- Email: [email protected]
Two-descent, two-torsors and local equivalence
Journal of Pure and Applied Algebra 143 (1999) 313–327
www.elsevier.com/locate/jpaa
Two-descent, two-torsors and local equivalence Luca Mauri, Myles Tierney ∗ Mathematics Department, Rutgers University, New Brunswick, NJ 08903, USA Dedicated to Michael Barr, on the occasion of his 60th birthday
Abstract We extend to dimension 2 the well-known equivalence in dimension 1 between objects locally c 1999 Elsevier Science isomorphic to a given one, descent data, Cech one-cocycles and torsors. B.V. All rights reserved. MSC: 55U10; 55P35; 18B40; 18G30
1. Introduction In this note we wish to outline an extension to dimension 2 of the well-known equivalence in dimension 1 between objects locally isomorphic to a given one, descent data, Cech one-cocycles and torsors. We begin by recalling the one-dimensional theory. This is mostly due to Grothendieck, and is treated in detail in [3, 4]. We then move to dimension 2. Here the de nition of two-dimensional descent data in the general case, without our normalization conditions, was given by Duskin in [2], though special cases can already be found in Rivano [8] and Hakim [5], suggesting that the idea was not unknown to Grothendieck. Duskin also established the 2-equivalence between objects locally equivalent to a given one and two-dimensional descent data. In our setting, in order to prove this we must show that epimorphisms in a topos E are e ective 2-descent morphisms for the 2- bration on E whose ber over X is the category of groupoids, which are stacks, in E=X . Except for the stack condition, this was a conjecture in [2]. Our de nition of 2-torsors is inspired by Breen [1], though we work internally inside a topos E, rather than externally with stacks de ned over E. It is a pleasure to thank Andre Joyal for several helpful discussions. ∗
Corresponding author. E-mail address: [email protected] (M. Tierney)
c 1999 Elsevier Science B.V. All rights reserved. 0022-4049/99/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 9 8 ) 0 0 1 1 7 - 0
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2. Dimension 1 In a topos E, let F be a xed object and U 1 a cover of 1. Another object X of E is said to be locally isomorphic to F via U if there is an isomorphism ’ : U × F → U × X over U . All such objects together with isomorphisms between them form the groupoid LocIso(U; F). We denote by
part of the simplicial nerve of the cover U , where d0 = 2 , d1 = 1 and s0 = (the diagonal) in dimension 1. In dimension 2, d0 = 23 d1 = 13 and d2 = 12 . A onedimensional (normalized) descent datum on U × F consists of an isomorphism f : d∗1 (U ×F) → d∗0 (U × F) over U × U , such that s0∗ f = id, and, over U × U × U, the triangle
commutes. If f and g are descent data on U × F, a morphism f → g is an automorphism a : F × U → F × U over U , such that f
d∗1 (U × F) −−−−−→ d∗0 (U × F) ∗ d∗0 a d1 a y y d∗1 (U × F) −−−−−→ d∗0 (U × F) g
commutes. Descent data and their morphisms form the groupoid Des1 (U × F). De ne a functor LocIso(U; F) → Des1 (U × F) as follows. For each X locally isomorphic to F via U , choose an isomorphism ’:U ×F → U ×X
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over U . Then the composite d∗ ’−1
d∗ ’
0 1 d∗1 (U × X ) = d∗0 (U × X ) −→ d∗0 (U × F) d∗1 (U × F) −→
is a descent datum on U × F. If Y is another such object with isomorphism :U ×F →U ×Y and if : X → Y is an isomorphism, then −1 (U × )’ : U × F → U × F is a morphism of the descent data associated to ’ and . It is easily seen that this is functorial. It is well known that the epimorphism U 1 is an e ective descent morphism for this bration (of objects over another) so that given a descent datum f : d∗1 (U × F) → d∗0 (U ×F) on U × F, there is an object X together with an isomorphism ’ : U × F → U × X over U , such that the descent datum determined by ’ is isomorphic to f. As a result, the functor above is an equivalence LocIso(U; F) ≈ Des1 (U × F): On the other hand, by the universal property of the group Aut(F) in E, descent data
are in 1–1 correspondence with morphisms U × U → Aut(F) which, because of the cocycle condition and normalization, are functors full(U ) → Aut(F), where full(U ) is the full equivalence relation on U regarded as a groupoid in E, and Aut(F) is considered to be a groupoid with one object. Morphisms of descent data correspond to natural transformations, yielding an equivalence Des1 (U × F) ≈ Fn(full(U ); Aut(F)); where Fn(full(U ); Aut(F)) is the category of functors and natural transformations. Let Gpd(E) be the category of groupoids in E and S(E) the category of simplicial objects in E. Let N : Gpd(E) → S(E) be the nerve functor, which is full and faithful. Natural transformations in Gpd(E) are given by functors 2 × G → H , which are sent by N to homotopies 4[1] × NG → NH , so N de nes an equivalence 1
Fn(full(U ); Aut(F)) ≈ C (U∗ ; Aut(F)); 1 1-cocycles whose objects are simplicial where C (U∗ ; Aut(F)) is the groupoid of Cech morphisms of U∗ = N full(U ) to N Aut(F), and whose morphisms are homotopies.
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A further equivalence is obtained by considering (one-dimensional) torsors. Recall that a torsor for Aut(F) is a nonempty object T (meaning T → 1 is epimorphic) provided with a (right) action a : T × Aut(F) → T which is free and transitive, i.e. such that (1 ; a) : T × Aut(F) → T × T is an isomorphism. Equivariant morphisms of torsors are automatically isomorphisms, so the category Tors1 (Aut(F)) is a groupoid. Let LocIso(F) denote the category of objects of E locally isomorphic to F (via some cover U 1), with isomorphisms as maps. If X is locally isomorphic to F, the object of isomorphisms Iso(F; X ) is an Aut(F) torsor with the action given by composition. If T is an Aut(F) torsor, T ⊗Aut(F) F is locally isomorphic to F (by localizing over T ). Thus, we have a pair of functors Iso(F;−)
LocIso(F)−−−−−→Tors1 (Aut(F)): ←−−−−− −⊗F Aut(F)
Furthermore, if X is locally isomorphic to F, and T is an Aut(F) torsor, we have natural maps T → Iso F; T ⊗ F Iso(F; X ) ⊗ F → X; Aut(F)
Aut(F)
induced by evaluation in the rst case, and hom-⊗ adjunction in the second. The rst is seen to be an isomorphism by localizing over Iso(F; X ), and the second is an isomorphism since it is an equivariant mapping of torsors. Thus, we have an equivalence LocIso(F) ≈ Tors1 (Aut(F)): 3. Dimension 2 In dimension 2 we want to classify groupoids locally equivalent to a given one, but this presents an immediate technical problem. Namely, in a topos E there are two kinds of equivalences. If G and H are groupoids in E, a functor f : G → H is called a weak equivalence, or categorical equivalence if it is full and faithful and essentially surjective. These are expressed by saying that
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is a pullback, and in the diagram
the map K → H0 is epimorphic. With this, of course, we are saying internally that every object of H is isomorphic to one in the image of f. f is a strong equivalence if there is a functor f0 : H → G such that f0 f ' idG and ff0 ' idH . In the presence of the axiom of choice, every weak equivalence is strong, but not in a general E. Also, the two classes of equivalences have quite di erent properties. For example, weak equivalences are not preserved by pullbacks in Gpd(E), but are re ected by pullbacks along an epimorphism of E. Strong equivalences are preserved under pullback, but not re ected. Some constructions result in weak equivalences some in strong, and in general we want strong. We deal with this problem in the following way. Recall [6] that a groupoid G in E is a stack, if for each weak equivalence e : A → B, any diagram
has a dotted ller making the resulting triangle commute up to isomorphism. Also, the weak equivalences are part of a Quillen homotopy structure on Gpd(E) in which the co brations are functors monomorphic on objects and the brant objects are strong stacks. Thus, G is a strong stack if for any co bration weak equivalence e : A ,→ B each diagram
has a dotted ller making the resulting triangle commute (not just up to isomorphism). In particular, if G is a groupoid in E, then factoring G → 1 as a co bration weak
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equivalence followed by a bration
we see that each G has a (strong) stack completion G∗ . What we need about stacks is the following. Proposition 1. Let e : G → H be a weak equivalence and G a stack. Then e is a strong equivalence. Proof. Fill in the following diagram:
obtaining e0 : H → G and an isomorphism : id G → e0 e. e0 is a weak equivalence since e and idG are. In the diagram
∼
∼
e0 : e0 (id H ) → e0 (ee0 ). But e0 is full and faithful, so e0 = e0 , where : id H → ee0 . We turn now to two-descent. Thus, let U 1 be a cover of 1, and
more of the simplicial nerve of U , with d0 = 234 , d1 = 134 , d2 = 124 and d3 = 123 in dimension 3. s0 = × U and s1 = U × . Let G be a groupoid in E=U , which amounts to a groupoid G in E, together with a functor G → dis(U ), where dis(U ) is the discrete groupoid in E with objects U . A two-dimensional (normalized) descent datum on G consists of a (strong) equivalence f : d∗1 (G) → d∗0 (G) in Gpd(E=U × U ) together with an isomorphism : id G → s0∗ f and an isomorphism in the triangle
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These should satisfy the equations d∗0 f = s1∗ ;
fd∗1 = s0∗
and the “cocycle condition”, which requires that the tetrahedron
in Gpd(E=U × U × U × U ) (drawn with even and odd faces detached) commutes: d∗0 · d∗2 = d∗3 · d∗1 : A 1-morphism of descent data (G; f; ; ) → (H; g; ; ) is a functor a : G → H in Gpd(E=U ) together with an isomorphism # in the square
of Gpd(E=U × U ) such that the cylinder
in Gpd(E=U ) and the prism
in Gpd(E=U × U × U ) commute.
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A 2-morphism
is a natural transformation
such that the cylinder
in Gpd(E=U × U ) commutes. These objects, 1-cells and 2-cells constitute a 2-category Des2 (U ), and there is an obvious 2-functor : Gpd(E) → Des2 (U ) given by (H) = (U × H; t; id; id), where t is the canonical switching. On morphisms (h) = (U × h; id) and ( ) = U × . Now denote by Stacks (E) the full subcategory of Gpd(E) whose objects are stacks, and by : Stacks(E) → Des2 (U ) the evident restriction of as de ned above. The objects of Des2 (U ) are now taken to be stacks G in E=U provided with a normalized two-dimensional descent datum. U 1 is said to be a morphism of e ective 2-descent if this is a 2-equivalence, i.e. a weak equivalence on the hom-categories such that each object of Des2 (U ) is strongly equivalent to one of the form (H). Theorem 2. Epimorphisms of E are morphisms of e ective 2-descent. Proof. We rst show that if G and H are stacks, the functor Stacks(G; H) → Des2 (U × G; U × H)
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induced by is a weak equivalence. For this it is easy to see that a functor f:U ×G→U × H compatible with the canonical descent data can be identi ed with a functor f : full(U ) × G → H: full(U ) → 1 is a weak equivalence, hence so is full(U ) × G → G. Thus, since H is a stack there is a dotted ller below making the triangle commute up to isomorphism.
Thus, the functor above is essentially surjective. Full and faithfulness follows since full(U ) × G → G is a weak equivalence. For the e ective part, let (G; f; ; ) be an object of Des2 (U ), where G is a stack. The normalized descent datum (f; ; ) on G de nes a Gpd(E)-valued pseudo-functor on full(U ), so we can apply the Grothendieck construction to obtain a Grothendieck bration p : E → full(U ). Here, for groupoids, Grothendieck bration means just (internal) path-lifting. The normalization condition implies that the pullback of p along the functor dis(U ) → full(U ), given by the identity on objects and on maps, is isomorphic to G. That is, there is a pullback diagram G y
−−−−−→
E p y
dis(U ) −−−−−→ full(U ) The groupoid we are looking for is E itself, for consider the diagram
full(U ) → 1 is a weak equivalence so 2 is, since weak equivalences are stable under product. Since idE is a weak equivalence q = (p; id) is also. Moreover, since 2 q = idE ,
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q is a strong equivalence with quasi inverse 2 . But 2 is not over full(U ), so we can not be sure q remains an equivalence when pulled back over dis(U ). However, q : E → full(U ) × E is a co bration weak equivalence and p is a Grothendieck bration, so over a boolean cover the diagram
has a dotted ller. It follows that q is a stable weak equivalence so that pulling back over dis(U ) yields a weak equivalence
which is strong, since G is a stack over U . To ensure that E is a stack, we can take the strong stack completion E → E∗ , for dis(U ) × E → dis(U ) × E∗ is a weak equivalence, which is strong since dis(U ) × E is a stack over dis(U ). We remark that the stack condition on G in the e ective part of the above proof is not necessary. In fact, if G is just a groupoid, the descent datum itself provides a quasi-inverse q0 to q which is over full(U ). However, in the rst part of the proof, stacks appear to be necessary. The functor induced by on the hom-groupoids will not be essentially surjective if H is not a stack. If G is a groupoid in E and U 1 is a cover of 1, we say another groupoid H is locally equivalent to G via U if there is a weak equivalence e : U × G → U × H over U . All such H together with weak equivalences and natural transformations form a 2-category wLocEq(U; G). Let LocEq(U; G∗ ) be the similarly de ned 2-category of stacks locally equivalent to G∗ , and let I : LocEq(U; G∗ ) → wLocEq(U; G) denote the inclusion. Then I is a 2-equivalence, since if H is locally equivalent to G, then H∗ is locally equivalent to G∗ , so H is weakly equivalent to a stack in the image of I . Note that all the weak equivalences in LocEq(U; G∗ ) are strong, including those in the de nition of local equivalence. Let F be a stack in E, U 1 a cover of 1, and LocEq(U; F) the 20 -groupoid of stacks locally equivalent to F via U , where by 20 -groupoid we mean a 2-category in which 1-cells are equivalences, and 2-cells are isomorphisms. Let Des2 (U × F) be the 20 -groupoid of 2-descent data on U × F, i.e. we restrict our attention to 1-cells which are equivalences. De ne a 2-functor LocEq(U; F) → Des2 (U × F)
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as follows. For each stack G locally equivalent to F, choose equivalences e:U ×F→U × G
and
e0 : U × G → U × F
over U , where e0 is an adjoint quasi-inverse to e, and isomorphisms : id → ee0 and 0 : id → e0 e. Let f be the composite d∗ e
d∗ e0
0 1 d∗1 (U × F) −→ d∗1 (U × G) = d∗0 (U × G) −→ d∗0 (U × F):
Then determines an isomorphism in
and 0 gives an isomorphism id → s0∗ f. For details concerning see [2]. If H is another such stack with equivalence d : U × F → U × H over U , and if ’ : G → H is an equivalence, then d0 · (U × ’) · e : U × F → U × F together with the # de ned in [2] is a 1-morphism of the descent data associated to e and d. Finally, if
is a 2-cell between equivalences on stacks locally equivalent to F, then d0 (U × )e is a 2-cell between the associated 1-morphisms of descent data. Since U 1 is a morphism of e ective 2-descent, it follows as in [2] that the 2-functor so de ned is a 2-equivalence LocEq(U; F) ≈ Des2 (U × F): Let Eq(F) be the groupoid of (weak) self equivalences of F. There is a weak equivalence
above Eq(F) where ev is evaluation. It is universal in the sense that for any groupoid G, functors G → Eq(F) are in 1–1 correspondence with weak equivalences
over G, where f is the pullback of (1 ; ev) along G → Eq(F).
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If F is a stack, as we are assuming, it is easy to see that Eq(F) is also. Furthermore, since stacks are stable under product, it follows that Eq(F) × F is a stack over Eq(F). Thus, the weak equivalence is strong. The classifying functor for an adjoint quasiinverse to (1 ; ev) provides a quasi-inverse assignment (−)0 : Eq(F) → Eq(F) making Eq(F) a strictly associative and unitary compact groupoid. Now, let (f; ; ) be a normalized 2-descent datum on U × F over U . Then
corresponds to a map U × U → Eq(F) which the normalization conditions, together with and the cocycle condition, say is part of a pseudo-functor full(U ) → Eq(F); where full(U ) is taken to be discrete in dimension 2, and Eq(F) is considered to be a 20 -groupoid with one object. 1-morphisms of descent data correspond to pseudo-natural transformations, and 2-morphisms to modi cations, so we obtain a 2-equivalence Des2 (U × F) ≈ Psd(full(U ); Eq(F)): Denote by 20 -Gpd(E) the category of 20 -groupoids in E with pseudo-functors as morphisms. In order to obtain the two-dimensional nerve functor N 2 : 20 -Gpd(E) → S(E); we must modify the de nition given in [7] to take account of the fact that our pseudofunctors no longer preserve units. This can be done, however, and the new N 2 is still full and faithful. It is easy to see that if ’; : G → H are pseudo-functors, then there is a pseudo-natural transformation : ’ → i there is a pseudo-functor h : 2 × G → H with h(0; −) = ’ and h(1; −) = . Since N 2 is full and faithful and takes pseudofunctors 2 × G → H to homotopies 4[1] × N 2 G → N 2 H, we obtain an isomorphism [N 2 G; N 2 H] ' 0 Psd(G; H) where [N 2 G; N 2 H] denotes the set of homotopy classes of simplicial maps. In particular, we obtain an isomorphism 2 H (U∗ ; Eq(F)) ' 0 Psd(full(U ); Eq(F)); 2 where H (U∗ ; Eq(F)) is [N 2 full(U ); N 2 Eq(F)].
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We turn now to torsors. A 2-torsor for Eq(F) is a non-empty stack T, meaning T0 1 is an epimorphism, together with a (right) action a : T × Eq(F) → T such that (1 ; a) : T × Eq(F) → T × T is an equivalence. A 1-morphism of torsors is an Eq(F)-equivariant functor. These are equivalences by localization, since any equivariant functor Eq(F) → Eq(F) is. a 2-morphism is a natural transformation compatible with the action of Eq(F) in the obvious sense. Let Tors2 (Eq(F)) denote the resulting 20 -groupoid. Write LocEq(F) for the 20 -groupoid of stacks locally equivalent to F via some cover U 1, with equivalences as 1-cells. Let G be a stack locally equivalent to F. Proposition 3. Eq(F; G) is an Eq(F) torsor. Proof. Eq(F; G) is non-empty since G is locally equivalent to F. Eq(F) acts on the right of Eq(F; G) by composition. Let (−)0 : Eq(F; G) → Eq(G; F) be the quasi-inverse assignment de ned as for Eq(F). Then a quasi-inverse to (1 ; a) : Eq(F; G) × Eq(F) → Eq(F; G) × Eq(F; G) is provided by the functor (−)0 × id
Eq(F; G) × Eq(F; G) −−−−−→ Eq(G; F) × Eq(F; G) (1 ; c)
−−−−−→ Eq(F; G) × Eq(F) where c denotes composition. Thus, we obtain a 2-functor Eq(F; −) : LocEq(F) → Tors2 (Eq(F)): Theorem 4. Eq(F; −) is a 2-equivalence. Proof. Let G and H be locally equivalent to F, and consider the functor : Eq(G; H) → Eq (Eq(F; G); Eq(F; H)) de ned by transposing the composition Eq(G; H) × Eq(F; G) → Eq(F; H); where Eq denotes equivariant equivalences. Let U 1 be a cover of 1 over which both G and H become equivalent to F. (−)×U is a logical functor, so × U becomes equivalent to Eq(F × U; F × U ) → Eq (Eq(F × U; F × U ); Eq(F × U; F × U ))
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which is an equivalence. It follows that is an equivalence, so Eq(F; −) induces equivalences on the hom-groupoids. To see that Eq(F; −) is 2-essentially surjective, let T be an Eq(F) torsor, and G a stack completion of T ⊗Eq(F) F. Thus, we have a weak equivalence e : T ⊗ F → G: Eq(F)
By transposition, e yields an equivariant functor f : T → hom(F; G): Let q : T × F → T ⊗Eq(F) F denote the quotient functor. Then the functor over T corresponding to f is
However, the diagram
commutes, where the two isomorphisms are canonical. It follows that (1 ; eq) is an equivalence, so that f is, in fact, an equivariant functor f : T → Eq(F; G) and, as such, an equivalence. References [1] L. Breen, Bitorseurs et cohomologie non abelienne, in: Pierre Cartier et al. (Eds.), The Grothendieck Festschrift, vol. I, Birkhauser, Basel, 1990, pp. 401– 476. [2] J. Duskin, An outline of a theory of higher dimensional descent, Bulletin de la Societe Mathematique de Belgique Ser. A 41(2) (1989) 249 –277. [3] J. Giraud, Cohomologie Non Abelienne, Springer, Berlin, 1971. [4] J. Giraud, Methode de la descente, Bulletin de la Societe Mathematique de France Supplement, Decembre 1964, Memoire 2.
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[5] M. Hakim, Topos Anneles et Schemas Relatifs, Springer, Berlin, 1972. [6] A. Joyal, M. Tierney, Strong stacks and classifying spaces, in: A. Carboni et al. (Eds.), Category Theory, Proceedings, Como 1990, Lecture Notes in Mathematics, vol. 1488, Springer, Berlin, 1991, pp. 213–236. [7] I. Moerdijk, Lectures on 2-dimensional groupoids, Rapport 175, Insitut de Mathematique Pure et Appliquee, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium, March 1990. [8] Neantro Saavedra Rivano, Categories Tannakiennes, Lecture Notes in Mathematics, vol. 265, Springer, Berlin, 1972.
www.elsevier.com/locate/jpaa
Two-descent, two-torsors and local equivalence Luca Mauri, Myles Tierney ∗ Mathematics Department, Rutgers University, New Brunswick, NJ 08903, USA Dedicated to Michael Barr, on the occasion of his 60th birthday
Abstract We extend to dimension 2 the well-known equivalence in dimension 1 between objects locally c 1999 Elsevier Science isomorphic to a given one, descent data, Cech one-cocycles and torsors. B.V. All rights reserved. MSC: 55U10; 55P35; 18B40; 18G30
1. Introduction In this note we wish to outline an extension to dimension 2 of the well-known equivalence in dimension 1 between objects locally isomorphic to a given one, descent data, Cech one-cocycles and torsors. We begin by recalling the one-dimensional theory. This is mostly due to Grothendieck, and is treated in detail in [3, 4]. We then move to dimension 2. Here the de nition of two-dimensional descent data in the general case, without our normalization conditions, was given by Duskin in [2], though special cases can already be found in Rivano [8] and Hakim [5], suggesting that the idea was not unknown to Grothendieck. Duskin also established the 2-equivalence between objects locally equivalent to a given one and two-dimensional descent data. In our setting, in order to prove this we must show that epimorphisms in a topos E are e ective 2-descent morphisms for the 2- bration on E whose ber over X is the category of groupoids, which are stacks, in E=X . Except for the stack condition, this was a conjecture in [2]. Our de nition of 2-torsors is inspired by Breen [1], though we work internally inside a topos E, rather than externally with stacks de ned over E. It is a pleasure to thank Andre Joyal for several helpful discussions. ∗
Corresponding author. E-mail address: [email protected] (M. Tierney)
c 1999 Elsevier Science B.V. All rights reserved. 0022-4049/99/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 9 8 ) 0 0 1 1 7 - 0
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2. Dimension 1 In a topos E, let F be a xed object and U 1 a cover of 1. Another object X of E is said to be locally isomorphic to F via U if there is an isomorphism ’ : U × F → U × X over U . All such objects together with isomorphisms between them form the groupoid LocIso(U; F). We denote by
part of the simplicial nerve of the cover U , where d0 = 2 , d1 = 1 and s0 = (the diagonal) in dimension 1. In dimension 2, d0 = 23 d1 = 13 and d2 = 12 . A onedimensional (normalized) descent datum on U × F consists of an isomorphism f : d∗1 (U ×F) → d∗0 (U × F) over U × U , such that s0∗ f = id, and, over U × U × U, the triangle
commutes. If f and g are descent data on U × F, a morphism f → g is an automorphism a : F × U → F × U over U , such that f
d∗1 (U × F) −−−−−→ d∗0 (U × F) ∗ d∗0 a d1 a y y d∗1 (U × F) −−−−−→ d∗0 (U × F) g
commutes. Descent data and their morphisms form the groupoid Des1 (U × F). De ne a functor LocIso(U; F) → Des1 (U × F) as follows. For each X locally isomorphic to F via U , choose an isomorphism ’:U ×F → U ×X
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over U . Then the composite d∗ ’−1
d∗ ’
0 1 d∗1 (U × X ) = d∗0 (U × X ) −→ d∗0 (U × F) d∗1 (U × F) −→
is a descent datum on U × F. If Y is another such object with isomorphism :U ×F →U ×Y and if : X → Y is an isomorphism, then −1 (U × )’ : U × F → U × F is a morphism of the descent data associated to ’ and . It is easily seen that this is functorial. It is well known that the epimorphism U 1 is an e ective descent morphism for this bration (of objects over another) so that given a descent datum f : d∗1 (U × F) → d∗0 (U ×F) on U × F, there is an object X together with an isomorphism ’ : U × F → U × X over U , such that the descent datum determined by ’ is isomorphic to f. As a result, the functor above is an equivalence LocIso(U; F) ≈ Des1 (U × F): On the other hand, by the universal property of the group Aut(F) in E, descent data
are in 1–1 correspondence with morphisms U × U → Aut(F) which, because of the cocycle condition and normalization, are functors full(U ) → Aut(F), where full(U ) is the full equivalence relation on U regarded as a groupoid in E, and Aut(F) is considered to be a groupoid with one object. Morphisms of descent data correspond to natural transformations, yielding an equivalence Des1 (U × F) ≈ Fn(full(U ); Aut(F)); where Fn(full(U ); Aut(F)) is the category of functors and natural transformations. Let Gpd(E) be the category of groupoids in E and S(E) the category of simplicial objects in E. Let N : Gpd(E) → S(E) be the nerve functor, which is full and faithful. Natural transformations in Gpd(E) are given by functors 2 × G → H , which are sent by N to homotopies 4[1] × NG → NH , so N de nes an equivalence 1
Fn(full(U ); Aut(F)) ≈ C (U∗ ; Aut(F)); 1 1-cocycles whose objects are simplicial where C (U∗ ; Aut(F)) is the groupoid of Cech morphisms of U∗ = N full(U ) to N Aut(F), and whose morphisms are homotopies.
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A further equivalence is obtained by considering (one-dimensional) torsors. Recall that a torsor for Aut(F) is a nonempty object T (meaning T → 1 is epimorphic) provided with a (right) action a : T × Aut(F) → T which is free and transitive, i.e. such that (1 ; a) : T × Aut(F) → T × T is an isomorphism. Equivariant morphisms of torsors are automatically isomorphisms, so the category Tors1 (Aut(F)) is a groupoid. Let LocIso(F) denote the category of objects of E locally isomorphic to F (via some cover U 1), with isomorphisms as maps. If X is locally isomorphic to F, the object of isomorphisms Iso(F; X ) is an Aut(F) torsor with the action given by composition. If T is an Aut(F) torsor, T ⊗Aut(F) F is locally isomorphic to F (by localizing over T ). Thus, we have a pair of functors Iso(F;−)
LocIso(F)−−−−−→Tors1 (Aut(F)): ←−−−−− −⊗F Aut(F)
Furthermore, if X is locally isomorphic to F, and T is an Aut(F) torsor, we have natural maps T → Iso F; T ⊗ F Iso(F; X ) ⊗ F → X; Aut(F)
Aut(F)
induced by evaluation in the rst case, and hom-⊗ adjunction in the second. The rst is seen to be an isomorphism by localizing over Iso(F; X ), and the second is an isomorphism since it is an equivariant mapping of torsors. Thus, we have an equivalence LocIso(F) ≈ Tors1 (Aut(F)): 3. Dimension 2 In dimension 2 we want to classify groupoids locally equivalent to a given one, but this presents an immediate technical problem. Namely, in a topos E there are two kinds of equivalences. If G and H are groupoids in E, a functor f : G → H is called a weak equivalence, or categorical equivalence if it is full and faithful and essentially surjective. These are expressed by saying that
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is a pullback, and in the diagram
the map K → H0 is epimorphic. With this, of course, we are saying internally that every object of H is isomorphic to one in the image of f. f is a strong equivalence if there is a functor f0 : H → G such that f0 f ' idG and ff0 ' idH . In the presence of the axiom of choice, every weak equivalence is strong, but not in a general E. Also, the two classes of equivalences have quite di erent properties. For example, weak equivalences are not preserved by pullbacks in Gpd(E), but are re ected by pullbacks along an epimorphism of E. Strong equivalences are preserved under pullback, but not re ected. Some constructions result in weak equivalences some in strong, and in general we want strong. We deal with this problem in the following way. Recall [6] that a groupoid G in E is a stack, if for each weak equivalence e : A → B, any diagram
has a dotted ller making the resulting triangle commute up to isomorphism. Also, the weak equivalences are part of a Quillen homotopy structure on Gpd(E) in which the co brations are functors monomorphic on objects and the brant objects are strong stacks. Thus, G is a strong stack if for any co bration weak equivalence e : A ,→ B each diagram
has a dotted ller making the resulting triangle commute (not just up to isomorphism). In particular, if G is a groupoid in E, then factoring G → 1 as a co bration weak
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equivalence followed by a bration
we see that each G has a (strong) stack completion G∗ . What we need about stacks is the following. Proposition 1. Let e : G → H be a weak equivalence and G a stack. Then e is a strong equivalence. Proof. Fill in the following diagram:
obtaining e0 : H → G and an isomorphism : id G → e0 e. e0 is a weak equivalence since e and idG are. In the diagram
∼
∼
e0 : e0 (id H ) → e0 (ee0 ). But e0 is full and faithful, so e0 = e0 , where : id H → ee0 . We turn now to two-descent. Thus, let U 1 be a cover of 1, and
more of the simplicial nerve of U , with d0 = 234 , d1 = 134 , d2 = 124 and d3 = 123 in dimension 3. s0 = × U and s1 = U × . Let G be a groupoid in E=U , which amounts to a groupoid G in E, together with a functor G → dis(U ), where dis(U ) is the discrete groupoid in E with objects U . A two-dimensional (normalized) descent datum on G consists of a (strong) equivalence f : d∗1 (G) → d∗0 (G) in Gpd(E=U × U ) together with an isomorphism : id G → s0∗ f and an isomorphism in the triangle
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These should satisfy the equations d∗0 f = s1∗ ;
fd∗1 = s0∗
and the “cocycle condition”, which requires that the tetrahedron
in Gpd(E=U × U × U × U ) (drawn with even and odd faces detached) commutes: d∗0 · d∗2 = d∗3 · d∗1 : A 1-morphism of descent data (G; f; ; ) → (H; g; ; ) is a functor a : G → H in Gpd(E=U ) together with an isomorphism # in the square
of Gpd(E=U × U ) such that the cylinder
in Gpd(E=U ) and the prism
in Gpd(E=U × U × U ) commute.
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A 2-morphism
is a natural transformation
such that the cylinder
in Gpd(E=U × U ) commutes. These objects, 1-cells and 2-cells constitute a 2-category Des2 (U ), and there is an obvious 2-functor : Gpd(E) → Des2 (U ) given by (H) = (U × H; t; id; id), where t is the canonical switching. On morphisms (h) = (U × h; id) and ( ) = U × . Now denote by Stacks (E) the full subcategory of Gpd(E) whose objects are stacks, and by : Stacks(E) → Des2 (U ) the evident restriction of as de ned above. The objects of Des2 (U ) are now taken to be stacks G in E=U provided with a normalized two-dimensional descent datum. U 1 is said to be a morphism of e ective 2-descent if this is a 2-equivalence, i.e. a weak equivalence on the hom-categories such that each object of Des2 (U ) is strongly equivalent to one of the form (H). Theorem 2. Epimorphisms of E are morphisms of e ective 2-descent. Proof. We rst show that if G and H are stacks, the functor Stacks(G; H) → Des2 (U × G; U × H)
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induced by is a weak equivalence. For this it is easy to see that a functor f:U ×G→U × H compatible with the canonical descent data can be identi ed with a functor f : full(U ) × G → H: full(U ) → 1 is a weak equivalence, hence so is full(U ) × G → G. Thus, since H is a stack there is a dotted ller below making the triangle commute up to isomorphism.
Thus, the functor above is essentially surjective. Full and faithfulness follows since full(U ) × G → G is a weak equivalence. For the e ective part, let (G; f; ; ) be an object of Des2 (U ), where G is a stack. The normalized descent datum (f; ; ) on G de nes a Gpd(E)-valued pseudo-functor on full(U ), so we can apply the Grothendieck construction to obtain a Grothendieck bration p : E → full(U ). Here, for groupoids, Grothendieck bration means just (internal) path-lifting. The normalization condition implies that the pullback of p along the functor dis(U ) → full(U ), given by the identity on objects and on maps, is isomorphic to G. That is, there is a pullback diagram G y
−−−−−→
E p y
dis(U ) −−−−−→ full(U ) The groupoid we are looking for is E itself, for consider the diagram
full(U ) → 1 is a weak equivalence so 2 is, since weak equivalences are stable under product. Since idE is a weak equivalence q = (p; id) is also. Moreover, since 2 q = idE ,
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q is a strong equivalence with quasi inverse 2 . But 2 is not over full(U ), so we can not be sure q remains an equivalence when pulled back over dis(U ). However, q : E → full(U ) × E is a co bration weak equivalence and p is a Grothendieck bration, so over a boolean cover the diagram
has a dotted ller. It follows that q is a stable weak equivalence so that pulling back over dis(U ) yields a weak equivalence
which is strong, since G is a stack over U . To ensure that E is a stack, we can take the strong stack completion E → E∗ , for dis(U ) × E → dis(U ) × E∗ is a weak equivalence, which is strong since dis(U ) × E is a stack over dis(U ). We remark that the stack condition on G in the e ective part of the above proof is not necessary. In fact, if G is just a groupoid, the descent datum itself provides a quasi-inverse q0 to q which is over full(U ). However, in the rst part of the proof, stacks appear to be necessary. The functor induced by on the hom-groupoids will not be essentially surjective if H is not a stack. If G is a groupoid in E and U 1 is a cover of 1, we say another groupoid H is locally equivalent to G via U if there is a weak equivalence e : U × G → U × H over U . All such H together with weak equivalences and natural transformations form a 2-category wLocEq(U; G). Let LocEq(U; G∗ ) be the similarly de ned 2-category of stacks locally equivalent to G∗ , and let I : LocEq(U; G∗ ) → wLocEq(U; G) denote the inclusion. Then I is a 2-equivalence, since if H is locally equivalent to G, then H∗ is locally equivalent to G∗ , so H is weakly equivalent to a stack in the image of I . Note that all the weak equivalences in LocEq(U; G∗ ) are strong, including those in the de nition of local equivalence. Let F be a stack in E, U 1 a cover of 1, and LocEq(U; F) the 20 -groupoid of stacks locally equivalent to F via U , where by 20 -groupoid we mean a 2-category in which 1-cells are equivalences, and 2-cells are isomorphisms. Let Des2 (U × F) be the 20 -groupoid of 2-descent data on U × F, i.e. we restrict our attention to 1-cells which are equivalences. De ne a 2-functor LocEq(U; F) → Des2 (U × F)
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as follows. For each stack G locally equivalent to F, choose equivalences e:U ×F→U × G
and
e0 : U × G → U × F
over U , where e0 is an adjoint quasi-inverse to e, and isomorphisms : id → ee0 and 0 : id → e0 e. Let f be the composite d∗ e
d∗ e0
0 1 d∗1 (U × F) −→ d∗1 (U × G) = d∗0 (U × G) −→ d∗0 (U × F):
Then determines an isomorphism in
and 0 gives an isomorphism id → s0∗ f. For details concerning see [2]. If H is another such stack with equivalence d : U × F → U × H over U , and if ’ : G → H is an equivalence, then d0 · (U × ’) · e : U × F → U × F together with the # de ned in [2] is a 1-morphism of the descent data associated to e and d. Finally, if
is a 2-cell between equivalences on stacks locally equivalent to F, then d0 (U × )e is a 2-cell between the associated 1-morphisms of descent data. Since U 1 is a morphism of e ective 2-descent, it follows as in [2] that the 2-functor so de ned is a 2-equivalence LocEq(U; F) ≈ Des2 (U × F): Let Eq(F) be the groupoid of (weak) self equivalences of F. There is a weak equivalence
above Eq(F) where ev is evaluation. It is universal in the sense that for any groupoid G, functors G → Eq(F) are in 1–1 correspondence with weak equivalences
over G, where f is the pullback of (1 ; ev) along G → Eq(F).
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If F is a stack, as we are assuming, it is easy to see that Eq(F) is also. Furthermore, since stacks are stable under product, it follows that Eq(F) × F is a stack over Eq(F). Thus, the weak equivalence is strong. The classifying functor for an adjoint quasiinverse to (1 ; ev) provides a quasi-inverse assignment (−)0 : Eq(F) → Eq(F) making Eq(F) a strictly associative and unitary compact groupoid. Now, let (f; ; ) be a normalized 2-descent datum on U × F over U . Then
corresponds to a map U × U → Eq(F) which the normalization conditions, together with and the cocycle condition, say is part of a pseudo-functor full(U ) → Eq(F); where full(U ) is taken to be discrete in dimension 2, and Eq(F) is considered to be a 20 -groupoid with one object. 1-morphisms of descent data correspond to pseudo-natural transformations, and 2-morphisms to modi cations, so we obtain a 2-equivalence Des2 (U × F) ≈ Psd(full(U ); Eq(F)): Denote by 20 -Gpd(E) the category of 20 -groupoids in E with pseudo-functors as morphisms. In order to obtain the two-dimensional nerve functor N 2 : 20 -Gpd(E) → S(E); we must modify the de nition given in [7] to take account of the fact that our pseudofunctors no longer preserve units. This can be done, however, and the new N 2 is still full and faithful. It is easy to see that if ’; : G → H are pseudo-functors, then there is a pseudo-natural transformation : ’ → i there is a pseudo-functor h : 2 × G → H with h(0; −) = ’ and h(1; −) = . Since N 2 is full and faithful and takes pseudofunctors 2 × G → H to homotopies 4[1] × N 2 G → N 2 H, we obtain an isomorphism [N 2 G; N 2 H] ' 0 Psd(G; H) where [N 2 G; N 2 H] denotes the set of homotopy classes of simplicial maps. In particular, we obtain an isomorphism 2 H (U∗ ; Eq(F)) ' 0 Psd(full(U ); Eq(F)); 2 where H (U∗ ; Eq(F)) is [N 2 full(U ); N 2 Eq(F)].
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We turn now to torsors. A 2-torsor for Eq(F) is a non-empty stack T, meaning T0 1 is an epimorphism, together with a (right) action a : T × Eq(F) → T such that (1 ; a) : T × Eq(F) → T × T is an equivalence. A 1-morphism of torsors is an Eq(F)-equivariant functor. These are equivalences by localization, since any equivariant functor Eq(F) → Eq(F) is. a 2-morphism is a natural transformation compatible with the action of Eq(F) in the obvious sense. Let Tors2 (Eq(F)) denote the resulting 20 -groupoid. Write LocEq(F) for the 20 -groupoid of stacks locally equivalent to F via some cover U 1, with equivalences as 1-cells. Let G be a stack locally equivalent to F. Proposition 3. Eq(F; G) is an Eq(F) torsor. Proof. Eq(F; G) is non-empty since G is locally equivalent to F. Eq(F) acts on the right of Eq(F; G) by composition. Let (−)0 : Eq(F; G) → Eq(G; F) be the quasi-inverse assignment de ned as for Eq(F). Then a quasi-inverse to (1 ; a) : Eq(F; G) × Eq(F) → Eq(F; G) × Eq(F; G) is provided by the functor (−)0 × id
Eq(F; G) × Eq(F; G) −−−−−→ Eq(G; F) × Eq(F; G) (1 ; c)
−−−−−→ Eq(F; G) × Eq(F) where c denotes composition. Thus, we obtain a 2-functor Eq(F; −) : LocEq(F) → Tors2 (Eq(F)): Theorem 4. Eq(F; −) is a 2-equivalence. Proof. Let G and H be locally equivalent to F, and consider the functor : Eq(G; H) → Eq (Eq(F; G); Eq(F; H)) de ned by transposing the composition Eq(G; H) × Eq(F; G) → Eq(F; H); where Eq denotes equivariant equivalences. Let U 1 be a cover of 1 over which both G and H become equivalent to F. (−)×U is a logical functor, so × U becomes equivalent to Eq(F × U; F × U ) → Eq (Eq(F × U; F × U ); Eq(F × U; F × U ))
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which is an equivalence. It follows that is an equivalence, so Eq(F; −) induces equivalences on the hom-groupoids. To see that Eq(F; −) is 2-essentially surjective, let T be an Eq(F) torsor, and G a stack completion of T ⊗Eq(F) F. Thus, we have a weak equivalence e : T ⊗ F → G: Eq(F)
By transposition, e yields an equivariant functor f : T → hom(F; G): Let q : T × F → T ⊗Eq(F) F denote the quotient functor. Then the functor over T corresponding to f is
However, the diagram
commutes, where the two isomorphisms are canonical. It follows that (1 ; eq) is an equivalence, so that f is, in fact, an equivariant functor f : T → Eq(F; G) and, as such, an equivalence. References [1] L. Breen, Bitorseurs et cohomologie non abelienne, in: Pierre Cartier et al. (Eds.), The Grothendieck Festschrift, vol. I, Birkhauser, Basel, 1990, pp. 401– 476. [2] J. Duskin, An outline of a theory of higher dimensional descent, Bulletin de la Societe Mathematique de Belgique Ser. A 41(2) (1989) 249 –277. [3] J. Giraud, Cohomologie Non Abelienne, Springer, Berlin, 1971. [4] J. Giraud, Methode de la descente, Bulletin de la Societe Mathematique de France Supplement, Decembre 1964, Memoire 2.
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[5] M. Hakim, Topos Anneles et Schemas Relatifs, Springer, Berlin, 1972. [6] A. Joyal, M. Tierney, Strong stacks and classifying spaces, in: A. Carboni et al. (Eds.), Category Theory, Proceedings, Como 1990, Lecture Notes in Mathematics, vol. 1488, Springer, Berlin, 1991, pp. 213–236. [7] I. Moerdijk, Lectures on 2-dimensional groupoids, Rapport 175, Insitut de Mathematique Pure et Appliquee, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium, March 1990. [8] Neantro Saavedra Rivano, Categories Tannakiennes, Lecture Notes in Mathematics, vol. 265, Springer, Berlin, 1972.