Ross Street: Characterization of bicategories of stacks, Category theory (Gummersbach 1981) LNM 962, 1982, MR0682967 (84d:18006)
Although the paper  was written in the setting of 2-categories, it was pointed out in the introduction of that paper how to modify the work in order to make it bicategorical. The purpose of the present paper is to make these modifications precise and to give an application.
The main theorem is a characterization of bicategories of stacks (= champs in French) in terms of limit, colimit, exactness, and size conditions on the bicategories: a bicategorical version of Giraud’s characterization of categories of sheaves .
On the way to this result a formula is given for the associated stack. The existence of the associated stack on a categorical site was proved by Giraud  using the associated category-valued sheaf construction and a strictification construction on fibrations. Exactness properties of the associated stack construction were not obvious from Giraud’s formula. The formula we give uses the obvious generalization to bicategories of the functor used in  for the associated sheaf. Exactness of is immediate: it preserves all finite bicategorical limits (biterminal objects, bipullbacks, and bicotensoring with finite categories in the sense of ). The associated stack is given by three applications of (recall that two are needed in the sheaf case).
The application we wish to present is really an application of the formula for the associated stack. We give an easy proof of the relationship between torsors and Čech cocyles. Combining this with a very general theorem giving the classifying property of torsors (they classify objects locally structure isomophic to some member of a given family of mathematical structures), we are able to deduce information about local structures in mathematics; for example, about vector bundles, locally finite objects in a topos, Azumaya algebras, and so on.
The notion of bicategory, homomorphism of bicategories, strong transformation and modification are those of Bénabou . We write for the bicategory of homomorphisms, strong transformations and modifications from the bicategory to the bicategory .
The notion of limit for bicategories is taken from Street . For homomorphisms , , the -indexed bilimit of is an object of satisfying an equivalence of homomorphisms:
As special cases we have biterminal objects, bipullback, biproduct, bicotensor product.
Suppose is a bicategory with finite bilimits. An arrow in is called f.f. when the functor
is fully faithful for all .
An arrow is called e.s.o. (short for “essentially surjective on objects”) when the following diagram is a bipullback for all f.f. arrows .
More generally, one can define what it means for a family of arrows into to be e.s.o. using a many-legged bipullback. A weak category in is a homomorphism of bicategories from the sketch ( = Gabriel theory) for the theory of categories into K which takes the distinguished cones to bilimits. There is an obvious notion of weak functor between weak categories. Given an arrow in we can form the following diagrams in which the squares containing 2-cells are bicomma object diagrams and the squares containing isomorphisms are bipullbacks.
We obtain two weak categories and , and a weak functor with the following properties:
(a) , is an identity, and is fully faithful.;
(b) the span from to is a discrete bifibration in the sense of Street ;
(c) is a weak equivalence relation on .
This leads us to define the congruence on to be a weak functor satisfying (a),(b),(c). So each arrow has a congruence associated with it.
A quotient for a congruence consists of an arrow and a 2-cell such that
is invertible, and
and which is biuniversal with these properties.
If has finite colimits then every congruence has a quotient.
An arrow is called a quotient map wen there exist a congruence on , and a 2-cell such that form a quotient for .
Proposition. Every quotient map is an e.s.o. (compare , (1.17))
Call regular when the following properties hold
– all finite bilimits exist
– each arrow is a composite where is fully faithful and is e.s.o.
– each pullback of an e.s.o. is e.s.o.
Theorem. In a regular bicategory every e.s.o. is a quotient map. (compare , (1.17))
Call exact when it is regular and each congruence is the congruence associated to some arrow. It follows that every congruence has a quotient in an exact bicategory. Fo all bicategories , the bicategory is exact.
A topology on a bicategory assigns to each object of , a set of f.f. arrows in satisfying the following conditions:
T0. the identity of is in ;
T1. for all in and all arrows in , there exists a bipullback
in which the top arrow is in ;
T2. if is in and is f.f. with the property that for each in the image of there exists a bipullback as in T1 with the top arrow in , then is equivalent to an arrow in .
A bisite is a bicategory together with a topology. A stack for such a bisite is a homomorphism of bicategories such that, for each in , an equivalence of categories is induced as follows:
The canonical topology on a bicategory is the largest topology for which the representable homomorphisms are all stacks. Write for the full sub-bicategory of consisting of the stacks for the bisite .
A bicategory is called a bitopos when there exists a bisite with small underlying bicategory such that there is a biequivalence: .
For a bisite , regard as an ordered set by taking when there exists a diagram:
If is small then, for each homomorphism , we can define a homomorphism by:
where runs over the directed set .
A homomorphism of bicategories which preserves finitary indexed bilimits will be called left exact. Since filtered colimits in Cat commute with finitary indexed bilimits, is a left exact homomorphism from to itself.
Theorem. For any small bisite , the left biadjoint of the inclusion
is obtained by applying three times and is hence left exact. If is faithful and is a stack then is the associated stack of . If is fully faithful and is a stack then is the associated stack of . (Compare [11;(3.8)].)
A set of objects of a bicategory is called e.s.o. generating when, for each object of , the set of arrows into with sources in the set, is e.s.o.
A bicategory is called lex-total when it has small homcategories and the Yoneda embedding
has a left-exact left biadjoint.
Bicoproducts in a bicategory are universal when they are preserved by bipullbacks. When any two distinct coprojections into a bicoproduct have a bi-initial bicomma object then the bicoproduct is disjoint. A set whose cardinality is no greater than the cardinality of the set of small sets is called moderate.
Theorem. The following conditions on a bicategory with small homcategories are equivalent:
(i) is a bitopos;
(ii) is lex-total and there exists a moderate set of jects of such that, for all in , there exists an e.s.o. with in
(iii) every canonical stack on K is representable and K has an e.s.o. generating small set of objects;
(iv) K is an exact bicategory which has disjoint universal small bicoproducts and has an e.s.o. generating small set of objects;
(v) there exists a small canonical bisite C with finitary indexed bilimits such that . (Compare ;(4.11)].)
Let denote a finitely complete category with coequalizers and such that each of the categories is cartesian closed. Let denote the 2-category of categories in . Let . Regard as a site by taking single regular epimorphisms into as covers of and generating the usual topology on . Regarding as a bicategory with only identity 2-cells, we obtain a bisite. The objects of which are stacks for this bisite will simply be called stacks in this section. Regard as contained in by taking objects of as discrete categories.
Regard as contained in by taking each category in to the representable . An object of is called admissible when, for all , with U,V in E, there is a bicomma object in .
Define by and on arrows is given by pulling back along them.
For each in there exists in satisfying:
For in , we can identify with the full subcategory of the spans in from to consisting of those spans for which the following is a pullback.
In standard topos terminology, .
For any admissible in , there is a yoneda arrow . For in , the yoneda arrow has component that functor which takes to the span .
Suppose is admissible and . The E-indexed colimit of is the pointwise left extension of along as below:
Here pointwiseness means that the left (Kan) extension property is stable under pullback along an arrow into .
Call cocomplete when it admits for all and with . In particular, is cocomplete for all in (see ,).
An object is locally isomorphic to a value of when there exist a regular epimorphism , an object of and an isomorphism .
Let be the full subcategory of consisting of such . Since the pullback of a regular epic is a regular epic, this defines an object of which is a subhomomorphism of .
for , an obejct which is locally isomorphic to a value of is called an -torsor.
Proposition. An object of isa stack if and only if it admits all colimits indexed by torsors. In particular, is a stack for all in .
Theorem on classification by torsors. Suppose is an admissible stack. Each in with in factors up to isomorphism as a composite of an arrow in which is the identity on objects and an arrow in whose components are fully faithful. the functor provides an equivalence
Theorem relating torsors and Čech cocycles. For each object in and of there is an equivalence
where runs over all regular epics into and denotes the category in determined by the kernel pair of .
Proof. Bunge  has shown that is the associated stack of . the colimit of the theorem is precisely the formula for as given in paragraph 2. There is a fully faithful arrow with a stack. So only one application of is needed to obtain an associated stack. So as required.
Take to be a tyopos with a natural numbers object . In the terminology of the last section, let denote the object of .
The objects of which are locally isomorphic to
a value of are the locally finite (Kuratowski/finite decidable) objects of .
The category in is the usual category of carfdinal-finite objects. The last two theorems give:
where denotes the chaotic category on the object of . Sionce is a topos in  and topos is an essentially algebraic notion, the filtered colimit on the right is a topos. This gives another proof that the locally finite objects in form a topos showing that the ideas invoved are basically cohomological (provided we allow cohomology with category-valued coefficients and not merely abelian-group-valued coefficients).
In the situation of paragraph 4, take to be a nice category of topological spaces. Restrict the regular epics to local homeomorphisms. Take to be the internalization of the theory of vector spaces over ; that is, is the category of modules in over the ring . Take to be the family of finite dimensional vector spaces.
Objects of locally isomorphic to a value of are vector bundles over .
is the category of matrices over as a category in .
The two theorems of paragraph 4 give equivalences:
where runs over surjective local homeomorphisms into . Thus we obtain an equivalence between the category of vector bundles over and the colimit of as e runs over surjective local homeomorphisms into . Now is a compact symmetric closed monoidal additive category with finite products and splitting idempotents. These properties are therefore inherited by the category of vector bundles. The result is a precise formulation of the clutching construction for vector bundles from which we can immediately deduce the property:
necessary for the construction of K-theory. The usual colimit involving the general linear group is also a consequence. The equivalence therefore brings together much of the introductory K-theory appearing in books such as , as an aspect of category-valued cohomology.
 M. Artin, A. Grothendieck and T.L. Verdier, editors, Théorie des topos et cohomologle étale des schemas, Lecture Notes in Math. 269 (Springer, Berlin,1972).
 M.F. Atiyah, K-Theory, Math. Lecture Notes Ser. No. 7 (Benjamin-Cummings, 1967).
 J. Bénabou, Introduction to bicategories, Lecture Notes in Math. 47 (Springer, Berlin, 1967) 1-77.
 J. Giraud, Cohomologie non abélienne, (Springer, Berlin, 1971).
 P. T. Johnstone, Topos Theory, (Academic Press, 1978)
 M. Karoubi, K-Theory: An introduction, Grundlehren der Mathematischen Wissenschaften, Band 226, (Springer Berlin,1978).
 R. H. Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle XXI (1980) 111-160, numdam.
 R. H. Street, Two dimensional sheaf theory, J. Pure and Appl. Algebra 24 (1982) 2Opp.