they consider but seems more appropriate for sites without all ﬁnite limits. Also, deﬁnition 4.14 makes sense in 2-categories other than Cat ( S ) or sub-2-categories thereof. 4.19. Pr oposition . L et ( S, J ) b e a ﬁnitely c omplete unary site with J satur ate d. Then a functor is a J -e quivalenc e if and only if it is a Bunge-Par´ e J -e quivalenc e. Pr oof . Let f : X / / Y b e a Bunge-P ar ´ e J -equiv alence, and consider the J -co v er given b y the map U := X 0 × Y 0 Y iso 1 / / Y 0 . Denote by ι : U / / Y iso 1 the pro jection on the second factor, by − ι the comp osite of ι with the inv ersion map ( − ) − 1 and by s 0 : U / / X 0 the pro jection on the ﬁrst factor. The arrow s 0 will b e the ob ject comp onen t of a functor s : Y [ U ] / / X , we need to deﬁne the arro w comp onen t s 1 . Consider the comp osite Y [ U ] 1 ' U × Y 0 Y 1 × Y 0 U ( s,ι ) × id × ( − ι,s ) − − − − − − − − − → ( X 0 × Y 0 Y iso 1 ) × Y 0 Y 1 × Y 0 ( Y iso 1 × Y 0 X 0 )  → X 0 × Y 0 Y 3 × Y 0 X 0 id × m × id − − − − − → X 0 × Y 0 Y 1 × Y 0 X 0 ' X 1 where the last isomorphism arises from f b eing fully faithful. It is clear that this commutes with source and target, b ecause these are given b y pro jection on the ﬁrst and last factor at each step. T o see that it resp ects identities and comp osition, one can use generalised elemen ts and the fact that the ι comp onen t will cancel with the − ι = ( − ) − 1 ◦ ι comp onen t. 16 W e deﬁne the natural isomorphism f ◦ s ⇒ j (here j : Y [ U ] / / Y is the canonical functor) to ha v e comp onen t ι as denoted ab o ve. Notice that the composite f 1 ◦ s 1 is just Y [ U ] 1 ' U × Y 0 Y 1 × Y 0 U ι × id ×− ι − − − − − → Y iso 1 × Y 0 Y 1 × Y 0 Y iso 1  → Y 3 m − → Y 1 . Since the arro w comp onen t of Y [ U ] / / Y is U × Y 0 Y 1 × Y 0 U pr 2 − − → Y 1 , ι is indeed a natural isomorphism using the diagram ( 2 ). Thus a Bunge-Par ´ e J -equiv alence is a J -equiv alence. In the other direction, given a J -equiv alence f : X / / Y , w e hav e a J -co v er j : U / / Y 0 and a map ( f , a ) : U / / X 0 × Y iso 1 suc h that j = ( t ◦ pr 2 ) ◦ ( f , a ). Since J is saturated, ( t ◦ pr 2 ) ∈ J and hence f is a Buge-Par ´ e J -equiv alence. W e can thus use deﬁnition 4.14 as we lik e, and it will still refer to the same sorts of w eak equiv alences that app ear in the literature. 5. Anafunctors W e no w let J b e a sub c anonic al singleton pretop ology on the am bien t category S . In this section we assume that C  → Cat ( S ) admits base c hange along arrows in the given pretop ology J . This is a slight generalisation of what is considered in [ Bartels 2006 ], where only C = Cat ( S ) is considered. 5.1. Definition . [ Makk ai 1996 , Bartels 2006 ] An anafunctor in ( S, J ) from a category X to a category Y consists of a J -cov er ( U / / X 0 ) and an internal functor f : X [ U ] / / Y . Since X [ U ] is an ob ject of C , an anafunctor is a span in C , and can b e denoted ( U, f ) : X − 7→ Y . 5.2. Example . F or an internal functor f : X / / Y in S , deﬁne the anafunctor ( X 0 , f ) : X − 7→ Y as the follo wing span X = ← − X [ X 0 ] f − → Y . W e will blur the distinction b et w een these tw o descriptions. If f = id : X / / X , then ( X 0 , id ) will be denoted simply b y id X . 5.3. Example . If U / / A is a co v er in ( S, J ) and B G is a group oid with one ob ject in S (i.e. a group in S ), an anafunctor ( U, g ) : disc ( A ) − 7→ B G is the same thing as a ˇ Cec h co cycle. 17 5.4. Definition . [Makk ai 1996, Bartels 2006] Let ( S , J ) b e a site and let ( U, f ) , ( V , g ) : X − 7→ Y b e anafunctors in S . A tr ansformation α : ( U, f ) ⇒ ( V , g ) from ( U, f ) to ( V , g ) is a natural transformation X [ U × X 0 V ] x x & & X [ U ] f & & α ⇒ X [ V ] g x x Y If α is a natural isomorphism, then α will b e called an isotr ansformation . In that case w e say ( U, f ) is isomorphic to ( V , g ). Clearly all transformations b et w een anafunctors b et w een in ternal group oids are isotransformations. 5.5. Example . Giv en functors f , g : X / / Y b et w een categories in S , and a natural transformation a : f ⇒ g , there is a transformation a : ( X 0 , f ) ⇒ ( X 0 , g ) of anafunctors, giv en by the comp onen t X 0 × X 0 X 0 = X 0 a − → Y 1 . 5.6. Example . If ( U, g ) , ( V , h ) : disc ( A ) − 7→ B G are tw o ˇ Cec h co cycles, a transformation b et w een them is a cob oundary on the co v er U × A V / / A . 5.7. Example . Let ( U, f ) : X − 7→ Y b e an anafunctor in S . There is an isotransfor- mation 1 ( U,f ) : ( U, f ) ⇒ ( U, f ) called the identity tr ansformation , giv en by the natural transformation with comp onen t U × X 0 U ' ( U × U ) × X 2 0 X 0 id 2 U × e − − − → X [ U ] 1 f 1 − → Y 1 (4) 5.8. Example . [ Makk ai 1996 ] Giv en anafunctors ( U, f ) : X / / Y and ( V , f ◦ k ) : X / / Y where k : V / / U is a cov er (o v er X 0 ), a r enaming tr ansformation ( U, f ) ⇒ ( V , f ◦ k ) is an isotransformation with comp onen t 1 ( U,f ) ◦ ( k × id) : V × X 0 U / / U × X 0 U / / Y 1 . (W e also call its inv erse for vertical comp osition a renaming transformation.) If k is an isomorphism, then it will itself b e referred to as a r enaming isomorphism . 18 W e deﬁne (following [Bartels 2006]) the comp osition of anafunctors as follows. Let ( U, f ) : X − 7→ Y and ( V , g ) : Y − 7→ Z b e anafunctors in the site ( S, J ). Their comp osite ( V , g ) ◦ ( U, f ) is the comp osite span deﬁned in the usual wa y . It is again a span in C : X [ U × Y 0 V ] x x f V & & X [ U ] | | f & & Y [ V ] x x g ! ! X Y Z The square is a pullbac k b y lemma 4.13 (whic h exists b ecause V / / Y 0 is a cov er), and the resulting span is an anafunctor b ecause V / / Y 0 , hence U × Y 0 V / / X 0 , are co v ers, and using the isomorphism ( 3 ). W e will sometimes denote the comp osite by ( U × Y 0 V , g ◦ f V ). Here we are using the fact w e hav e sp eciﬁed pullbacks of co v ers in S . Without this w e w ould not end up with a bicategory (see theorem 5.16 ), but what [ Makk ai 1996 ] calls an anabic ate gory . This is similar to a bicategory , but comp osition and other structural maps are only anafunctors, not functors. Consider the sp ecial case when V = Y 0 , so that ( Y 0 , g ) is just an ordinary functor. Then there is a renaming transformation (the iden tit y transformation!) ( Y 0 , g ) ◦ ( U, f ) ⇒ ( U, g ◦ f ), using the equality U × Y 0 Y 0 = U (b y remark 4.12 ). If w e let g = id Y , then w e see that ( Y 0 , id Y ) is a strict unit on the left for anafunctor comp osition. Similarly , considering ( V , g ) ◦ ( Y 0 , id ), we see that ( Y 0 , id Y ) is a t w o-sided strict unit for anafunctor comp osition. In fact, w e hav e also prov ed 5.9. Lemma . Given two functors f : X / / Y , g : Y / / Z in S , their c omp osition as anafunctors is e qual to their c omp osition as functors: ( Y 0 , g ) ◦ ( X 0 , f ) = ( X 0 , g ◦ f ) . As a concrete and relev ant example of a renaming transformation w e can consider the triple comp osition of anafunctors ( U, f ) : X − 7→ Y , ( V , g ) : Y − 7→ Z , ( W , h ) : Z − 7→ A. The tw o p ossibilities of comp osing these are  ( U × Y 0 V ) × Z 0 W , h ◦ ( g f V ) W  and  U × Y 0 ( V × Z 0 W ) , h ◦ g W ◦ f V × Z 0 W  . 19 5.10. Lemma . The unique isomorphism ( U × Y 0 V ) × Z 0 W ' U × Y 0 ( V × Z 0 W ) c ommuting with the various pr oje ctions is a r enaming isomorphism. The isotr ansformation arising fr om this r enaming tr ansformation is c al le d the associator . A simple but useful criterion for describing isotransformations where one of the ana- functors inv olved is a functor is as follows. 5.11. Lemma . A n anafunctor ( V , g ) : X − 7→ Y is isomorphic to a functor ( X 0 , f ) : X − 7→ Y if and only if ther e is a natur al isomorphism X [ V ] } } g ! ! X f 9 9 ∼ ⇒ Y Just as there is a v ertical comp osition of natural transformations b et w een in ternal functors, there is a vertical comp osition of transformations b et ween in ternal anafunctors [ Bartels 2006 ]. This is where the sub canonicit y of J will b e used in order to construct a map lo cally ov er some cov er. Consider the follo wing diagram X [ U × X 0 V × X 0 W ] u u ) ) X [ U × X 0 V ] y y ) ) X [ V × X 0 W ] u u & & X [ U ] f + + a ⇒ X [ V ] g   b ⇒ X [ W ] h s s Y W e can form a natural transformation b et w een the leftmost and the rightmost comp osites as functors in S . This will ha v e as its comp onen t the arro w e ba : U × X 0 V × X 0 W id × ∆ × id − − − − − → U × X 0 V × X 0 V × X 0 W a × b − − → Y 1 × Y 0 Y 1 m − → Y 1 in S . Notice that the ˇ Cec h group oid of the co v er U × X 0 V × X 0 W / / U × X 0 W (5) is U × X 0 V × X 0 V × X 0 W ⇒ U × X 0 V × X 0 W , 20 with source and target arising from the tw o pro jections V × X 0 V / / V . Denote this pair of parallel arro ws by s, t : U V 2 W ⇒ U V W for brevity . In [ Bartels 2006 ], section 2.2.3, we ﬁnd the comm uting diagram U V 2 W t / / s   U V W e ba   U V W e ba / / Y 1 (6) (this can b e c hec k ed b y using generalised elements) and so w e hav e a functor ˇ C ( U × X 0 V × X 0 W ) / / disc( Y 1 ) . Our pretop ology J is assumed to b e sub canonical, so example 4.6 giv es us a unique arrow ba : U × X 0 W / / Y 1 , which is the data for the comp osite of a and b . 5.12. Remark . In the sp ecial case that U × X 0 V × X 0 W / / U × X 0 W is split (e.g. is an isomorphism), the comp osite transformation has U × X 0 W / / U × X 0 V × X 0 W e ba − → Y 1 as its comp onen t arrow. In particular, this is the case if one of a or b is a renaming transformation. 5.13. Example . Let ( U, f ) : X − 7→ Y b e an anafunctor and U 00 j 0 − → U 0 j − → U successiv e reﬁnemen ts of U / / X 0 (e.g isomorphisms). Let ( U 0 , f U 0 ) and ( U 00 , f U 00 ) denote the comp osites of f with X [ U 0 ] / / X [ U ] and X [ U 00 ] / / X [ U ] resp ectiv ely . The arrow U × X 0 U 00 id U × j ◦ j 0 − − − − − → U × X 0 U / / Y 1 is the comp onen t for the composition of the isotransformations ( U, f ) ⇒ ( U 0 , f U 0 ) , ⇒ ( U 00 , f U 00 ) describ ed in example 5.8 . Thus we can see that the comp osite of renaming transformations asso ciated to isomorphisms φ 1 , φ 2 is simply the renaming transformation asso ciated to their comp osite φ 1 ◦ φ 2 . This can be used to sho w that the asso ciator satisﬁes the necessary coherence conditions. 5.14. Example . If a : f ⇒ g , b : g ⇒ h are natural transformations b et w een functors f , g , h : X / / Y in S , their comp osite as transformations b et w een anafunctors ( X 0 , f ) , ( X 0 , g ) , ( X 0 , h ) : X − 7→ Y . is just their comp osite as natural transformations. This uses the equalit y X 0 × X 0 X 0 × X 0 X 0 = X 0 × X 0 X 0 = X 0 , whic h is due to our c hoice in remark 4.12 of canonical pullbac ks. Ev en though we don’t hav e pseudoinv erses for w eak equiv alences of in ternal categories, one might guess that the lo cal splitting guaranteed to exist b y deﬁnition is actually more than just a splitting of sorts. This is in fact the case, if we use anafunctors. 21 5.15. Lemma . L et f : X / / Y b e a J -e quivalenc e in S . Ther e is an anafunctor ( U, ¯ f ) : Y − 7→ X and isotr ansformations ι : ( X 0 , f ) ◦ ( U, ¯ f ) ⇒ id Y  : ( U, ¯ f ) ◦ ( X 0 , f ) ⇒ id X Pr oof . W e hav e the anafunctor ( U, ¯ f ) by deﬁnition as f is J -lo cally split. Since the anafunctors id X , id Y are actually functors, w e can use lemma 5.11 . Using the sp ecial case of anafunctor comp osition when the second is a functor, this tells us that ι will b e given b y a natural isomorphism X f   Y [ U ] / / ¯ f < < Y   with comp onen t ι : U / / Y 1 . Notice that the comp osite f 1 ◦ ¯ f 1 is just Y [ U ] 1 ' U × Y 0 Y 1 × Y 0 U ι × id ×− ι − − − − − → Y 1 × Y 0 Y 1 × Y 0 Y 1  → Y 3 m − → Y 1 . Since the arro w comp onen t of Y [ U ] / / Y is U × Y 0 Y 1 × Y 0 U pr 2 − − → Y 1 , ι is indeed a natural isomorphism using the diagram (2). The other isotransformation  is b et w een ( X 0 × Y 0 U, ¯ f ◦ pr 2 ) and ( X 0 , id X ), and is giv en b y the comp onen t  : X 0 × X 0 X 0 × Y 0 U = X 0 × Y 0 U id × ( ¯ f 0 ,ι ) − − − − − → X 0 × Y 0 ( X 0 × Y 0 Y 1 ) ' X 2 0 × Y 2 0 Y 1 ' X 1 The diagram ( X 0 × Y 2 0 U ) 2 × X 2 0 X 1 '   pr 2 / / X 1 '   U × Y 0 X 1 × Y 0 U − ι × f × ι   ( X 0 × Y 0 Y 1 ) × Y 0 Y 1 × Y 0 ( Y 1 × Y 0 X 0 ) id × m × id / / X 0 × Y 0 Y 1 × Y 0 X 0 comm utes (a fact which can b e chec k ed using generalised elemen ts), and using ( 2 ) we see that  is natural. 22 The ﬁrst half of the follo wing theorem is prop osition 12 in [ Bartels 2006 ], and the second half follo ws b ecause all the constructions of categories in v olv ed in dealing with anafunctors outlined ab o ve are still ob jects of C . 5.16. Theorem . [ Bartels 2006 ] F or a site ( S, J ) wher e J is a sub c anonic al single- ton pr etop olo gy, internal c ate gories, anafunctors and tr ansformations form a bic ate gory Cat ana ( S, J ) . If we r estrict attention to a ful l sub-2-c ate gory C which admits b ase change for arr ows in J , we have an analo gous ful l sub-bic ate gory C ana ( J ) . In fact the bicategory C ana ( J ) fails to b e a strict 2-category only in the sense that the asso ciator is given b y the non-iden tit y isotransformation from lemma 5.10 . All the other structure is strict. There is a strict 2-functor C ana ( J ) / / Cat ana ( S, J ) which is an inclusion on ob jects and fully faithful in the strictest sense, namely b eing the identit y functor on hom-categories. The following is the main result of this section, and allo ws us to relate anafunctors to the lo calisations considered in the next section. 5.17. Pr oposition . Ther e is a strict, identity-on-obje cts 2-functor α J : C / / C ana ( J ) sending J -e quivalenc es to e quivalenc es, and c ommuting with the r esp e ctive inclusions into Cat ( S ) and Cat ana ( S, J ) . Pr oof . W e deﬁne α J to b e the identit y on ob jects, and as describ ed in examples 5.2 , 5.5 on 1-arrows and 2-arrows (i.e. functors and transformations). W e need ﬁrst to show that this gives a functor C ( X , Y ) / / C ana ( J )( X , Y ). This is precisely the con ten t of example 5.14 . Since the iden tit y 1-cell on a category X in C ana ( J ) is the image of the identit y functor on S in C , α J resp ects iden tit y 1-cells. Also, lemma 5.9 tells us that α J resp ects comp osition. That α J sends J -equiv alences to equiv alences is the con ten t of lemma 5.15 . The 2-category C is lo cally small (i.e. enric hed in small categories) if S itself is lo cally small (i.e. enriched in sets), but a priori the collection of anafunctors X − 7→ Y do not constitute a set for S a large category . 5.18. Pr oposition . L et ( S, J ) b e a lo c al ly smal l, sub c anonic al unary site satisfying WISC and let C admit b ase change along arr ows in J . Then C ana ( J ) is lo c al ly essential ly smal l. Pr oof . Giv en an ob ject A of S , let I ( A ) b e a weakly initial set for J / A . Consider the lo cally full sub-2-category of C ana ( J ) with the same ob jects, and arro ws those anafunctors ( U, f ) : X − 7→ Y suc h that U / / X 0 is in I ( X 0 ). Every anafunctor is then isomorphic, b y example 5.8 , to one in this sub-2-category . The collection of anafunctors ( U, f ) : X − 7→ Y for a ﬁxed U forms a set, b y lo cal smallness of C , and similarly the collection of transformations b et w een a pair of anafunctors forms a set b y lo cal smallness of S . 23 Examples of locally small sites ( S, J ) where C ana ( J ) is not kno wn to be lo cally essen tially small are the category of sets from the mo del of ZF used in [ v an den Berg 2012 ], the model of ZF constructed in [ Rob erts 2013 ] and the top os from prop osition 3.23 . W e note that lo cal essen tial smallness of C ana ( J ) seems to b e a condition just slightly w eak er than WISC. 6. Lo calising bicategories at a class of 1-cells Ultimately we are in teresting in inv erting all J -equiv alences in C and so need to discuss what it means to add the formal pseudoin v erses to a class of 1-cells in a 2-category – a pro cess kno wn as lo c alisation . This was done in [ Pronk 1996 ] for the more general case of a class of 1-cells in a bicategory , where the resulting bicategory is constructed and its univ ersal prop erties examined. The application in lo c. cit. is to show the equiv alence of v arious bicategories of stac ks to lo calisations of 2-categories of smo oth, top ological and algebraic group oids. The results of this article can b e seen as one-half of a generalisation of these results to more general sites. 6.1. Definition . [ Pronk 1996 ] Let B b e a bicategory and W ⊂ B 1 a class of 1-cells. A lo c alisation of B with r esp e ct to W is a bicategory B [ W − 1 ] and a weak 2-functor U : B / / B [ W − 1 ] suc h that U sends elements of W to equiv alences, and is univ ersal with this prop ert y i.e. precomp osition with U giv es an equiv alence of bicategories U ∗ : H om ( B [ W − 1 ] , D ) / / H om W ( B , D ) , where H om W denotes the sub-bicategory of weak 2-functors that send elemen ts of W to equiv alences (call these W -inverting , abusing notation sligh tly). The universal prop ert y means that W -in v erting weak 2-functors F : B / / D factor, up to an equiv alence, through B [ W − 1 ], inducing an essen tially unique weak 2-functor e F : B [ W − 1 ] / / D . 6.2. Definition . [ Pronk 1996 ] Let B b e a bicategory with a class W of 1-cells. W is said to admit a right c alculus of fr actions if it satisﬁes the follo wing conditions 2CF1. W con tains all equiv alences 2CF2. a) W is closed under comp osition b) If a ∈ W and there is an isomorphism a ∼ ⇒ b then b ∈ W 2CF3. F or all w : A 0 / / A, f : C / / A with w ∈ W there exists a 2-commutativ e square P v   g / / A 0 w   C f / / A ' z  24 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-arrow β : f ◦ v ⇒ g ◦ v suc h that α ◦ v = w ◦ β . Moreov er: when α is an isomorphism, w e require β to b e an isomorphism to o; when v 0 and β 0 form another suc h pair, there exist 1-cells u, u 0 suc h that v ◦ u and v 0 ◦ u 0 are in W , and an isomorphism  : v ◦ u ⇒ v 0 ◦ u 0 suc h that the following diagram comm utes: f ◦ v ◦ u β ◦ u + 3 f ◦  '   g ◦ v ◦ u g ◦  '   f ◦ v 0 ◦ u 0 β 0 ◦ u 0 + 3 g ◦ v 0 ◦ u 0 (7) F or a bicategory B with a calculus of right fractions, [ Pronk 1996 ] constructs a lo cal- isation of B as a bicategory of fractions; the 1-arro ws are spans and the 2-arro ws are equiv alence classes of bicategorical spans-of-spans diagrams. F rom now on w e shall refer to a calculus of right fractions as simply a calculus of fractions, and the resulting lo calisation constructed by Pronk as a bicategory of fractions. Since B [ W − 1 ] is deﬁned only up to equiv alence, it is of great in terest to know when a bicategory D , in whic h elemen ts of W are sent to equiv alences by a 2-functor B / / D , is equiv alent to B [ W − 1 ]. In particular, one might b e in terested in ﬁnding suc h an equiv alent bicategory with a simpler description than that whic h app ears in [Pronk 1996]. 6.3. Proposition . [ Pr onk 1996 ] A we ak 2-functor F : B / / D which sends elements of W to e quivalenc es induc es an e quivalenc e of bic ate gories e F : B [ W − 1 ] ∼ − → D if the fol lowing c onditions hold EF1. F is essential ly surje ctive, EF2. F or every 1-c el l f ∈ D 1 ther e ar e 1-c el ls w ∈ W and g ∈ B 1 such that F g ∼ ⇒ f ◦ F w , EF3. F is lo c al ly ful ly faithful. Thanks are due to Matthieu Dup on t for p oin ting out (in p ersonal communication) that prop osition 6.3 actually only holds in the one direction, not in b oth, as claimed in lo c. cit. The following is useful in showing a weak 2-functor sends weak equiv alences to equiv a- lences, b ecause this condition only needs to b e c hec k ed on a class that is in some sense coﬁnal in the weak equiv alences. 25 6.4. Pr oposition . L et V ⊂ W b e two classes of 1-c el ls in a bic ate gory B such that for al l w ∈ W , ther e exists v ∈ V and s ∈ W and an invertible 2-c el l a w   b v / / s ? ? c . '   Then a we ak 2-functor F : B / / D that sends elements of V to e quivalenc es also sends elements of W to e quivalenc es. Pr oof . In the following the coherence arro ws will b e presen t, but unlab elled. It is enough to prov e that if in a bicategory D with a class of maps M (in our case M = F ( W )) such that for all w ∈ M there is an equiv alence v and an isomorphism α , a w   b v / / s ? ? c ' α   where s ∈ M , then all elemen ts of M are also equiv alences. Let ¯ v b e a pseudoin v erse for v and let j = s ◦ ¯ v . Then there is sequence of isomorphisms w ◦ j ⇒ ( w ◦ s ) ◦ ¯ v ⇒ v ◦ ¯ v ⇒ I . Since s ∈ M , there is an equiv alence u , t ∈ M and an isomorphism β giving the follo wing diagram d t   u / / a w   b v / / s ? ? c . α   β   Let ¯ u b e a pseudoin v erse of u . W e kno w from the ﬁrst part of the pro of that we ha v e a pseudosection k = t ◦ ¯ u of s , with an isomorphism s ◦ k ⇒ I . W e then ha v e the follo wing sequence of isomorphisms: j ◦ w = ( s ◦ ¯ v ) ◦ w ⇒ (( s ◦ ¯ v ) ◦ w ) ◦ ( s ◦ k ) ⇒ s ◦ (( ¯ v ◦ v ) ◦ ( t ◦ ¯ u )) ⇒ ( s ◦ t ) ◦ u ⇒ ¯ u ◦ u ⇒ I . Th us all elemen ts of M are equiv alences. 26 7. 2-categories of in ternal categories admit bicategories of fractions In this section we pro v e the result that C  → Cat ( S ) admits a calculus of fractions for the J -equiv alences, where J is a singleton pretop ology on S . The following is the ﬁrst main theorem of the pap er, and subsumes a num b er of other, similar theorems throughout the literature (see section 8 for details). 7.1. Theorem . L et S b e a c ate gory with a singleton pr etop olo gy J . Assume the ful l sub-2-c ate gory C  → Cat ( S ) admits b ase change along maps in J . Then C admits a right c alculus of fr actions for the class W J of J -e quivalenc es. Pr oof . W e show the conditions of deﬁnition 6.2 hold. 2CF1. An in ternal equiv alence is clearly J -lo cally split. Lemma 4.17 giv es us the rest. 2CF2. a) That the comp osition of fully faithful functors is again fully faithful is trivial. Consider the comp osition g ◦ f of t w o J -lo cally split functors, Y [ U ]   u   Z [ V ]   v   X f / / Y g / / Z     By lemma 4.13 the functor u pulls bac k to a functor Z [ U × Y 0 V ] / / Z [ V ]. The comp osite Z [ U × Y 0 V ] / / Z is fully faithful with ob ject comp onen t in J , hence g ◦ f is J -locally split. b) Lemma 4.17 tells us that fully faithful functors are closed under isomorphism, so we just need to show J -lo cally split functors are closed under isomorphism. Let w , f : X / / Y b e functors and a : w ⇒ f b e a natural isomorphism. First, let w b e J -lo cally split. It is immediate from the diagram Y [ U ]   u   X w ' ' f 7 7 Y   a   that f is also J -lo cally split. 2CF3. Let w : X / / Y b e a J -equiv alence, and let f : Z / / Y b e a functor. F rom the deﬁnition of J -lo cally split, w e hav e the diagram Y [ U ]   u   X w / / Y   27 W e can use lemma 4.13 to pull u bac k along f to get a 2-comm uting diagram Z [ U × Y 0 Z 0 ] v % % x x Y [ U ]   u ! ! Z f y y X w / / Y   with v ∈ W J as required. 2CF4. Since J -equiv alences are representably fully faithful, giv en Y w X f = = g ! ! ⇓ a Z Y w > > where w ∈ W J , there is a unique a 0 : f ⇒ g such that Y w X f = = g ! ! ⇓ a Z Y w > > = X f $ $ g : : ⇓ a 0 Y w / / Z . The existence of a 0 is the ﬁrst half of 2CF4, where v = id X . Note that if a is an isomorphism, so if a 0 , since w is represen tably fully faithful. Given v 0 : W / / X ∈ W J suc h that there is a transformation X f W v 0 = = v 0 ! ! ⇓ b Y X g > > 28 satisfying X f W v 0 = = v 0 ! ! ⇓ b Y w / / Z X g > > = Y w W v 0 / / X f = = g ! ! ⇓ a Z Y w > > = W v 0 / / X f $ $ g : : ⇓ a 0 Y w / / Z , (8) then uniqueness of a 0 , together with equation (8) giv es us X f W v 0 = = v 0 ! ! ⇓ b Y X g > > = W v 0 / / X f $ $ g : : ⇓ a 0 Y . This is precisely the diagram ( 7 ) with v = id X , u = v 0 , u 0 = id W and  the identit y 2-arro w. Hence 2CF4 holds. The pro of of theorem 7.1 is written using only the language of 2-categories, so can b e generalised from C to other 2-categories. This approac h will b e tak en up in [Rob erts B]. The second main result of the pap er is that w e wan t to know when this bicategory of fractions is equiv alent to a bicategory of anafunctors, as the latter bicategory has a muc h simpler construction. 7.2. Theorem . L et ( S, J ) b e a sub c anonic al unary site and let the ful l sub-2-c ate gory C  → Cat ( S ) admit b ase change along arr ows in J . Then ther e is an e quivalenc e of bic ate gories C ana ( J ) ' C [ W − 1 J ] under C . Pr oof . Let us show the conditions in prop osition 6.3 hold. T o b egin with, the 2-functor α J : C / / C ana ( J ) sends J -equiv alences to equiv alences b y prop osition 5.17. EF1. α J is the iden tit y on 0-cells, and hence surjectiv e on ob jects. 29 EF2. This is equiv alen t to showing that for any anafunctor ( U, f ) : X − 7→ Y there are functors w , g such that w is in W J and ( U, f ) ∼ ⇒ α J ( g ) ◦ α J ( w ) − 1 where α J ( w ) − 1 is some pseudoin v erse for α J ( w ). Let w b e the functor X [ U ] / / X and let g = f : X [ U ] / / Y . First, note that X [ U ] } } = # # X X [ U ] is a pseudoin v erse for α J ( w ) =     X [ U ][ U ] = y y # # X [ U ] X     . Then the comp osition α J ( f ) ◦ α J ( w ) − 1 is X [ U × U U × U U ] w w ' ' X Y , whic h is just ( U, f ) (recall w e ha v e the equalit y U × U U × U U = U b y remark 4.12 ). EF3. If a : ( X 0 , f ) ⇒ ( X 0 , g ) is a transformation of anafunctors for functors f , g : X / / Y , it is giv en by a natural transformation f ⇒ g : X = X [ X 0 × X 0 X 0 ] / / Y . Hence we get a unique natural transformation a : f ⇒ g suc h that a is the image of a 0 under α J . W e no w give a series of results follo wing from this theorem, using basic prop erties of pretop ologies from section 3. 7.3. Cor ollar y . When J and K ar e two sub c anonic al singleton pr etop olo gies on S such that J un = K un , for example J c oﬁnal in K , ther e is an e quivalenc e of bic ate gories C ana ( J ) ' C ana ( K ) . The class of maps in T op of the form ` U i / / X for an op en cov er { U i } of X form a singleton pretopology . This is because O is a sup er extensive pretop ology (see the appendix). Giv en a site with a sup erextensiv e pretop ology J , we ha v e the follo wing result which is useful when J is not a singleton pretop ology (the singleton pretop ology q J is deﬁned analogously to the case of T op , details are in the app endix). 30 7.4. Corollar y . L et ( S, J ) b e a sup er extensive site wher e J is a sub c anonic al pr etop olo gy. Then C [ W − 1 J un ] ' C ana ( q J ) . Pr oof . This essen tially follows b y lemma A.9. Ob viously this can b e combined with previous results, for example if K is coﬁnal in q J , for J a non-singleton pretop ology , K -anafunctors lo calise C at the class of J un -equiv alences. Finally , giv en WISC we ha v e a b ound on the size of the hom-categories, up to equiv a- lence. 7.5. Theorem . L et ( S, J ) b e a sub c anonic al unary site satisfying WISC with S lo c al ly smal l and let C  → Cat ( S ) admit b ase change along arr ows in J . Then any lo c alisation C [ W − 1 J ] is lo c al ly essential ly smal l. Recall that this lo calisation can b e chosen suc h that the class of ob jects is the same as the class of ob jects of C , and so it is not necessary to consider additional set-theoretic mec hanisms for dealing with large (2-)categories here. W e note that the issue of size of lo calisations is not touched on in [ Pronk 1996 ]. even though suc h issues are commonly addressed in lo calisation of 1-categories. If we ha v e a sp eciﬁed b ound on the hom-sets of S and also know that some WISC κ holds, then we can put sp eciﬁc bounds on the size of the hom-categories of the lo calisation. This is imp ortan t if examining ﬁne size requirements or implications for lo calisation theorems such as these, for example higher versions of lo cally presen table categories. 8. Examples The simplest example is when w e tak e the trivial singleton pretopology tr iv , where co v ering families are just single isomorphisms: tr iv -equiv alences are internal equiv alences and, up to equiv alence, lo calisation at W triv do es nothing. It is w orth p oin ting out that if w e lo calise at W triv un , which is equiv alent to considering anafunctors with source leg having a split epimorphism for its ob ject comp onen t, then b y corollary 7.3 this is equiv alen t to lo calising at W triv , so C ana ( tr iv un ) ' C ana ( tr iv ) ' C . The ﬁrst non-trivial case is that of a regular category with the canonical singleton pretop ology c . This is the setting of [ Bunge-P ar ´ e 1979 ]. Recall that W B P J is the class of Bunge-Par ´ e J -equiv alences (deﬁnition 4.18 ). F or no w, let C denote either Cat ( S ) or Gp d ( S ). 8.1. Proposition . L et ( S, J ) b e a ﬁnitely c omplete unary site with J satur ate d. Then we have C [( W B P J ) − 1 ] ' C [ W − 1 J ] This is merely a restatement of the fact Bunge-P ar ´ e J -equiv alences and ordinary J -equiv alences coincide in this case. 31 8.2. Cor ollar y . The c anonic al singleton pr etop olo gy c on a ﬁnitely c omplete c ate gory S is satur ate d. Henc e W B P c = W c for this site, and C [( W B P c ) − 1 ] ' C [ W − 1 c ] ' C ana ( c ) W e can combine this corollary with corollary 7.3 so that the localisation of either Cat ( S ) or Gp d ( S ) at the Bunge-P ar ´ e weak equiv alences can b e calculated using J -anafunctors for J coﬁnal in c . W e note that c do es not satisfy WISC in general (see prop osition 3.23 and the commen ts following), so the lo calisation might not b e lo cally essentially small. The previous corollaries deal with the case when w e are in terested in the 2-categories consisting of all of the internal categories or group oids in a site. Ho w ev er, for many applications of internal categories/groupoids it is not suﬃcient to tak e all of Cat ( S ) or Gp d ( S ). One widely used example is that of Lie group oids, which are group oids internal to the category of (ﬁnite-dimensional) smo oth manifolds suc h that source and target maps are submersions (more on these below). Other examples are used in the theory of algebraic stac ks, namely group oids internal to sc hemes or algebraic spaces. Other types of such pr esentable stac ks use group oids in ternal to some site with sp eciﬁed conditions on the source and target maps. Although it is not cov ered explicitly in the literature, it is p ossible to consider presentable stac ks of categories, and this will b e taken up in future w ork [Rob erts A]. W e th us need to furnish examples of sub-2-categories C , sp eciﬁed b y restricting the sort of maps that are allow ed for source and target, that admit base change along some class of arro ws. The follo wing lemma gives a suﬃciency condition for this to b e so. 8.3. Lemma . L et Cat M ( S ) b e deﬁne d as the ful l sub-2-c ate gory of Cat ( S ) with obje cts those c ate gories such that the sour c e and tar get maps b elong to a singleton pr etop olo gy M . Then Cat M ( S ) admits b ase change along arr ows in M , as do es the c orr esp onding 2-c ate gory Gp d M ( S ) of gr oup oids. Pr oof . Let X b e an ob ject of Cat M ( S ) and f : M / / X 0 ∈ M . In the following diagram, all the squares are pullbacks and all arro ws are in M . X [ M ] 1   / / s 0 ! ! t 0 * * X 1 × X 0 M / /   M   M × X 0 X 1   / / X 1 / /   X 0 M / / X 0 The maps mark ed s 0 , t 0 are the source and target maps for the base change along f , so X [ M ] is in Cat M ( S ). The same argumen t holds for group oids v erbatim. 32 In practice one often only w an ts base change along a subclass of M , such as the class of op en cov ers sitting inside the class of op en maps in T op . W e can then apply theo erems 7.1 and 7.2 to the 2-categories Cat M ( S ) and Gp d M ( S ) with the classes of M -equiv alences, and indeed to sub-2-categories of these, as w e shall in the examples b elo w. W e shall fo cus of a few concrete cases to show ho w the results of this pap er subsume similar results in the literature pro v ed for sp eciﬁc sites. The category of smo oth manifolds is not ﬁnitely complete so the lo calisation results in this section so far do not apply to it. There are t w o wa ys around this. The ﬁrst is to expand the category of manifolds to a category of smo oth spaces whic h is ﬁnitely complete (or even cartesian closed). In that case all the results one has for ﬁnitely complete sites can b e applied. The other is to take careful note of whic h ﬁnite limits are actually needed, and show that all constructions work in the original category of manifolds. There is then a h ybrid approac h, which is to w ork in the expanded category , but p oin t out which results/constructions actually fall inside the original category of manifolds. Here we shall tak e the second approach. First, let us pin down some deﬁnitions. 8.4. Definition . Let Diﬀ b e the category of smo oth, ﬁnite-dimensional manifolds. A Lie c ate gory is a category in ternal to Diﬀ where the source and target maps are submersions (and hence the required pullbacks exist). A Lie gr oup oid is a Lie category whic h is a group oid. A pr op er Lie group oid is one where the map ( s, t ) : X 1 / / X 0 × X 0 is prop er. An ´ etale Lie group oid is one where the source and target maps are lo cal diﬀeomorphisms. By lemma 8.3 the 2-categories of Lie categories, Lie group oids and proper Lie group oids admit base c hange along an y of the following classes of maps: op en co v ers ( qO ), surjectiv e lo cal diﬀeomorphisms ( ´ et ), surjective submersions ( S ubm ). The 2-categories of ´ etale Lie group oids and prop er ´ etale Lie group oids admit base change along arro ws in ´ et and S ubm . W e should note that we ha v e qO coﬁnal in ´ et , whic h is coﬁnal in S ubm . W e can th us apply the main results of this pap er to the sites ( Diﬀ , O ), ( Diﬀ , qO ), ( Diﬀ , ´ et ) and ( Diﬀ , S ubm ) and the 2-categories of Lie categories, Lie group oids, prop er Lie goupoids and so on. How ev er, the deﬁnition of w eak equiv alence w e ha v e here, in volving J -lo cally split functors, is not one that appp ears in the Lie group oid literature, whic h is actually Bunge-Par ´ e S ubm -equiv alence. How ev er, w e hav e the following result: 8.5. Proposition . A functor f : X / / Y b etwe en Lie c ate gories is a S ubm -e quivalenc e if and only if it is a Bunge-Par´ e S ubm -e quivalenc e. Before we pro v e this, w e need a lemma prov ed b y Ehresmann. 8.6. Lemma . [ Ehr esmann 1959 ] F or any Lie c ate gory X , the subset of invertible arr ows, X iso 1  → X 1 is an op en submanifold. Hence there is a Lie group oid X iso and an identit y-on-ob jects functor X iso / / X whic h is universal for functors from Lie group oids. In particular, a natural isomorphism b et w een functors with co domain X is given b y a comp onen t map that factors through X iso 1 , and the induced source and target maps X iso 1 / / X 0 are submersions. 33 Pr oof . (prop osition 8.5 ) F ull faithfulness is the same for b oth deﬁnitions, so we just need to show that f is S ubm -lo cally split if and only if it is essen tially S ubm -surjectiv e. W e ﬁrst show the forward implication. The sp ecial case of a qO -equiv alence b et w een Lie group oids is a small generalisation of the pro of of prop osition 5.5 in [ Mo erdijk-Mr ˇ cun 2003 ], whic h states than an in ternal equiv alence of Lie group oids is a Bunge-Par ´ e S ubm -equiv alence. Since qO is coﬁnal in S ubm , a S ubm -equiv alence is a qO -equiv alence, hence a Bunge-Par ´ e S ubm -equiv alence. F or the case when X and Y are Lie categories, we use the fact that we can deﬁne X 0 × Y 0 Y iso 1 and that the lo cal sections constructed in Moerdijk-Mrˇ cun’s pro of factor through this manifold to set up the pro of as in the group oid case. F or the rev erse implication, the construction in the ﬁrst half of the proof of prop osition 4.19 go es through v erbatim, as all the pullbacks used inv olve submersions. The need to lo calise the category of Lie group oids at W S ubm w as p erhaps ﬁrst noted in [ Pradines 1989 ], where it w as noted that something other than the standard construction of a category of fractions w as needed. Ho w ev er Pradines lac k ed the necessary 2-categorical lo calisation results. Pronk considered the sub-2-category of ´ etale Lie group oids, also lo calised at W S ubm , in order to relate these group oids to diﬀerentiable ´ etendues [ Pronk 1996 ]. Lerman discusses the 2-category of orbifolds qua stacks [ Lerman 2010 ] and argues that it should b e a lo calisation of the 2-category of prop er ´ etale Lie group oids (again at W S ubm ). These three cases use diﬀeren t constructions of the 2-categorical localisation: Pradines used what he called mer omorphisms , whic h are equiv alence classes of butterﬂy-lik e diagrams and are related to Hilsum-Sk andalis morphisms, Pronk in troduces the tec hniques outlined in this paper, and Lerman uses Hilsum-Sk andalis morphisms, also known as righ t principal bibundles. In terestingly , [ Colman 2010 ] considers this lo calisation of the 2-category of Lie group oids then considers a further lo calisation, not giv en by the results of this pap er. 2 Colman in essence sho ws that the full sub-2-category of top ologically discrete group oids, i.e. ordinary small group oids, is a lo calisation at those in ternal functors which induce an equiv alence on fundamental group oids. Our next example is that of top ological groupoids, whic h corresp ond to v arious ﬂav ours of stac ks on the category T op . The idea of w eak equiv alences of top ological group oids predates the case of Lie groupoids, and [ Pradines 1989 ] credits it to Haeﬂiger, v an Est, and [ Hilsum-Sk andalis 1987 ]. In particular the ﬁrst tw o w ere ultimately in terested in deﬁning the fundamental group of a foliation, that is to say , of the top ological group oid asso ciated to a foliation, considered up to weak eqiv alence. Ho w ev er more recen t examples ha v e fo cussed on topological stacks, or v arian ts thereon. In particular, in parallel with the algebraic and diﬀeren tiable cases, the top ological stacks for whic h there is a go od theory corresp ond to those top ological groupoids with conditions on their source and target maps. Aside from ´ etale top ological group oids (which were considered by [ Pronk 1996 ] in relation to ´ etendues), the real adv ances here hav e come from 2 In fact this is the only 2-categorical lo calisation result inv olving internal categories or group oids known to the author to not b e cov ered by theorem 7.1 or its sequel [Rob erts B]. 34 w ork of No ohi, starting with [ No ohi 2005a ], who axiomatised the concept of lo c al ﬁbr ation and asked that the source and target maps of top ological group oids are lo cal ﬁbrations. 8.7. Definition . A singleton pretop ology LF in T op is called a class of lo c al ﬁbr ations if the follo wing conditions hold: 3 1. LF con tains the op en em b eddings 2. LF is stable under copro ducts, in the sense that ` i ∈ I X i / / Y is in LF if each X i / / Y is in LF 3. LF is lo cal on the target for the op en cov er pretop ology . That is, if the pullbac k of a map f : X / / Y along an op en co v er of Y is in LF , then f is in LF . Conditions 1. and 2. tell us that qO ⊂ LF , and that LF is q J for some superextensive pretop ology J con taining the op en em b eddings as singleton ‘co v ering’ families (b ew are the misleading terminology here: co v ering families are not assumed to b e join tly surjective). Note that LF will not b e subcanonical, b y condition 1. As an example, given an y of the follo wing pretop ologies K : • Serre ﬁbrations, • Hurewicz ﬁbrations, • op en maps, • split maps, • pro jections out of a cartesian pro duct, • isomorphisms; one can deﬁne a class of lo cal ﬁbrations b y choosing those maps whic h are in K on pulling back to an op en cov er of the co domain. Such maps are then called lo c al K . As an example of the usefulness of this concept, the top ological stac ks corresp onding to top ological group oids with lo cal Hurewicz ﬁbrations as source and target hav e a nicely b eha v ed homotop y theory . The case of ´ etale group oids corresp onds to the last named class of maps, whic h give us lo cal isomorphisms, i.e. ´ etale maps. W e can then apply lemma 8.3 and theorem 7.1 to the 2-category Grp LF ( T op ) to lo calise at the class W qO (as qO ⊂ LF ), or an y other singleton pretop ology contained in LF , using anafunctors whenev er this pretop ology is sub canonical. Note that if C satisﬁes WISC, so will the corresp onding LF , although this is probably not necessary to consider in the presence of full AC. A slightly diﬀeren t approach is tak en in [ Carc hedi 2012 ], where the author introduces a new pretop ology on the category C GH of compactly generated Hausdorﬀ spaces. W e giv e a deﬁnition equiv alent to the one in lo c cit . 3 W e hav e pack aged the conditions in a wa y sligh tly diﬀerent to [ No ohi 2005a ], but the deﬁnition is in fact iden tical. 35 8.8. Definition . A (not necessarily op en) cov er { V i  → X } i ∈ I is called a C G -co v er if for an y map K / / X from a compact space K , there is a ﬁnite op en cov er { U j  → K } whic h reﬁnes the co v er { V i × X K / / K } i ∈ I . C G -cov ers form a pretop ology C G on C GH . Compactly generated stacks then corresp ond to group oids in C GH suc h that source and target maps are in the pretop ology C G un . Again, we can lo calise Gp d C G ( C GH ) at W C G un using lemma 8.3 and theorem 7.1 , and anafunctors can be again pressed in to service. W e no w arrive at the more in v olv ed case of algebraic stac ks (cf. the contin ually growing [ Stac ks pro ject ] for the exten t of the theory of algebraic stac ks), which were the ﬁrst presen table stacks to b e deﬁned. There are some subtleties ab out the site of deﬁnition for algebraic stac ks, and p o w erful represen tabilit y theorems, but we can restrict to three main cases: group oids in the category of aﬃne sc hemes Aﬀ = Ring op ; groupoids in the category Sc h of sc hemes; and group oids in the category AlgSp of algebraic spaces. Algebraic spaces reduce to algebraic stac ks on Sc h represen ted b y group oids with trivial automorphism groups, and the category of schemes is a sub category of S h ( Aﬀ ), so we shall just consider the case when our am bien t category is Aﬀ . In any case, all the sp ecial prop erties of classes of maps in all three sites are ultimately deﬁned in terms of prop erties of ring homomorphisms. Note that group oids in Aﬀ are exactly the same thing as cogroup oid ob jects in Ring , whic h are more commonly known as Hopf algebr oids . Despite the p ossibly unfamiliar language used b y algebraic geometry , algebraic stacks reduce to the following semiformal deﬁnition. W e ﬁx three singleton pretop ologies on our site Aﬀ : J , E and D suc h that E and D are lo cal on the target for the pretop ology J . An algebraic stac k then is a stack on Aﬀ for the pretop ology J whic h ‘corresp onds’ to a group oid X in Aﬀ suc h that source and target maps b elong to E and ( s, t ) : X 1 / / X 2 0 b elongs to D . W e reco v er the algebraic stac ks b y lo calising the 2-category of suc h group oids at W E (this claim of course needs substantiating, something w e will not do here for reasons of space, referring rather to [ Pronk 1996 , Sc h¨ appi 2012 ] and the forthcoming [ Rob erts A ]). In practice, D can b e something like closed maps (to reco v er Hausdorﬀ-like conditions) or all maps, and E consists of either smo oth or ´ etale maps, corresp onding to Artin and Deligne-Mumford stac ks respectively . J is then something lik e the ´ etale top ology (or rather, the singleton pretop ology asso ciated to it, as the ´ etale top ology is sup erextensiv e), and w e can apply lemma 8.3 to see that base change exists along J , along with the fact that asking for ( s, t ) ∈ D is automatically stable under forming the base change. In practice, a v ariety of com binations of J, E and D are used, as well as passing from Aﬀ to Sc h and AlgSp , so there are v arious compatibilities to chec k in order to know one can apply theorem 7.1. A ﬁnal application w e shall consider is when our ambien t category consists of algebraic ob jects. As mentioned in section 2, a n um b er of authors hav e considered lo calising group oids in Mal’tsev, or Barr-exact, or protomo dular, or semi-ab elian categories, which are hallmarks of categories of algebraic ob jects rather than spatial ones, as w e hav e b een considering so far. In the case of group oids in Grp (whic h, as in any Mal’tsev category , coincide with the internal categories) it is a w ell-kno wn result that they can b e describ ed using cr osse d 36 mo dules . 8.9. Definition . A cr osse d mo dule (in Grp ) is a homomorphism t : G / / H together with a homomorphism α : H / / Aut ( G ) suc h that t is H -equiv arian t (using the conjugation action of H on itself ), and such that the comp osition α ◦ t : G / / Aut ( G ) is the action of G on itself b y conjugation. A crossed mo dule is often denoted, when no confusion will arise, by ( G / / H ). A morphism ( G / / H ) / / ( K / / L ) of crossed mo dules is a pair of maps G / / K and H / / L making the obvious square commute, and comm uting with all the action maps. Similar deﬁnitions hold for groups in ternal to cartesian closed categories, and ev en just ﬁnite-pro duct categories if one replaces H / / Aut ( G ) with its transp ose H × G / / G . Ultimately of course there is a deﬁnition for crossed mo dules in semiab elian categories (e.g. [ AMMV 2010 ]), but we shall consider just groups. There is a natural deﬁnition of 2-arro w b et w een maps of crossed mo dules, but the sp eciﬁcs are not imp ortan t for the presen t purp oses, so w e refer to [ No ohi 2005c , deﬁnition 8.5] for details. The 2-categories of group oids in ternal to Grp and crossed mo dules are equiv alent, so we shall just w ork with the terminology of the latter. Giv en the result that crossed mo dules corresp ond to p oin ted, connected homotopy 2- t yp es, it is natural to ask if all maps of such arise from maps b et w een crossed modules. The answ er is, p erhaps unsurprisingly , no, as one needs maps which only we akly preserve the group structure. One can either write down the deﬁnition of some generalised form of map ([ No ohi 2005c , deﬁnition 8.4]), or lo calise the 2-category of crossed modules ([ No ohi 2005c ] considers a mo del structure on the category of crossed modules). T o localise the 2-category of crossed mo dules we can consider the singleton pretop ology epi on Grp consisting of the epimorphisms, and lo calise Gp d ( Grp ) at W epi . There are p oten tially interesting sub-2-categories of crossed mo dules that one migh t w an t to consider, for example, the one corresponding to nilp otent p oin ted connected 2-t yp es. These are crossed mo dules t : G / / H where the cok ernel of t is a nilpotent group and the (canonical) action of cok er t on k er t is nilpotent. The corresp ondence b et w een suc h crossed mo dules and the corresp onding internal group oids is a nice exercise, as well as seeing that this 2-category admits base c hange for the pretop ology epi . A. Sup erextensiv e sites The usual sites of top ological spaces, manifolds and sc hemes all share a common prop ert y: one can (generally) take copro ducts of cov ering families and end up with a co v er. In this app endix we gather some results that generalise this fact, none of which are esp ecially deep, but help provide examples of bicategories of anafunctors. Another reference for sup erextensiv e sites is [Sh ulman 2012]. A.1. Definition . [ CL W 1993 ] A ﬁnitary (resp. inﬁnitary ) extensive category is a category with ﬁnite (resp. small) copro ducts suc h that the following condition holds: let I 37 b e a a ﬁnite set (resp. an y set), then, given a collection of comm uting diagrams X i / /   Z   A i / / ` i ∈ I A i , one for each i ∈ I , the squares are all pullbac ks if and only if the collection { X i / / Z } i ∈ I forms a copro duct diagram. In such a category there is a strict initial ob ject: giv en a map A / / 0 with 0 initial, w e hav e A ' 0. A.2. Example . T op is inﬁnitary extensiv e. Ring op , the category of aﬃne sc hemes, is ﬁnitary extensive. In T op w e can take an op en co v er { U i } I of a space X and replace it with the single map ` I U i / / X , and w ork just as b efore using this new sort of co v er, using the fact T op is extensive. The sort of sites that mimic this b eha viour are called sup er extensive . A.4. Definition . (Bartels-Sh ulman) A sup er extensive site is an extensiv e category S equipp ed with a pretop ology J containing the families ( U i / / a I U i ) i ∈ I and such that all cov ering families are b ounded; this means that for a ﬁnitely extensive site, the families are ﬁnite, and for an inﬁnitary site, the families are small. The pretop ology in this instance will also b e called sup erextensiv e. A.5. Example . Giv en an extensiv e category S , the extensive pr etop olo gy has as co v ering families the b ounded collections ( U i / / ` I U i ) i ∈ I . The pretop ology on any sup erextensiv e site contains the extensive pretop ology . A.6. Example . The category T op with its usual pretop ology of op en cov ers is a sup erextensiv e site. A.7. Example . An elemen tary top os with the coheren t pretop ology is ﬁnitary su- p erextensiv e, and a Grothendieck top os with the canonical pretop ology is inﬁnitary sup erextensiv e. Giv en a sup erextensiv e site ( S, J ), one can form the class q J of arro ws of the form ` I U i / / A for cov ering families { U i / / A } i ∈ I in J (more precisely , all arro ws isomorphic in S/ A to such arro ws). A.8. Pr oposition . The class q J is a singleton pr etop olo gy, and is sub c anonic al if and only if J is. 38 Pr oof . Since isomorphisms are co v ers for J they are co v ers for q J . The pullbac k of a q J -co v er ` I U i / / A along B / / A is a q J -co v er as copro ducts and pullbac ks commute b y deﬁnition of an extensive category . Now for the third condition w e use the fact that in an extensive category a map f : B / / a I A i implies that B ' ` I B i and f = ` i f i . Giv en q J -co v ers ` I U i / / A and ` J V j / / ( ` I U i ), w e see that ` J V j ' ` I W i for some ob jects W i . By the previous p oin t, the pullback a I U k × ` I U i 0 W i is a q J -co v er of U i , and hence ( U k × ` I U i 0 W i / / U k ) i ∈ I is a J -co v ering family for eac h k ∈ I . Thus ( U k × ` I U i 0 W i / / A ) i,k ∈ I is a J -cov ering family , and so a J V j ' a k ∈ I a i ∈ I U k × ` I U i 0 W i ! / / A is a q J -cov er. The map ` I U i / / A is the co equaliser of ` I × I U i × A U j ⇒ ` I U i if and only if A is the colimit of the diagram in deﬁnition 3.3 . Hence ( ` I U i / / A ) is eﬀective if and only if ( U i / / A ) i ∈ I is eﬀective Notice that the original sup erextensiv e pretop ology J is generated by the union of q J and the extensiv e pretop ology . One reason w e are interested in sup erextensiv e sites is the follo wing. A.9. Lemma . In a sup er extensive site ( S, J ) , we have J un = ( q J ) un . This means w e can replace the singleton pretop ology J un (e.g. lo cal-section-admitting maps of top ological spaces) with the singleton pretop ology q J (e.g. disjoin t unions of op en cov ers) when deﬁning anafunctors. This mak es for muc h smaller pretop ologies in practice. One class of extensiv e categories which are of particular in terest is those that also ha v e ﬁnite/small limits. These are called lextensive . F or example, T op is inﬁnitary lextensiv e, as is a Grothendiec k top os. In con trast, an elementary top os is in general only ﬁnitary lextensiv e. W e end with a lemma ab out WISC. A.10. Lemma . If ( S, J ) is a sup er extensive site, ( S, J ) satisﬁes WISC if and only if ( S, q J ) do es. One reason for wh y superextensive sites are so useful is the following result from [Sc h¨ appi 2012]. 39 A.11. Proposition . [ Sc h¨ appi 2012 ] L et ( S, J ) b e a sup er extensive site, and F a stack for the extensive top olo gy on S . Then the asso ciate d stack e F on the site ( S, q J ) is also the asso ciate d stack for the site ( S, J ) . As a corollary , since ev ery weak 2-functor F : S / / Gp d for extensive S represen ted b y an internal groupoid is automatically a stack for the extensiv e top ology , we see that we only need to stackify F with resp ect to a singleton pretop ology on S . This will b e applied in [Rob erts A]. References O. Abbad, S. Man to v ani, G. Metere, and E.M. Vitale, Butterﬂies ar e fr actions of we ak e quivalenc es , preprint (2010). Av ailable from http://users.mat.unimi.it/users/metere/ . P . Aczel, The typ e the or etic interpr etation of c onstructive set the ory , Logic Collo quium ’77, Stud. Logic F oundations Math., vol. 96, North-Holland, 1978, pp 55–66. E. Aldro v andi, B. No ohi, Butterﬂies I: Morphisms of 2-gr oup stacks , Adv. Math., 221 , issue 3 (2009), pp 687–773, E. Aldro v andi, B. Noohi, Butterﬂies II: T orsors for 2-gr oup stacks , Adv. Math., 225 , issue 2 (2010), pp 922–976, T. Bartels, Higher gauge the ory I: 2-Bund les , Ph.D. thesis, Univ ersit y of California Riv erside, 2006, [arXiv:math.CT/0410328]. J. Baez and A. Lauda, Higher dimensional algebr a V: 2-gr oups , Theory and Application of Categories 12 (2004), no. 14, pp 423–491. J. Baez and M. Makk ai, Corr esp ondenc e on the c ate gory the ory mailing list, January 1997 , a v ailable at http://www.mta.ca/ ~ cat- dist/catlist/1999/anafunctors . J. B´ enab ou, L es distributeurs , rapp ort 33, Universit ´ e Catholique de Louv ain, Institut de Math ´ ematique Pure et Appliqu´ ee, 1973. J. B´ enab ou, Th´ eories r elatives ` a un c orpus , C. R. Acad. Sci. P aris S´ er. A-B 281 (1975), no. 20, Ai, A831–A834. J. B´ enab ou, Some r emarks on 2-c ate goric al algebr a , Bulletin de la So ci ´ et ´ e Math´ ematique de Belgique 41 (1989), pp 127–194. J. B ´ enab ou, Anafunctors versus distributors , email to Mic hael Sh ulman, p osted on the category theory mailing list 22 January 2011, av ailable from http://article.gmane.org/gmane.science.mathematics.categories/6485 . 40 B. v an den Berg, Pr e dic ative top oses , preprin t (2012), M. Breck es, Ab elian metamorphosis of anafunctors in butterﬂies , preprin t (2009). M. Bunge and R. P ar ´ e, Stacks and e quivalenc e of indexe d c ate gories , Cah. T op ol. G ´ eom. Diﬀ´ er. 20 (1979), no. 4, pp 373–399. A. Carb oni, S. Lack, and R. F. C. W alters, Intr o duction to extensive and distributive c ate gories , J. Pure Appl. Algebra 84 (1993), pp 145–158. D. Carchedi, Comp actly gener ate d stacks: a c artesian-close d the ory of top olo gic al stacks , Adv. Math. 229 (2012), no. 6, pp 3339–33397, H. Colman, On the homotopy 1-typ e of Lie gr oup oids , Appl. Categ. Structures 19 , Issue 1 (2010), pp 393–423, [arXiv:math/0612257]. H. Colman and C. Costo y a, A Quil len mo del structur e for orbifolds , preprint (2009). Av ailable from http://faculty.ccc.edu/hcolman/ . W. G. Dwyer and D. M. Kan, Simplicial lo c alizations of c ate gories , J. Pure Appl. Algebra 17 (1980), no. 3, pp 267–284. A. Dold, Partitions of unity in the the ory of ﬁbr ations , Ann. Math. 78 (1963), no. 2, pp 223–255. C. Ehresmann, Cat´ egories top olo giques et c at ´ egories diﬀ´ er entiables , Collo que G ´ eom. Diﬀ. Globale (Bruxelles, 1958), Cen tre Belge Rec h. Math., Louv ain, 1959, pp 137–150. C. Ehresmann, Cat ´ egories structur ´ ees , Annales de l’Ecole Normale et Sup erieure 80 (1963), pp 349–426. T. Everaert, R.W. Kieb oom, and T. v an der Linden, Mo del structur es for homotopy of internal c ate gories , Theory and Application of Categories 15 (2005), no. 3, pp 66–94. P . Gabriel and M. Zisman, Calculus of fr actions and homotopy the ory , Springer-V erlag, 1967. M. Hilsum and G. Sk andalis, Morphismes-orient ´ es d‘epsac es de feuil les et fonctorialit ` e en th´ eorie de Kasp ar ov (d’apr` es une c onje ctur e d’A. Connes) , Ann. Sci. ´ Ecole Norm. Sup. 20 (1987), pp 325–390. P . Johnstone, Sketches of an elephant, a top os the ory c omp endium , Oxford Logic Guides, v ol. 43 and 44, The Clarendon Press Oxford Universit y Press, 2002. A. Joy al and M. Tierney , Str ong stacks and classifying sp ac es , Category theory (Como, 1990), Lecture Notes in Math., v ol. 1488, Springer, 1991, pp 213–236. A. Karaglia, Emb e dding p osets into c ar dinals with D C κ , preprin t (2012), [ ]. 41 G. M. Kelly , Complete functors in homolo gy. I. Chain maps and endomorphisms Pro c. Cam- bridge Philos. So c. 60 (1964), pp 721–735. E. Lerman, Orbifolds as stacks? , L’Enseign Math. (2) 56 (2010), no. 3-4, pp 315–363, J. Lurie, Higher T op os The ory , Annals of Mathematics Studies 170 , Princeton Univ ersit y Press, 2009. Av ailable from http://www.math.harvard.edu/ ~ lurie/ . S. MacLane and I. Mo erdijk, She aves in ge ometry and lo gic , Springer-V erlag, 1992. M. Makk ai, A voiding the axiom of choic e in gener al c ate gory the ory , J. Pure Appl. Algebra 108 (1996), pp 109–173. Av ailable from http://www.math.mcgill.ca/makkai/ . S. Manto v ani, G. Metere and E. M. Vitale, Pr ofunctors in Mal’c ev c at- e gories and fr actions of functors , preprin t (2012). Av ailable from http://perso.uclouvain.be/enrico.vitale/research.html . I. Mo erdijk and J. Mr ˇ cun, Intr o duction to foliations and lie gr oup oids , Cambridge studies in adv anced mathematics, vol. 91, Cambridge Univ ersit y Press, 2003. J. Mrˇ cun, The Hopf algebr oids of functions on ´ etale gr oup oids and their princip al Morita e quivalenc e , J. Pure Appl. Algebra 160 (2001), no. 2-3, pp 249–262. B. No ohi, F oundations of top olo gic al stacks I , preprin t (2005), [ arXiv:math.A G/0503247 ]. B. No ohi, On we ak maps b etwe en 2-gr oups , preprin t (2005), [arXiv:math/0506313]. B. No ohi, Notes on 2-gr oup oids, 2-gr oups and cr osse d-mo dules , preprin t (2005) [arXiv:math/0512106]. J. Pradines, Morphisms b etwe en sp ac es of le aves viewe d as fr actions , Cah. T opol. G ´ eom. Diﬀ´ er. Cat ´ eg. 30 (1989), no. 3, pp 229–246, D. Pronk, Etendues and stacks as bic ate gories of fr actions , Comp ositio Math. 102 (1996), no. 3, pp 243–303. D. M. Rob erts, Con(ZF+ ¬ WISC) , preprint, (2013). D. M. Rob erts, A l l pr esentable stacks ar e stacks of anafunctors , forthcoming (A). D. M. Rob erts, Strict 2-sites, J -sp ans and lo c alisations , forthcoming (B). D. Sc h¨ appi, A char acterization of c ate gories of c oher ent she aves of c ertain algebr aic stacks , preprin t (2012), M. Shulman, Exact c ompletions and smal l she aves , Theory and Application of Categories, 27 (2012), no. 7, pp 97–173. 42 The Stacks pro ject authors, Stacks pr oje ct , http://stacks.math.columbia.edu . R. Street, Fibr ations in bic ate gories , Cah. T op ol. G ´ eom. Diﬀ´ er. Cat´ eg. 21 (1980), pp 111–160. J.-L. T u, P . Xu and C. Laurent-Gengoux Twiste d K-the ory of diﬀer entiable stacks , Ann. Sci. ´ Ecole Norm. Sup. (4) 37 (2004), no. 6, pp 841–910, [ arXiv:math/0306138 ]. E. M. Vitale, Bipul lb acks and c alculus of fr actions , Cah. T op ol. G ´ eom. Diﬀ ´ er. Cat ´ eg. 51 (2010), no. 2, pp 83–113. Av ailable from http://perso.uclouvain.be/enrico.vitale/ . Scho ol of Mathematic al Scienc es, University of A delaide A delaide, SA 5005 A ustr alia Email: david.roberts@adelaide.edu.au

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