Zoran Skoda
Characterization of Bicategories of Stacks

Ross Street: Characterization of bicategories of stacks, Category theory (Gummersbach 1981) LNM 962, 1982, MR0682967 (84d:18006)


Although the paper [11] 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 [1].

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 [5] 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 LL used in [1] for the associated sheaf. Exactness of LL is immediate: it preserves all finite bicategorical limits (biterminal objects, bipullbacks, and bicotensoring with finite categories in the sense of [9]). The associated stack is given by three applications of LL (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.

1. Regular and exact bicategories.

The notion of bicategory, homomorphism of bicategories, strong transformation and modification are those of Bénabou [3]. We write Hom(A,B)Hom(A,B) for the bicategory of homomorphisms, strong transformations and modifications from the bicategory AA to the bicategory EE.

The notion of limit for bicategories is taken from Street [9]. For homomorphisms F:ACatF : A\to Cat, S:AKS : A \to K, the FF-indexed bilimit of SS is an object {F,S}\{F,S\} of KK satisfying an equivalence of homomorphisms:

K(,{F,S})=Hom(A,Cat)(F,(,S)). K(-,\{F,S\}) = Hom(A,Cat) (F,(-,S)).

As special cases we have biterminal objects, bipullback, biproduct, bicotensor product.

Suppose KK is a bicategory with finite bilimits. An arrow m:XYm: X\to Y in KK is called f.f. when the functor

K(K,m):K(K,X)K(K,Y) K(K,m) : K(K,X)\to K(K,Y)

is fully faithful for all KK.

An arrow e:ABe : A\to B is called e.s.o. (short for “essentially surjective on objects”) when the following diagram is a bipullback for all f.f. arrows m:XYm: X\to Y.

K(B,X) K(B,m) K(B,Y) K(e,X) bicart K(e,Y) K(A,X) K(A,m) K(AY)\array{ K(B,X) & \stackrel{K(B,m)}\to & K(B,Y)\\ K(e,X)\downarrow & bicart &\downarrow K(e,Y)\\ K(A,X) &\stackrel\to{K(A,m)} & K(A Y) }

More generally, one can define what it means for a family of arrows into BB to be e.s.o. using a many-legged bipullback. A weak category TT in KK 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 f:ABf : A\to B in KK we can form the following diagrams in which the squares containing 2-cells are bicomma object diagrams and the squares containing isomorphisms are bipullbacks.

E 2 2 d 2 E 1 2 d 1 A d 0 d 0 f E 1 2 f A B d 0 f A f B \array{ E^2_2 & \stackrel{d_2}\rightarrow & E^2_1 &\stackrel{d_1}\rightarrow & A\\ d_0\downarrow &\cong &d_0\downarrow &&\downarrow f\\ E^2_1 &\stackrel{f}\rightarrow & A &\Rightarrow& B\\ d_0\downarrow&\Rightarrow&\downarrow f &&\\ A&\stackrel{f}\rightarrow & B && }
E 1 1 d 1 A d 0 f A f B\array{E^1_1&\stackrel{d_1}\to& A\\ d_0\downarrow &\Rightarrow&\downarrow f\\ A&\stackrel{f}\rightarrow & B}

We obtain two weak categories E 2:E 2 2E 1 2E O 2E^2 : E^2_2\to E^2_1 \to E^2_O and E 1:E 1 1E O 1E^1 : E^1_1 \to E^1_O, and a weak functor j:E 1E 2j : E^1 \longrightarrow E^2 with the following properties:

(a) E 0 1=E 0 2=AE^1_0 = E^2_0 = A, j 0j_0 is an identity, and j 1:E 1 1E 1 2j_1 : E^1_1\longrightarrow E^2_1 is fully faithful.;

(b) the span (d 0,E 1 2,d 1)(d_0,E^2_1,d_1) from AA to AA is a discrete bifibration in the sense of Street [9];

(c) E 1E^1 is a weak equivalence relation on AA.

This leads us to define the congruence on AA to be a weak functor j:E 1E 2j: E^1\to E^2 satisfying (a),(b),(c). So each arrow f:ABf:A\to B has a congruence associated with it.

A quotient for a congruence j:E 1E 2j:E^1\to E^2 consists of an arrow g:AXg:A\to X and a 2-cell γ:gd ogd 1\gamma: g d_o\Rightarrow g d_1 such that

E 1 1 j 1 E 1 2 d 1 A d 0 γ g A g X\array{ E^1_1&\stackrel{j_1}\to & E^2_1&\stackrel{d_1}\to &A\\ &&d_0\downarrow&\stackrel{\gamma}\Rightarrow&\downarrow g\\ && A&\stackrel{g}\to& X }

is invertible, and

E 2 2 d 2 E 1 2 d 1 A d 0 d 0 g E 1 2 d 1 A g X γ g 1 A g X 1 X=E 1 1 d 1 E 1 2 d 1 A d 0 γ g A g X\array{ E^2_2 & \stackrel{d_2}\rightarrow & E^2_1 &\stackrel{d_1}\rightarrow & A\\ d_0\downarrow &\cong &d_0\downarrow &\Rightarrow &\downarrow g\\ E^2_1 &\stackrel{d_1}\rightarrow & A &\stackrel{g}\to& X\\ \downarrow&\stackrel{\gamma}\Rightarrow&\downarrow g &\cong&\downarrow 1\\ A&\stackrel{g}\rightarrow & X &\stackrel{1}\to& X } \,\,\,\,\,\,\, =\,\,\,\,\,\,\,\, \array{ E^1_1&\stackrel{d_1}\to & E^2_1&\stackrel{d_1}\to & A\\ &&d_0 \downarrow&\stackrel{\gamma}\Rightarrow&\downarrow g\\ && A&\stackrel{g}\to& X }

and which is biuniversal with these properties.

If KK has finite colimits then every congruence has a quotient.

An arrow q:AQq:A\to Q is called a quotient map wen there exist a congruence EE on AA, and a 2-cell τ:qd 0qd 1\tau : q d_0 \Leftarrow q d_1 such that Q,d,τQ, d, \tau form a quotient for EE.

Proposition. Every quotient map is an e.s.o. (compare [11], (1.17))

Call KK regular when the following properties hold

– all finite bilimits exist

– each arrow ff is a composite mem e where mm is fully faithful and ee 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 [11], (1.17))

Call KK 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 CC, the bicategory Hom(C op,Cat)Hom(C^{op},Cat) is exact.

2. Bitoposes.

A topology on a bicategory CC assigns to each object UU of CC, a set CovUCov U of f.f. arrows RC(,U)R \to C(-,U) in Hom(C op,Cat)Hom(C^{op},Cat) satisfying the following conditions:

T0. the identity of C(,U)C(-,U) is in CovUCov U;

T1. for all RC(,U)R \to C(-,U) in CovUCov U and all arrows u:VUu: V\to U in CC, there exists a bipullback

S C(,V) R C(,U)\array{ S &\to& C(-,V)\\ \downarrow &&\downarrow\\ R &\to &C(-,U) }

in which the top arrow is in CovVCov V;

T2. if RC(,U)R'\to C(-,U) is in CovUCov U and RC(,U)R\to C(-,U) is f.f. with the property that for each u:VUu: V\to U in the image of RVC(V,U)R'V \to C(V,U) there exists a bipullback as in T1 with the top arrow in CovVCov V, then RC(,U)R\to C(-,U) is equivalent to an arrow in CovUCov U.

A bisite is a bicategory together with a topology. A stack for such a bisite is a homomorphism of bicategories F:C opCatF : C^{op}\to Cat such that, for each RC(,U)R\to C(-,U) in CovUCov U, an equivalence of categories is induced as follows:

Hom(C op,Cat)(C(,U),F)=Hom(C op,cat)(R,F). Hom(C^{op},Cat) (C(-,U),F) = Hom(C^{op},cat) (R,F).

The canonical topology on a bicategory is the largest topology for which the representable homomorphisms are all stacks. Write StackCStack C for the full sub-bicategory of Hom(C op,Cat)Hom(C^{op},Cat) consisting of the stacks for the bisite CC.

A bicategory KK is called a bitopos when there exists a bisite CC with small underlying bicategory such that there is a biequivalence: KStackCK\sim Stack C.

For a bisite CC, regard CovUCov U as an ordered set by taking RSR\leq S when there exists a diagram:

R S C(,u) .\array{ R & \to & S\\ \downarrow &\cong &\downarrow\\ &C(-,u)&. }

If CC is small then, for each homomorphism P:C opCatP: C^{op}\to Cat, we can define a homomorphism LP:C opCatL P : C^{op}\to Cat by:

(LP)U=colim RHom(C op,Cat)(R,P) (L P)U = colim_R Hom(C^{op},Cat) (R,P)

where RR runs over the directed set (CovU) op(Cov U)^{op}.

A homomorphism of bicategories which preserves finitary indexed bilimits will be called left exact. Since filtered colimits in Cat commute with finitary indexed bilimits, LL is a left exact homomorphism from Hom(c op,cat)Hom(c^{op},cat) to itself.

Theorem. For any small bisite CC, the left biadjoint of the inclusion

StackCHom(C op,Cat) Stack C \hookrightarrow Hom(C^{op},Cat)

is obtained by applying LL three times and is hence left exact. If PFP\to F is faithful and FF is a stack then L 2PL^2 P is the associated stack of PP. If PFP\to F is fully faithful and FF is a stack then LPL P is the associated stack of PP. (Compare [11;(3.8)].)

3. Characterization theorem.

A set of objects of a bicategory CC is called e.s.o. generating when, for each object UU of CC, the set of arrows into UU with sources in the set, is e.s.o.

A bicategory KK is called lex-total when it has small homcategories and the Yoneda embedding

Y:KHom(K op,Cat) Y: K - Hom(K^{op},Cat)

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 KK with small homcategories are equivalent:

(i) KK is a bitopos;

(ii) KK is lex-total and there exists a moderate set \mathcal{M} of jects of KK such that, for all XX in KK, there exists an e.s.o. MXM \to X with MM in \mathcal{M}

(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 KStackCK\sim Stack C . (Compare [11];(4.11)].)

4. Application to torsors.

Let EE denote a finitely complete category with coequalizers and such that each of the categories E/UE/U is cartesian closed. Let KK denote the 2-category of categories in EE. Let F=Hom(E op,Cat)F = Hom(E^{op},Cat). Regard EE as a site by taking single regular epimorphisms into UU as covers of UU and generating the usual topology on EE. Regarding EE as a bicategory with only identity 2-cells, we obtain a bisite. The objects of FF which are stacks for this bisite will simply be called stacks in this section. Regard EE as contained in KK by taking objects of EE as discrete categories.

Regard EE as contained in KK by taking each category AA in EE to the representable E(,A)E(-,A). An object XX of FF is called admissible when, for all x:UXx: U\to X, y:VXy: V\to X with U,V in E, there is a bicomma object x/yx/y in EE.

Define SFS\in F by SU=E/US U= E/U and SS on arrows is given by pulling back along them.

For each XX in FF there exists PXPX in FF satisfying:

F(Y,PX)=F(X op×Y,S) F(Y,PX) = F(X^{op}\times Y,S)

For AA in KK, we can identify (𝒫A)U(\mathcal{P}A)U with the full subcategory of the spans ApEqUA\stackrel{p}\leftarrow E\stackrel{q}\to U in FF from UU to AA consisting of those spans for which the following is a pullback.

E 1 d 1 E 0 p 1 p 0 A1 d 1 A 0\array{ E_1 & \stackrel{d_1}\to& E_0\\ p_1\downarrow &&\downarrow p_0\\ A1 &\stackrel{d_1}\to & A_0 }

In standard topos terminology, (𝒫A)U=E A op×U(\mathcal{P}A)U = E^{A^{op}\times U}.

For any admissible XX in FF, there is a yoneda arrow Y X:XPXY_X : X\to PX. For AA in KK, the yoneda arrow Y AY_A has component Y AU:E(U,A)(𝒫A)UY_A U : E(U,A)\to (\mathcal{P}A)U that functor which takes a:UAa: U\to A to the span AA/aUA \leftarrow A/a \rightarrow U.

Suppose AA is admissible and E(𝒫A)UE\in(\mathcal{P}A)U. The E-indexed colimit colim(E,f)colim(E,f) of f:AXf : A\to X is the pointwise left extension of fpfp along qq as below:

E p U p colim(E,f) A f X\array{ E & \stackrel{p}\rightarrow & U\\ p\downarrow &&\downarrow colim(E,f)\\ A &\stackrel{f}\rightarrow & X }

Here pointwiseness means that the left (Kan) extension property is stable under pullback along an arrow into UU.

Call XX cocomplete when it admits colim(E,f)colim(E,f) for all EE and f:AXf:A\to X with AKA\in K. In particular, 𝒫B\mathcal{P} B is cocomplete for all BB in KK (see [8],[10]).

An object zXUz \in X U is locally isomorphic to a value of f:AXf:A\to X when there exist a regular epimorphism e:VUe:V\to U, an object aa of AVA V and an isomorphism (Xe)zf Va(X e) z \cong f_V a.

V e U a z A f X\array{ V & \stackrel{e}\rightarrow & U\\ a\downarrow &\cong &\downarrow z\\ A &\stackrel{f}\rightarrow & X }

Let Loc X(f)ULoc_X(f)U be the full subcategory of XUX U consisting of such zz. Since the pullback of a regular epic is a regular epic, this defines an object Loc X(f)Loc_X(f) of FF which is a subhomomorphism of XX.

for AKA\in K, an obejct E(𝒫A)UE\in (\mathcal{P} A) U which is locally isomorphic to a value of y A:A𝒫Ay_A : A\to \mathcal{P} A is called an AA-torsor.

A/a d 1 V d 0 p.b. A p E q U\array{ &&A/a & \stackrel{d_1}\to & V\\ &d_0 \swarrow &\downarrow &p.b.&\downarrow \\ A&\stackrel{p}\to&E &\stackrel{q}\to & U }

Put TorA=Loc 𝒫A(y A)Tor A= Loc_{\mathcal{P}A}(y_A).

Proposition. An object XX of FF isa stack if and only if it admits all colimits indexed by torsors. In particular, 𝒫B\mathcal{P}B is a stack for all BB in KK.

Theorem on classification by torsors. Suppose XFX\in F is an admissible stack. Each x:WXx:W\to X in FF with WW in EE factors up to isomorphism as a composite of an arrow WX[x]W\to X[x] in KK which is the identity on objects and an arrow i:X[x]Xi:X[x]\to X in FF whose components are fully faithful. the functor Ecolim(E,i)E\mapsto colim(E,i) provides an equivalence

TorX[x]Loc X(x). Tor X[x] \cong Loc_X(x).

Theorem relating torsors and Čech cocycles. For each object AA in KK and UU of EE there is an equivalence

(TorA)Ucolim e:VUK(er U(e),A) (Tor A) U\cong colim_{e: V\to U} K(er_U(e),A)

where e:VtUe:V\t U runs over all regular epics into UU and er U(e)er_U(e) denotes the category in EE determined by the kernel pair of ee.

Proof. Bunge [4] has shown that TorATor A is the associated stack of AA. the colimit of the theorem is precisely the formula for LAL A as given in paragraph 2. There is a fully faithful arrow y A:A𝒫Ay_A:A\to \mathcal{P}A with 𝒫A\mathcal{P}A a stack. So only one application of LL is needed to obtain an associated stack. So TorALATor A\cong L A as required.

5. Finiteness in a topos.

Take EE to be a tyopos with a natural numbers object NN. In the terminology of the last section, let FinSN=E/NFin \in S N = E/N denote the object N×N+NsucNN\times N \stackrel{+}\to N\stackrel{suc}\to N of E/NE/N.

The objects ZZ of S1S1 which are locally isomorphic to
a value of Fin:NSFin:N\to S are the locally finite (Kuratowski/finite decidable) objects of EE.

The category S[Fin]S[Fin] in EE is the usual category E finE_{fin} of carfdinal-finite objects. The last two theorems give:

Tor(E fin)Loc(Fin) Tor(E_{fin}) \cong Loc(Fin)
Tor(E fin)1colim Ronto1K(R c,E fin) Tor(E_{fin})1 \cong colim_{R\onto 1} K(R_c, E_{fin})

where R cR_c denotes the chaotic category on the object RR of EE. Sionce E finE_{fin} is a topos in EE [6] 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 EE 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).

6. Vector bundles

In the situation of paragraph 4, take EE to be a nice category of topological spaces. Restrict the regular epics to local homeomorphisms. Take XFX\in F to be the internalization of the theory of vector spaces over R\mathbf{R}; that is, XUXU is the category of modules in E/UE/U over the ring R×UU\mathbf{R}\times U\to U. Take Euc:NXEuc : \mathbf{N}\to X to be the family R n\mathbf{R}^n of finite dimensional vector spaces.

Objects ZZ of XUXU locally isomorphic to a value of EucEuc are vector bundles over UU.

X[Euc]X[Euc] is the category Mat(R)Mat(\mathbf{R}) of matrices over R\mathbf{R} as a category in EE.

The two theorems of paragraph 4 give equivalences:

Tor(Mat(R))Loc(Euc)=(vectorbundles) Tor (Mat(\mathbf{R})) \cong Loc(Euc) = (vector bundles)
Tor(Mat(R))Ucolim VeUK(er U(e),Mat(R)) Tor (Mat(\mathbf{R}))U \cong colim_{V\stackrel{e}\to U} K(er_U(e),Mat(\mathbf{R}))

where ee runs over surjective local homeomorphisms into UU. Thus we obtain an equivalence between the category of vector bundles over UU and the colimit of K(er U(e),Mat(R)K(er_U(e),Mat(\mathbf{R}) as e runs over surjective local homeomorphisms into UU. Now Mat(R)Mat(\mathbf{R}) 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:

STSTimpliesSS S \oplus T \cong S' \oplus T implies S\cong S'

necessary for the construction of K-theory. The usual colimit involving the general linear group GL(n,R)GL(n,\mathbf{R}) is also a consequence. The equivalence therefore brings together much of the introductory K-theory appearing in books such as [2],[7] as an aspect of category-valued cohomology.


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[3] J. Bénabou, Introduction to bicategories, Lecture Notes in Math. 47 (Springer, Berlin, 1967) 1-77.

[4] M. Bunge, Stack completions and Morita equivalence for categories in a topos, Cahiers de topologie et géométrie différentielle XX-4 (1979) 401-436, MR558106, numdam.

[5] J. Giraud, Cohomologie non abélienne, (Springer, Berlin, 1971).

[6] P. T. Johnstone, Topos Theory, (Academic Press, 1978)

[7] M. Karoubi, K-Theory: An introduction, Grundlehren der Mathematischen Wissenschaften, Band 226, (Springer Berlin,1978).

[8] R. Street, Cosmoi of internal categories, Transactions Amer. Math. Soc. 258 (1980) 271-318, MR82a:18007, doi

[9] R. H. Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle XXI (1980) 111-160, numdam.

[10] R. H. Street, Conspectus of variable categories, J. Pure and Appl. Algebra 21 (1981) 307-338, MR84i:18006, doi90021-9)

[11] R. H. Street, Two dimensional sheaf theory, J. Pure and Appl. Algebra 24 (1982) 2Opp.

Revised on July 20, 2011 15:29:27 by Zoran Škoda (