Homology, Homotopy and Applications, vol.3(2), 2001, pp.361–384

STACKS AND THE HOMOTOPY THEORY OF SIMPLICIAL SHEAVES

J.F. JARDINE

(communicated by Gunnar Carlsson)

Abstract Stacks are described as sheaves of groupoids G satisfying an effective descent condition, or equivalently such that the classifying object BG satisfies descent. The set of simplicial sheaf homotopy classes [∗, BG] is identified with equivalence classes of acyclic homotopy colimits fibred over BG, generalizing the classical relation between torsors and non-abelian cohomology. Group actions give rise to quotient stacks, which appear as parameter spaces for the separable transfer construction in special cases.

Introduction

This paper arose from a personal need for a collection of technical statements about homotopy theoretic objects related to stacks in connection with a project on transfer maps for presheaves of spectra. This project is briefly described in Example 18 below: there is a very general pairing which is defined for finite maps of integral Noetherian schemes and arbitrary presheaves of spectra on the category of schemes, and this map specializes to a transfer in the case where a certain stack has a global section.

The present paper defines stacks (from several points of view) as sheaves of groupoids satisfying an effective descent condition, and then discusses the basic homotopy theoretic properties of their associated classifying simplicial sheaves. It is shown that any sheaf of groupoids G has a stack completion map η : G → St(G) such that St(G) is a stack (Lemma 9), and that the induced map η : BG → BG St(G) of classifying simplicial sheaves is a local weak equivalence (Lemma 7). The stack completion can be constructed either geometrically (following Giraud [4]) by putting in the missing cocycles, or by using the strong stack completion functor of Joyal and Tierney [12]. Either way, the object B St(G) satisfies descent, which means that every globally fibrant model B St(G) → X in the simplicial presheaf category is a pointwise weak equivalence in the sense that all maps B St(G)(U) → X(U) of sections are weak equivalences of simplicial sets (Theorem 6). The Joyal-Tierney strong stack completion G^{∧} of a sheaf of groupoids G, in fact, produces a simplicial

This research was supported by NSERC. Received February 4, 2001, revised August 15, 2001; published on September 13, 2001. 2000 Mathematics Subject Classification: Primary 18G50; Secondary 18F20, 18G30, 14A20. Key words and phrases: groupoids, stacks, simplicial sheaves, descent. \begin{matrix}c \\ ◯ \end{matrix} 2001, J.F. Jardine. Permission to copy for private use granted.

sheaf BG^{∧} which is globally fibrant on the nose. More generally, there is a case to be made for defining a stack to be a presheaf of groupoids G such that BG satisfies descent in the above sense. These results are proved in Sections 2 and 3 of this paper, after an identification of the topos theoretic notion of an internal equivalence G → H of sheaves of groupoids with a morphism which induces a local weak equivalence BG → BH of simplicial sheaves (Lemma 1) in Section 1.

Section 4 contains homotopy classification results which expand on the identification of the non-abelian cohomology object H^{1}(∗, G) for sheaves of groups G with the set morphisms [∗, BG] from the terminal object to BG in the homotopy category of simplicial sheaves [8]. Torsors for a sheaf of groupoids G are defined by internal functors Y → BG such that Y is weakly equivalent to a point. An internal functor is defined here to be a simplicial sheaf map such that Y is given by a homotopy colimit over G in each section. From this point of view, it is essentially no surprise that the set of path components of the category of G-torsors is isomorphic to [∗, BG] for arbitrary sheaves of groupoids G (Theorem 14).

In the case where G is a sheaf of groups and N is a sheaf carrying a G-action, Theorem 16 identifies path components of the quotient stack with morphisms [∗, EG×_{G} N] in the homotopy category taking values in the associated Borel construction, or classifying space of the translation groupoid. This result is exactly the technical device that is required for the transfer project, and it specializes to results of Breen [2] which classify various classes of bitorsors. The result is proven by means of a direct cocycle argument.

The results displayed here certainly do not tell the end of the story about stacks and homotopy theory. There is no attempt, for example, to discuss higher nonabelian cohomology objects, and the range of applications given here is very focussed. Stacks have been vigorously studied for many years, and the results of this paper are not particularly new in the sense that most of them are known in some other form. The synthesis presented here, however, is original and the results of this paper have not yet been collected together elsewhere.

1. Internal equivalences

This paper makes rather vigorous use of the homotopy theory of simplicial sheaves and presheaves on a small Grothendieck site C. The site C is a category equipped with a collection of subfunctors R ⊂ hom( , U), called covering sieves, of all contravariant functors represented by objects U ∈ C. The covering sieves satisfy various axioms: all functors hom( , U) should themselves be covering, covering sieves are closed under pullback (if R ⊂ hom( , U) is covering and ϕ : V → U is a morphism of C, then ϕ^{−1}R is a covering sieve of V), and the local character axiom should be satisfied. The local character axiom says that if R is a sieve and S is a covering sieve such that ϕ^{−1}R is covering for all members ϕ of S, then R is covering. Covering families in the sense of pretopologies generate sieves, and the standard axioms for a pretopology ensure that the associated sieves are covering sieves for a topology.

Recall that presheaves on C are just contravariant set-valued functors, while sheaves are presheaves F satisfying a limit condition

F (U) ≅ \underleftarrow{lim}_{ϕ:V →U∈R} F (V)

for all covering sieves R ⊂ hom(U, ) of C.

A simplicial presheaf (respectively sheaf) is a simplicial object in the category of presheaves (respectively sheaves) on C; a simplicial presheaf is alternatively just a contravariant functor on C taking values in simplicial sets. The categories of simplicial presheaves and simplicial sheaves have standard closed model structures in which the cofibrations are monomorphisms and the weak equivalences (called local weak equivalences) are maps which induce isomorphisms in all sheaves of homotopy groups, for all local choices of base points. In the presence of enough points, a local weak equivalence of simplicial presheaves or simplicial sheaves is a map which induces a weak equivalence of simplicial sets in all stalks. A fibration for the theory, called a global fibration, is a map which has the right lifting property with respect to all maps which are simultaneously cofibrations and weak equivalences.

The existence of the closed model structure for simplicial sheaves is a result of Joyal [11], while the closed model structure for simplicial presheaves came a little later [7]. The canonical map η : X → X̃ from a simplicial presheaf X to its associated simplicial sheaf is a local weak equivalence; it follows that the categories of simplicial sheaves and simplicial presheaves are models for the same homotopy category.

Global fibrations are somewhat mysterious objects; they are constructed from sets of generating cofibrations by transfinite inductions that involve cardinality counts which depend on the size of the site C. A useful rule of thumb is that globally fibrant objects behave like injective resolutions, and are the subject of descent theory: a simplicial presheaf X is said to satisfy descent if there is a local weak equivalence X → GX with GX globally fibrant, such that all maps in sections X(U) → GX(U) are weak equivalences of simplicial sets. In general, a local weak equivalence X → Y with Y globally fibrant is said to be a globally fibrant model for X. Any map Z → W which induces weak equivalences in all sections is said to be a pointwise (or sectionwise) weak equivalence. In particular, a simplicial presheaf X satisfies descent if some and hence any globally fibrant model for X is a pointwise weak equivalence.

We shall also encounter local fibrations [6] in this paper. In the presence of enough points, a local fibration is a map of simplicial presheaves f : X → Y which induces Kan fibrations in all stalks. More generally, a local fibration is characterized by a local right lifting property with respect to all inclusions Λ_{k}^{n} ⊂ Δ^{n} of horns in simplices: f : X → Y is a local fibration if the lifting problem

\xymatrix{Λ_{k}^{n} \ar[r]^{α} \ar[d] & X(U) \ar[d]^{f} \\ Δ^{n} \ar[r]_{β} \ar@{.>}[ur] & Y (U)}

has a local solution in the sense that there is some covering sieve R ⊂ hom( , U) such that the liftings exist in all diagrams

\xymatrix{Λ_{k}^{n} \ar[r]^{ϕ^{∗}α} \ar[d] & X(V) \ar[d]^{f} \\ Δ^{n} \ar[r]_{ϕ^{∗}β} \ar[ur] & Y (V)}

associated to ϕ : V → U in R.

It is a theorem [6, 7] that a map is a local fibration and a local weak equivalence if and only it has the local right lifting with respect to all inclusions ∂Δ^{n} ⊂ Δ^{n}. Such maps are also, variously, called acyclic local fibrations or hypercovers.

Now suppose, for the moment, that f : G → H is an ordinary functor of small groupoids. Then it’s well known and easy to see that the functor f induces a weak equivalence f_{∗} : BG → BH of the associated classifying simplicial sets if and only if f is an equivalence of categories. Put a different way, this means that the following two conditions hold:

1) the diagram of functions

\xymatrix{Arr(G) \ar[r]^{f} \ar[d]_{(s,t)} & Arr(H) \ar[d]^{(s,t)} \\ Ob(G) × Ob(G) \ar[r]_{f×f} & Ob(H) × Ob(H)}  (1)

is a pullback, and

2) for every object x of H there is an object y of G and a morphism x → f(y)

in H. Here, s = d_{1} is the source map and t = d_{0} is the target.

The first condition says that the functor f is fully faithful, while the second says that objects of H should lift to G up to isomorphism. In the context of classifying spaces, the first condition says simultaneously that f induces a monomomorphism π_{0}BG → π_{0}BH and an isomorphism on π_{1}(BG, x) → π_{1}(BH, f(x)) on all possible automorphism groups, while the second says that the map π_{0}BG → π_{0}BH is surjective. Every groupoid is equivalent to a disjoint union of groups, whose classifying spaces have no higher homotopy groups, so that these two conditions are enough to show that BG → BH is a weak equivalence.

Condition 2) above can be translated into diagram theoretic terms. Consider the picture

\xymatrix{Ob(G) ×_{Ob(H)} Arr(H) \ar[r]^{f_{∗}} \ar[d]_{t_{∗}} & Arr(H) \ar[r]^{s} \ar[d]^{t} & Ob(H) \\ Ob(G) \ar[r]_{f} & Ob(H)}

Then condition 2) above is equivalent to the following:

2’) the composite function

Ob(G) ×_{Ob(H)} Arr(H) ⟶^{f_{∗}} Arr(H) ⟶^{s} Ob(H)  (2)

is surjective. Conditions 1) and 2’) are easily translated into the topos theoretic setting, as

has been done in several places [3], [12]. Say that a map f : G → H of sheaves of groupoids (on a small site C) is an internal equivalence if the diagram (1) is a pullback of sheaves, and the composite map (2) is an epimorphism of sheaves. Note that (2) is an epimorphism if and only if for every section x ∈ Ob(H)(U) there is a covering sieve R for U such that ϕ^{∗}(x) is isomorphic to an object in the image of the functor G(V) → H(V) for every ϕ : V → U in R.

Lemma 1. Suppose that f : G → H is a morphism of sheaves of groupoids. Then the map f induces a local weak equivalence of simplicial sheaves f_{∗} : BG → BH if and only if f is an internal equivalence.

Proof. Condition 1) is equivalent to the assertion that f is fully faithful in the sense that the induced maps f : G(x, y) → H(fx, fy) is an isomorphism of sheaves on C ↓ U for all U ∈ C and for all x, y ∈ Ob(G)(U). This means that the map G(x, y)(U) → H(fx, fy)(U) is a bijection for all U ∈ C and all x, y ∈ Ob(G)(U).

Suppose that f_{∗} : BG → BH is a weak equivalence of simplicial sheaves. Then all induced maps f : G(x, x) → H(fx, fx) are isomorphisms of sheaves on C ↓ U for all U ∈ C and for all x ∈ Ob(G)(U). If G(x, y)(U) ≠ ∅, then there is a morphism α : x → y in G(U) and there is a commutative diagram

\xymatrix{G(x, x) \ar[r]^{f_{∗}} \ar[d]_{α_{∗}}^{≅} & H(fx, fx) \ar[d]^{f(α)_{∗}}_{≅} \\ G(x, y) \ar[r]_{f_{∗}} & H(fx, fy)}

so that the induced map f_{∗} : G(x, y) → H(fx, fy) is an isomorphism of sheaves. If H(fx, fy)(U) ≠ ∅, then since π_{0}BG → π_{0}BH is an isomorphism of sheaves there is a covering sieve R ⊂ hom( , U) such that ϕ^{∗}(x) and ϕ^{∗}(y) are in the same component of G(V) for all ϕ : V → U in R. It follows that G(x, y)(V) ≠ ∅ for all ϕ ∈ R, and the isomorphisms

G(x, y)(V) ⟶^{f} H(fx, fy)(V)

induce an isomorphism G(x, y)(U) → H(fx, fy)(U). If H(fx, fy)(U) = ∅ then G(x, y)(U) = ∅. It follows that G(x, y)(U) → H(fx, fy)(U) is an isomorphism for all x, y ∈ Ob(G)(U) and for all U ∈ C, giving condition 1). Condition 2) is a consequence of the fact that the map π_{0}BG → π_{0}BH is an isomorphism and hence an epimorphism of sheaves under the assumption that f_{∗} : BG → BH is a weak equivalence.

For the converse, it is evident that all maps

G(x, x) → H(fx, fx)

are isomorphisms under the assumption that the first condition is satisfied, so that f induces an isomorphism in all fundamental groups for all local choices of base point. The second condition implies that the induced map π_{0}BG → π_{0}BH is an epimorphism of sheaves. To see that this map is a monomorphism, it suffices to show that the underlying presheaf map π_{0}^{p}BG → π_{0}^{p}BH is a monomorphism of presheaves, but this is a consequence of the assumption that all maps

G(x, y)(U) → H(fx, fy)(U)

are isomorphisms. All sheaves of higher homotopy groups for BG and BH are trivial.

2. Stacks

Suppose that C is a small Grothendieck site. A stack is normally defined to be a sheaf of groupoids G on C which satisfies the effective descent condition.

The effective descent condition (Definition 2 below) can be a bit of a mouthful — three equivalent descriptions are given here. Suppose that R ⊂ hom( , U) is a covering sieve, and use the notation R for the full subcategory of C ↓ U whose objects are the morphisms ϕ : V → U of the sieve. In this notation, there is a canonical functor R → C which is defined by sending ϕ : V → U to the object V, and G|_{R} will denote the restriction of a sheaf or presheaf along this functor.

Following Giraud [4], an effective descent datum x = {x_{ϕ}} : R → G on R with coefficients in G consists of objects x_{ϕ} ∈ G(V) for each ϕ : V → U in R, and isomorphisms c_{ψ} : ψ^{∗}x_{ϕ} → x_{ϕψ} for each composable pair

W ⟶^{ψ} V ⟶^{ϕ} U

in R, such that c_{1} = 1 and for each sequence

W′ ⟶^{ω} W ⟶^{ψ} V ⟶^{ϕ} U,

the diagram

[figure: ω^{∗}ψ^{∗}x_{ϕ}; ω^{∗}c_{ψ}; ω^{∗}x_{ϕψ}; c_{ω}; (ψω)^{∗}x_{ϕ}; c_{ψω}; x_{ϕψω}]

commutes.

Alternatively, an effective descent datum can be viewed as a pseudo-natural transformation R → G|_{R}, where R is identified with a functor taking values in discrete categories, and so a morphism of descent data is most naturally a homotopy of pseudo-natural transformations. This means that a morphism f : x → y of descent data on R consists of morphisms f_{ϕ} : x_{ϕ} → y_{ϕ} in G(V), for each ϕ : V → U in R such that the diagrams

\xymatrix{ψ^{∗}x_{ϕ} \ar[r]^{ψ^{∗}f_{ϕ}} \ar[d]_{c_{ψ}} & ψ^{∗}y_{ϕ} \ar[d]^{c_{ψ}} \\ x_{ϕψ} \ar[r]_{f_{ϕψ}} & y_{ϕψ}}

Note that the collection of morphisms x → y of effective descent data x, y : hom( , U) → G can be identified up to isomorphism with the collection of morphisms x_{U} → y_{U} in G(U).

Finally, there is a homotopy theoretic description of effective descent data which is quite persuasive even though it depends on the smallness of the underlying site C. Observe that every covering sieve R ⊂ hom( , U) determines a functor R → Shv(C) which is defined by sending an object ϕ : V → U of R to (the sheaf associated to) V = hom( , V). The associated translation category E_{R} is best described in terms of its corresponding classifying object BE_{R}, which has sheaf of n-simplices given by the disjoint union

⨆_{ϕ_{0}→···→ϕ_{n}} V_{0}

indexed over strings of morphisms in R between objects ϕ_{i} : V_{i} → U. In other words, BE_{R} is the homotopy colimit of the objects appearing in the covering sieve R. From this point of view, an effective descent datum is a functor E_{R} → G, or equivalently a simplicial sheaf map BE_{R} → BG, and a morphism of effective descent data is a natural transformation E_{R} × 1 → G, or a simplicial homotopy BE_{R} × Δ^{1} → BG.

Write hom(E_{R}, G) for the groupoid of effective descent data on R with coefficients in G. Observe that the associated nerve B hom(E_{R}, G) is isomorphic to the simplicial function space hom(BE_{R}, BG). This function space is a Kan complex, since classifying spaces of groupoids are Kan complexes.

Let E_{U} denote the translation category associated to the covering sieve hom( , U) which is generated by the identity map 1_{U} on U. The set of path components π_{0} hom(E_{U}, G) can be identified with the set π_{0}G(U) of path components of the groupoid G(U), for any groupoid G. In effect, the functor G(U) → hom(E_{U}, G) which is defined by x ↦ {ϕ^{∗}x} is an equivalence of categories.

Definition 2. A sheaf of groupoids G satisfies the effective descent condition if the restriction functor

hom(E_{U}, G) → hom(E_{R}, G)

is an equivalence of categories for every covering sieve R ⊂ hom(U, ), and all objects U ∈ C.

This definition is equivalent to the classical statement, which is the requirement that every effective descent datum on a covering sieve R ⊂ hom( , U) taking values in G is determined up to isomorphism by some object of G(U). Note that the composite functor

G(U) → hom(E_{U}, G) → hom(E_{R}, G)

is already fully faithful, since G is a sheaf. The effective descent condition is therefore equivalent to the requirement that

π_{0} hom(E_{U}, G) → π_{0} hom(E_{R}, G)

is a bijection, or that the function

π_{0}G(U) → π_{0} hom(E_{R}, G)

is bijective.

The set of simplicial homotopy classes of maps π(BE_{R}, BG) coincides with the set π_{0}(E_{R}, G) of functors mod natural isomorphism. Thus, yet another way of stating the effective descent condition for G is to require that every simplicial map BE_{R} → BG extends to a map BE_{U} → BG which is unique up to naive homotopy.

Suppose that {ϕ_{i} : V_{i} → U} is a covering family for the object U of C, and that G is a stack. Write V_{•} for the Čech resolution of U which is associated to the covering. A simplicial sheaf map α : V_{•} → BG consists of objects α_{i} ∈ G(V_{i}) and morphisms

α_{i,j} : α_{i}|_{V_{i}×_{U} V_{j}} → α_{j}|_{V_{i}×_{U} V_{j}}

of G(V_{i} ×_{U} V_{j}), which satisfy the obvious cocycle condition (ie. composition law) in G(V_{i} ×_{U} V_{j} ×_{U} V_{k}). We also require that the morphisms α_{i,i} are identities. Write hom(V_{•}, BG) for the groupoid of cocyles and isomorphisms of cocycles. The Čech resolution V_{•} is the nerve of a groupoid, so that the nerve B hom(V_{•}, BG) coincides with the function space hom(V_{•}, BG), and this function space is a Kan complex.

Suppose that R is the covering sieve associated to the family {V_{i} → U}. Then R consists of those maps ϕ : V → U which factor through some V_{i}. It is plain that any simplicial sheaf map BE_{R} → BG (for an arbitrary sheaf of groupoids G) restricts to a cocycle V_{•} → BG: the elements x_{i} ∈ G(U_{i}) are defined by x_{i} = x_{ϕ_{i}}, and the map α_{i,j} is defined by the composite isomorphism

pr_{i}^{∗}x_{i} → x_{p_{i,j}} ← pr_{j}^{∗}x_{j}

arising from the diagram

\xymatrix{V_{i} ×_{U} V_{j} \ar[r]^{pr_{j}} \ar[d]_{pr_{i}} \ar[dr]^{p_{i,j}} & V_{j} \ar[d]^{ϕ_{j}} \\ V_{i} \ar[r]_{ϕ_{i}} & U}

Restriction plainly preserves isomorphism of the data, so we get a functor

hom(E_{R}, G) → hom(V_{•}, BG)

and hence a function

π(BE_{R}, BG) → π(V_{•}, BG)

relating naive homotopy classes.

Lemma 3. Suppose that R is the covering sieve which is generated by a covering family {ϕ_{i} : V_{i} → U}. Then the restriction function π(BE_{R}, BG) → π(V_{•}, BG) is a bijection for any sheaf of groupoids G.

Proof. Suppose that the cocycle α : V_{•} → BG is defined by objects α_{i} ∈ G(V_{i}) and isomorphisms α_{i,j} in G(V_{i} ×_{U} V_{j}). Choose specific factorizations

\xymatrix{V \ar[r]^{ψ} \ar[dr]_{ϕ} & V_{i} \ar[d]^{ϕ_{i}} \\  & U}

for all ϕ : V → U in R, where ψ is the identity map if ϕ = ϕ_{i} for some i. Define an object x_{ϕ} ∈ G(V) by x_{ϕ} = ψ^{∗}α_{i} for each ϕ. If τ : W → V is any other morphism, then the isomorphisms α_{i,j} induce a morphism τ^{∗}x_{ϕ} → x_{ϕτ} in G(W), and the collection of all such elements and morphisms determines an effective descent datum {x_{ϕ}} : R → G on account of the cocycle conditions for α. The assignment {x_{i}} ↦ {x_{ϕ}} defines a functor hom(V_{•}, BG) → hom(E_{R}, G) which is inverse to the restriction functor up to isomorphism.

We can now prove the following:

Proposition 4. Suppose that G is a stack, and that U is an object of C. Then there is an isomorphism

[U, BG] ≅ π_{0}G(U).

Proof. There is an isomorphism

[U, BG] ≅ \underrightarrow{lim}_{V} π(V, BG),

where the colimit is indexed over hypercovers V → K(U, 0) of the object U. We can assume [6] that these hypercovers are of the form V_{•} → U for some sheaf epimorphism V → U, and then a cofinality argument allows us to presume that the sheaf epi ⨆ V_{i} → U arises from some covering family {V_{i} → U}. The canonical map

π_{0}G(U) → π(V_{•}, BG)

is the composite

π_{0}G(U) ≅ π_{0} hom(E_{U}, G) → π_{0} hom(E_{R}, G) → π_{0} hom(V_{•}, BG)

if R is the sieve which is generated by the covering family {V_{i} → U}. This composite is a bijection by the assumption that G is a stack, along with Lemma 3

Recall that a map f : X → Y of simplicial presheaves on C is said to be a pointwise weak equivalence if f induces a weak equivalence f : X(U) → Y (U) of simplicial sets for all objects U ∈ C.

Lemma 5. Suppose that G is a sheaf of groupoids and that x is a global section of Ob(G). Suppose that BG → X is a globally fibrant model of BG. Then the induced map Ω_{x}BG → Ω_{x}X is a pointwise weak equivalence.

Proof. The section x is a global choice of basepoint x : ∗ → BG for BG, and Ω_{x}BG is the corresponding loop object. Let the presheaf of groupoids G_{x} be the connected component of x in G in all sections, and observe that the inclusion G(x, x) → G_{x} is an equivalence of groupoids in all sections. Then there is an isomorphism Ω_{x}BG ≅ Ω_{x}BG_{x}, and there are pointwise weak equivalences of simplicial presheaves

Ω_{x}BG_{x} ← ΩBG(x, x) → P BG(x, x) ×_{BG(x,x)} EG(x, x) ← G(x, x)

The “simplicial” object G(x, x) is a sheaf concentrated in degree 0 and is therefore globally fibrant. It follows that every local weak equivalence Ω_{x}BG → Y with Y globally fibrant must be a pointwise weak equivalence. The map Ω_{x}BG → Ω_{x}X is such a map.

The following is the main result of this section. In colloquial terms, it asserts that classifying spaces of stacks satisfy descent.

Theorem 6. Suppose that G is a stack of groupoids on a small site C, and that f : BG → X is a globally fibrant model of its associated classifying object. Then f is a pointwise weak equivalence.

Proof. Consider the restricted map BG|_{U} → X|_{U} on the site C ↓ U. This map is a local weak equivalence, and X|_{U} is globally fibrant. The induced map in global sections is BG(U) → X(U). To show that this map is a weak equivalence of simplicial sets, it suffices to show that it induces an isomorphism π_{0}BG(U) ≅ π_{0}X(U) and to show that for any choice of base point x ∈ BG(U) the induced map Ω_{x}BG(U) → Ω_{x}X(U) is a weak equivalence of simplicial sets (note that X(U) is already a Kan complex, so this makes sense). The statement about path components is a consequence of Proposition 4, while the statement about loop spaces is a consequence of Lemma 5.

3. Stack completion

Suppose that R is a covering sieve for U and that G is a sheaf of groupoids. Recall that the groupoid hom(E_{R}, G) has as objects all effective descent data x = {x_{ϕ}} for R, and all morphisms (or isomorphisms) x → y of descent data, defined in the obvious way. Every refinement S ⊂ R of covering sieves determines a functor hom(E_{R}, G) → hom(E_{S}, G) defined by restriction x ↦ x|_{S} of descent data. Write

St^{p}(G)(U) = \underrightarrow{lim}_{R} hom(E_{R}, G)

for the filtered colimit in the category of groupoids of all such categories of effective descent data. Every map ϕ : V → U in the site C and every covering sieve R of U together determine a functor

ϕ^{∗} : hom(E_{R}, G) → hom(E_{ϕ^{−1}R}, G),

again by restriction of descent data. This functor respects the restriction functors coming from an inclusion S ⊂ R of covering sieves, and so ϕ : V → U induces a functor ϕ^{∗} : St^{p}(G)(U) → St^{p}(G)(V). Write St(G) for the sheaf of groupoids associated to the presheaf St^{p}(G). There is a natural morphism

η : G → St(G)

of sheaves of groupoids.

The object St(G) is the stack completion of G (or “associated stack” in [4]), but we have to show that the name makes sense. We shall prove that

1) the induced map η_{∗} : BG → B St(G) is a local weak equivalence of simplicial sheaves, and

2) the sheaf of groupoids St(G) is a stack.

Lemma 7. Suppose that G is a sheaf of groupoids. Then the induced map η_{∗} : BG → B St(G) is a local weak equivalence of simplicial sheaves.

Proof. We verify the conditions 1) and 2’) for η — see Lemma 1.

We have already seen that the functor G(U) → hom(E_{R}, G) is fully faithful, and η is a filtered colimit of such functors so that η is fully faithful, giving condition 1).

Suppose that x = {x_{ϕ}} is an object of hom(E_{R}, G), and suppose that ϕ : V → U is a morphism of R. Then the covering sieve

ϕ^{−1}(R) = {α : W → V | ϕ · α ∈ R}

coincides with hom( , V). It follows that the class ϕ^{∗}(x) lifts to G(V) up to isomorphism for all ϕ : V → U in R. Every element of Ob(St(G))(U) lifts locally to Ob(St^{p}G), and we have just seen that every element of Ob(St^{p}G)(V) lifts locally up to isomorphism to an element of Ob(G). It follows that every element of Ob(StG)(U) lifts locally up to isomorphism to an element of Ob(G), and condition 2’) is verified.

Lemma 8. Suppose that G is a sheaf of groupoids, and suppose that S ⊂ R ⊂ hom( , U) are covering sieves of U. Then the restriction functor

hom(E_{R}, G) → hom(E_{S}, G)

is fully faithful.

Proof. Suppose that f : x|_{S} → y|_{S} is a morphism of restricted descent data. Suppose that ϕ : V → U is a morphism of R, and let ψ : W → V be a morphism of the covering sieve ϕ^{−1}S for V. Then there are morphisms F_{ψ} : ψ^{∗}x_{ϕ} → ψ^{∗}y_{ϕ} which are uniquely determined by the commutativity of the diagram

\xymatrix{ψ^{∗}x_{ϕ} \ar[r]^{F_{ψ}} \ar[d]_{c_{ψ}} & ψ^{∗}y_{ϕ} \ar[d]^{c_{ψ}} \\ x_{ϕψ} \ar[r]_{f_{ϕψ}} & y_{ϕψ}}

The maps F_{ψ} are compatible with composition in ϕ^{−1}S, and therefore uniquely determine a map F_{ϕ} : x_{ϕ} → y_{ϕ}.

Seeing that the diagrams

\xymatrix{γ^{∗}x_{ϕ} \ar[r]^{γ^{∗}F_{ϕ}} \ar[d]_{c_{γ}} & γ^{∗}y_{ϕ} \ar[d]^{c_{γ}} \\ x_{ϕγ} \ar[r]_{F_{ϕγ}} & y_{ϕγ}}

commute is a matter of applying the same principle that creates the maps F_{ϕ}: one shows that the diagram commutes after applying ω^{∗}, where ω is any map such that ϕγω is a member of the sieve S, and this is a consequence of the cocycle condition for x and y together with the definition of the morphisms F_{ϕ}.

We have shown that the restriction map

hom(E_{R}, G)(x, y) → hom(E_{S}, G)(x|_{S}, y|_{S})

is surjective. The maps f_{ϕ} : x_{ϕ} → y_{ϕ} are uniquely determined by the restriction to S, by the same argument, so the restriction map is also a monomorphism.

Lemma 9. Suppose that G is a sheaf of groupoids. Then the sheaf of groupoids St(G) is a stack.

Proof. We have to show that the functor St(G)(U) → hom(E_{R}, St(G)) is an equivalence of categories, for every U ∈ C and for every covering sieve R ⊂ hom( , U). We already know that the functor

H(U) → hom(E_{R}, H)

is fully faithful for arbitrary sheaves of groupoids H. It suffices to show that every object of hom(E_{R}, St(G)) lifts to St(G)(U) up to isomorphism.

Suppose that x = {x_{ϕ}} is an object of hom(E_{R}, St(G)). Then there is a refinement S ⊂ R such that the restriction of x to S comes equipped with isomorphisms d_{ψ} : η(z_{ψ}) → x_{ψ} for each ψ : W → U in S — this follows from Lemma 7. The functor η : G → St(G) is fully faithful, so that a unique descent datum structure in G for the sieve S is induced on the elements z_{ψ} by the descent datum structure on x|_{S}. In effect, given the string

W ⟶^{ω} V ⟶^{ψ} U

the cocycle map c_{ω} : ω^{∗}z_{ψ} → x_{ψω} is the unique morphism which makes the following diagram of isomorphisms commute:

[figure: η(ω^{∗}z_{ψ}); η(c_{ω}); ω^{∗}η(z_{ψ}); ω^{∗}d_{ψ}; ω^{∗}x_{ψ}; c_{ω}; η(z_{ψω}); d_{ψω}; x_{ψω}]

The diagram of canonical functors

\xymatrix{G(U) \ar[r]^{η} \ar[d] & St(G)(U) \ar[d] \\ hom(E_{S}, G) \ar[r]_{η_{∗}} \ar@{-->}[ur] & hom(E_{S}, St(G))}  (3)

commutes, and we have just seen that x|_{S} is in the image of

η_{∗} : hom(E_{S}, G) → hom(E_{S}, St(G))

up to isomorphism. It follows that x|_{S} lifts to an element y ∈ St(G)(U) up to isomorphism. Finally y|_{R} and x restrict to isomorphic elements on S, so that there is an isomorphism y|_{R} → x by Lemma 8.

Lemma 10. Suppose that G is a sheaf of groupoids. Then the simplicial presheaf B St^{p}(G) satisfies descent, in the sense that any globally fibrant model B St^{p}(G) → X is a pointwise weak equivalence.

Proof. It suffices to show that the map η_{∗} : π_{0} St^{p}(G)(U) → π_{0} St(G)(U) which is induced by the associated sheaf map is a bijection, and that all presheaves of homomorphisms of St^{p}(G) are sheaves. These two conditions imply that η : B St^{p}(G) → B St(G) is a pointwise weak equivalence, and then one can invoke Theorem 6.

There is a commutative diagram of canonical functors

\xymatrix{G(U) \ar[r] \ar[d] & St^{p}(G)(U) \ar[r] \ar[d] & St(G)(U) \ar[d] \\ hom(E_{R}, G) \ar[r] \ar[ur] & hom(E_{R}, St^{p}(G)) \ar[r] & hom(E_{R}, St(G))}

for any covering sieve R of U (this diagram refines (3)). For any object x ∈ St(G)(U) there is a covering sieve R such that the corresponding descent datum {ϕ^{∗}x} is isomorphic to a descent datum {η(z_{ϕ})} where {z_{ϕ}} is a descent datum in G. But then there is an element y ∈ St^{p}(G)(U) such that η(y) and x restrict to isomorphic elements in hom(E_{R}, St(G)). It follows that there is an isomorphism η(y) ≅ x. If two elements x, y ∈ G(U) become isomorphic in St^{p}(G)(U) then there is an isomorphism x|_{R} → y|_{R} in hom(E_{R}, G) for some covering sieve R of U. The functor G(U) → hom(E_{R}, G) is fully faithful, so that x ≅ y in G(U) since G is a sheaf. It follows that the map

η_{∗} : π_{0} St^{p}(G)(U) → π_{0} St(G)(U)

is a bijection.

For the sheaf property, we start with a preliminary case. Suppose that x, y ∈ hom(E_{R}, G), and that there are compatible maps α_{ϕ} : x|_{ϕ^{−1}R} → y|_{ϕ^{−1}R} for all ϕ ∈ R. Then for all ϕ : V → U in R, ϕ^{−1}R = {1_{V} }, so that α_{ϕ} uniquely determines an isomorphism α_{ϕ} : x_{ϕ} → y_{ϕ} ∈ G(V). Any γ : W → V, determines a commutative diagram

\xymatrix{γ^{∗}x_{ψ} \ar[r]^{γ^{∗}α_{ψ}} \ar[d]_{c} & γ^{∗}y_{ψ} \ar[d]^{c} \\ x_{ψγ} \ar[r]_{α_{ψγ}} & y_{ψγ}}

where the cocycle structures come from R, since the maps α_{ψ} are compatible with restriction. The family {α_{ϕ}} therefore determines a unique map x → y in hom(E_{R}, G).

Suppose that S is a covering sieve of U, and suppose that there are maps α_{ψ} : x|_{ψ^{−1}R} → y|_{ψ^{−1}R} for ψ ∈ S, and that these maps are compatible with restriction. Then for all ψ : V → U in the covering sieve R ∩ S, the compatible family of morphisms α_{ψ} uniquely determines a morphism α : x|_{R∩S} → y|_{R∩S} of hom(E_{R∩S}, G). If ϕ ∈ S and γ : W → V ∈ ϕ^{−1}R, then ϕγ ∈ R ∩ S and there is a commutative diagram of functors

\xymatrix{hom(E_{R}, G) \ar[r]^{res} \ar[d]_{ϕ^{∗}} & hom(E_{R∩S}, G) \ar[d]^{(ϕγ)^{∗}} \\ hom(E_{ϕ^{−1}R}, G) \ar[r]_{γ^{∗}} & hom(E_{W} , G)}

The restriction functor is fully faithful by Lemma 8, so that there is a unique morphism β : x → y such that β_{R∩S} = α. But then

γ^{∗}ϕ^{∗}(β) = (ϕγ)^{∗}(α) = α_{ϕγ} = γ^{∗}α_{ϕ}

for all γ, so that ϕ^{∗}(β) = α_{ϕ} for all ϕ ∈ S. The morphism β is already determined by its restriction to R ∩ S, so there is a unique β : x → y in hom(E_{R}, G) such that ϕ^{∗}(β) = α_{ϕ} for all ϕ ∈ S.

More generally, if we have x, y ∈ hom(E_{R}, G) and a compatible family of morphisms ω_{ϕ} : x_{ϕ} → y_{ϕ} in St^{p}(G)(V) for all ϕ : V → U in the covering sieve S, then in the diagram

\xymatrix{hom(E_{R}, G) \ar[r] \ar[d]_{ϕ^{∗}} & St^{p}(G)(U) \ar[d]^{ϕ^{∗}} \\ hom(E_{ϕ^{−1}R}, G) \ar[r] & St^{p}(G)(V)}

the canonical functor hom(E_{ϕ^{−1}R}, G) → St^{p}(G)(V) is fully faithful by Lemma 8, so that there is a unique morphism α_{ϕ} of hom(E_{ϕ^{−1}R}, G) which maps to ω_{ϕ}, and the family {α_{ϕ}} is compatible. It therefore follows from the previous case that the presheaf of morphisms St^{p}(G)(x, y) on C ↓ U is a sheaf.

Joyal and Tierney show [12] that there is a closed model structure on the category of sheaves of groupoids on a small Grothendieck site C for which the cofibrations are the functors G → H that are injective on objects, and the weak equivalences are the internal equivalences. They say that a sheaf of groupoids G is a strong stack if it is fibrant for this closed model structure, and a strong stack completion H → H^{∧} for a sheaf of groupoids H is a choice of fibrant model within their closed model structure.

Observe that if f : X → Y is a local weak equivalence of simplicial sheaves on C, then the associated morphism f_{∗} : π(X) → π(Y) of sheaves of fundamental groupoids is an internal equivalence. Also, if f is a cofibration of simplicial presheaves, then the map f_{∗} : π(X) → π(Y) is injective on objects, and hence is a cofibration of sheaves of groupoids in the sense of Joyal and Tierney. The fundamental groupoid construction is left adjoint to the classifying space construction, so the simplicial sheaf BG is globally fibrant if G is a strong stack. It follows that any strong stack completion H → H^{∧} induces a globally fibrant model BH → BH^{∧} of the classifying simplicial sheaf BH of H. The strong stack completion H^{∧} is itself a stack, since we have the following:

Lemma 11. Any covering sieve R ⊂ hom( , U) of an object U induces a local weak equivalence BE_{R} → K(U, 0) of simplicial presheaves.

The lemma implies that the induced map

BH^{∧}(U) ≅ hom(K(U, 0), BH^{∧}) → hom(BE_{R}, BH^{∧})

is a weak equivalence of simplicial sets since BH^{∧} is globally fibrant.

Proof of Lemma 11. Suppose that W ∈ C, and consider the induced map of W - sections

⨆_{ϕ_{0}→···→ϕ_{n}} hom(W, V_{0}) → hom(W, U).

The fibre F_{ϕ} over a fixed morphism ϕ : W → U is the nerve of the category of factorizations

\xymatrix{ & V \ar[d]^{ψ} \\ W \ar[ur] \ar[r]_{ϕ} & U}

of ϕ with ψ ∈ R. If ϕ : W → U is a member of R and then this category is non-empty and has an initial object, namely the picture

\xymatrix{ & W \ar[d]^{ϕ} \\ W \ar[ur]^{1} \ar[r]_{ϕ} & U}

Find a factorization of the canonical map

\xymatrix{BE_{R} \ar[d]_{i} \ar[r] & K(U, 0) \\ F \ar[ur]_{π}}

where i is a pointwise weak equivalence and π is a pointwise Kan fibration. In W - sections, and in the case where the fibre F_{ϕ} of the canonical map over ϕ : W → U is non-empty, then the fibre of π over ϕ is a fibrant model of F_{ϕ}, and if ϕ is in R this space is contractible. It follows that all lifting problems

\xymatrix{∂Δ^{n} \ar[r] \ar[d] & F (W) \ar[d]^{π} \\ Δ^{n} \ar[r] & K(U, 0)(W)}

have solutions if n > 0. But also every vertex ϕ : W → U lifts locally to BE_{R} since ϕ^{−1}(R) is a covering sieve for W. It follows that the map π : F → K(U, 0) is a local trivial fibration, and hence a local weak equivalence.

We now have the following corollaries of Theorem 6, Lemma 7 and Lemma 9

Corollary 12. Suppose that G is a stack on the site C. Then any strong stack completion G → G^{∧} induces a pointwise weak equivalence BG → BG^{∧}. In other words, the groupoid G(U) is equivalent to G^{∧}(U) in each section.

Corollary 13. Suppose that G is a sheaf of groupoids on C. Then the stack completion St(G) is pointwise equivalent to any strong stack completion G^{∧} of G.

We’re henceforth going to take the most inclusive point of view, and say that a presheaf of groupoids G is a stack if the associated simplicial presheaf BG satisfies descent. A morphism G → H of presheaves of groupoids is an internal equivalence if the associated map BG → BH is a local weak equivalence of simplicial presheaves. Finally, a stack completion of a presheaf of groupoids H is an internal equivalence H → H′ such that H′ is a stack. Stack completions exist from several points of view: one is entitled to use the composites

H ⟶^{η} H̃ → H̃′

where η is the associated sheaf functor, and H̃′ can be any of the strong stack completion H̃^{∧}, the stack completion St(H̃) or the “pre-stack completion” St^{p}(H̃). Any two stack completions of a presheaf of groupoids H are pointwise equivalent by the standard tricks [7].

The cocycle approach which is implicit in the construction H ↦ St^{p}(H) appears most often in geometric applications. If, for example, G is a sheaf of groups, then the usual argument relating cocyles and torsors specializes to the assertion that the groupoid St^{p}(G)(U) is equivalent to the groupoid of G-torsors Tors(U, G) over an object U in C.

4. Some classification results

Suppose that G is a sheaf of groups on C, and recall that a left G-torsor is a sheaf P equipped with a free left G-action G × P → P, such that the sheaf P/G is isomorphic to the terminal sheaf ∗, and such that there is a sheaf epi U → ∗ with an associated G-equivariant isomorphism

G × U ≅ U × P.

Recall that it suffices to take U = ⨆_{i} U_{i}, where U_{i} is a set of representables, by standard nonsense. Furthermore, triviality of P over U in the sense of the isomorphism above amounts to the existence of a section σ : U → P of P, for then the displayed isomorphism is defined by the map

(g, u) ↦ (u, g · σ(u)).

In particular, the G-torsor P is trivial, or equivariantly isomorphic to G, if and only if it has a global section.

The non-abelian cohomology object H^{1}(C, G) is the set of isomorphism classes of G-torsors on the site C, as usual.

It is shown in [8] that there is an identification

H^{1}(C, G) ≅ [∗, BG],  (4)

where the thing on the right is morphisms in the homotopy category of simplicial sheaves (or presheaves) on the site C. The proof of this result starts with the Verdier hypercovering characterization

[∗, BG] ≅ \underrightarrow{lim}_{V} π(V, BG)

of morphisms in the homotopy category (BG is locally fibrant), where π( , ) indicates simplicial homotopy classes of maps and the filtered colimit is indexed over the hypercovers (or local trivial fibrations) V → ∗. Simplicial homotopy classes of maps V → BG can be formally identified with homotopy classes of functors π(V) → G of sheaves of groupoids, where π(V) is the fundamental groupoid of V in the sheaf category, and then the heart of the demonstration of the identification (4) is to show that map V → cosk_{0} V_{0} is isomorphic to the canonical map V → Bπ(V), or that π(V) is isomorphic to the trivial groupoid on the sheaf V_{0}. Homotopy classes of functors π(V) → G may then be identified with equivalence classes of cocycles cosk_{0} V_{0} → BG for the covering V_{0} → ∗, and so there is an isomorphism

\underrightarrow{lim}_{V} π(V, BG) ≅ \underrightarrow{lim}_{V_{0}} π(cosk_{0} V_{0}, BG).

Thus, [∗, BG] is identified with the Čech invariant Ȟ^{1}(C, G), which is well known to coincide with the set of isomorphism classes of G-torsors.

Suppose that I is a small category. A set-valued functor X : I → Sets determines a simplicial set EX and a map π : EX → BI such that if 0 : 0 → n is the ordinal number map which picks out the object 0 then the diagram

\xymatrix{EX_{n} \ar[r]^{0^{∗}} \ar[d]_{π} & EX_{0} \ar[d]^{π} \\ BI_{n} \ar[r]_{0^{∗}} & BI_{0}}

is a pullback. In effect, EX is the homotopy colimit, with

EX_{n} = ⨆_{i_{0}→···→i_{n}} X(i_{0}),

and π is the obvious map.

Conversely, if π : Y → BI is a simplicial set map such that all diagrams

\xymatrix{Y_{n} \ar[r]^{0^{∗}} \ar[d]_{π} & Y_{0} \ar[d]^{π} \\ BI_{n} \ar[r]_{0^{∗}} & BI_{0}}  (5)

are pullbacks, then Y ≅ EX for some functor X : I → Sets. To see this, set X(i) = π^{−1}(i) for the map Y_{0} → BI_{0}. Also

Y_{n} = ⨆_{\underline{i}=i_{0}→···→i_{n}} π^{−1}(\underline{i})

The pullback squares (5) induce isomorphisms π^{−1}(\underline{i}) ≅ π^{−1}(i_{0}) = X(i_{0}) of fibres, which together determine an isomorphism over BI of the simplicial set Y with a simplicial set having n-simplices of the form

⨆_{i_{0}→···→i_{n}} X(i_{0})

such that all diagrams

\xymatrix{X(i_{0}) \ar[r]^{1} \ar[d]_{in\underline{i}} & X(i_{0}) \ar[d]^{in_{i_{0}}} \\ ⨆_{i_{0}→···→i_{n}} X(i_{0}) \ar[r]_{0^{∗}} & ⨆_{i} X(i)}

commute. The map

d_{0} : ⨆_{α:i_{0}→i_{1}} X(i_{0}) → ⨆_{i} X(i)

defines functions α_{∗} : X(i_{0}) → X(i_{1}) on summands. Any ordinal number map θ : m → n such that θ(0) = 0 necessarily induces a function

θ^{∗} : ⨆_{i_{0}→···→i_{n}} X(i_{0}) → ⨆_{j_{0}→···→j_{m}} X(j_{0})

which restricts to identity maps on summands, by comparison of pullbacks. It follows that an arbitrary ordinal number map γ : r → n induces a function

⨆_{i_{0}→···→i_{n}} X(i_{0}) → ⨆_{j_{0}→···→j_{r}} X(j_{0})

which is induced on summands by the maps X(i_{0}) → X(i_{θ(0)}) determined by the morphism i_{0} → i_{θ(0)} in the underlying index category I. The assignments of α_{∗} :

X(i) → X(j) to the maps α : i → j therefore determine a functor X : I → Sets such that EX ≅ Y.

It follows that the category of functors X : I → Sets and natural transformations is equivalent to the category of simplicial set maps π : Y → BI satisfying the pullback condition (5), with fibrewise maps over BI as morphisms. The equivalence arises from the homotopy colimit construction.

This equivalence is natural, and therefore gives an internal description of sheafvalued functors I → Shv(C) defined on a small sheaf of categories I over a site C.

Suppose now that G is a sheaf of groupoids on C. A G-torsor on C can be defined to be a simplicial sheaf map Y → BG which satisfies the pullback condition (5), and such that the canonical map Y → ∗ is a local weak equivalence. From the development above, Y is the nerve EX of a sheaf of groupoids, so that the condition that Y → ∗ is equivalent to requiring that G acts transitively and effectively on the sheaf Y_{0}. If G is a sheaf of groups, then this definition says that a G-torsor is the G-space Y_{0} and

Y ≅ EG ×_{G} Y_{0} ≃ ∗,

which is equivalent to the classical definition. The set of G-torsors and natural transformations (or deck transformations) between them forms a category which will be denoted by Tors(∗, G).

The sheaf of path components π_{0}G of BG determines a local fibration BG → π_{0}G. Take a global section x ∈ π_{0}G(∗). Then the fibre over x is the classifying object BG_{x} of a sheaf of groupoids G_{x} which we shall call the path component of x. Note that G_{x} is locally connected. If Y → BG is a G-torsor, then Y is locally connected so that it factors through a unique path component BG_{x} ⊂ BG of G. If α : Y → Y′ is a morphism of G-torsors, then Y and Y′ factor through the same path component BG_{x}. It follows that

Tors(∗, G) = ⨆_{x∈π_{0}G(∗)} Tors(∗, G_{x}).

If G_{x} has a global section y, then the inclusion G_{x}(y, y) → G_{x} is an internal equivalence, and any map θ : Y → Y′ of G_{x}-torsors pulls back to a morphism θ_{∗} of G_{x}(y, y)-torsors. This map θ^{∗} of group-valued torsors must be an isomorphism, so that θ is an isomorphism as well since G_{x} is locally connected. It follows that every morphism of G-torsors is an isomorphism, for arbitrary sheaves of groupoids G.

Every G-torsor Y → BG has within it a hypercover Y → ∗, and every morphism Y → Y′ of torsors is a morphism of hypercovers. It follows that the map

Ob Tors(∗, G) → [∗, BG]

defined by sending Y → BG to the composite ∗ ⟵^{≃} Y → BG in the homotopy category factors through a function

π_{0} Tors(∗, G) → [∗, BG]

Theorem 14. The function π_{0} Tors(∗, G) → [∗, BG] is a natural bijection for all sheaves of groupoids G.

Proof. All sheaf theoretic cocycles V_{•} → BG and all G-torsors Y → BG are of the form Bf : BI → BG where f : I → G is a functor defined on a groupoid I which is trivial in the sense that I(x, x) = ∗ for all local choices of objects x and I is locally connected.

Suppose given such a functor f : I → G, and make the homotopy colimit construction

\xymatrix{BI(U) \ar[d]_{f} & ⨆_{x_{0}→···→x_{n}} B(f ↓ x_{0}) \ar[l]_{α}^{≃} \ar[d] \ar[r]^{β} & ⨆_{x_{0}→···→x_{n}} π_{0}B(f ↓ x_{0}) \ar[d]^{h(f)} \\ BG(U) & BG(U) \ar[l]^{1} \ar[r]_{1} & BG(U)}

on the presheaf level in each section. The map α is the standard weak equivalence associated to the functor f : BI(U) → BG(U), and the map β is induced by the simplicial set maps B(f ↓ x_{0}) → π_{0}B(f ↓ x_{0}) — these maps are weak equivalences since all components of the groupoid I(U) are trivial. The presheaf map h(f) therefore represents a G-torsor. This map is canonically isomorphic to f : BI → BG if f is a G-torsor, or homotopy colimit of sheaves over G: in that case I = EX for some functor X, and there is a natural isomorphism π_{0}B(f ↓ x_{0}) ≅ X(x_{0}).

Given a cocycle g : V_{•} → BG, the associated G-torsor h(g) represents an element mapping to the class of g via the function

π_{0} Tors(∗, G) → [∗, BG].

If f : Y → BG is a G-torsor then h(f) ≅ f in the category Tors(∗, G). Further, any homotopy BI × Δ^{1} → BG extends to a map BI × BG(1) → BG where G(1) is the trivial groupoid on two objects. If I is locally connected and trivial, then so is I × G(1). Thus, any homotopy g ≃ g′ of cocycles determines torsors h(g) and h(g′) which are in the same component of Tors(∗, G). It follows that G-torsors f and f′ are in the same component of Tors(∗, G) if the corresponding cocycles are homotopic up to refinement.

Remark 15. The homotopy colimit construction in the proof of Theorem 14 specializes to the standard construction of a G-torsor from a G-cocycle, in the case where G is a sheaf of groups.

The last major result of this section identifies the stack completion of a Borel construction.

Suppose that G is a sheaf of groups and that N is a sheaf on C carrying a Gaction. A G-torsor over N consists of a G-torsor P, together with a G-equivariant map p : P → N. There is a groupoid Tors(C, G)/N of such things, whose objects are the G-torsors over N, and whose morphisms are the G-equivariant commutative diagrams

\xymatrix{P \ar[dd]_{θ}^{≅} \ar[dr]^{p} \\  & N \\ P′ \ar[ur]_{p′}}

The groupoid Tors(C, G)/N is the groupoid of global sections of a stack

Tors(C, G)/N,

which is usually called the quotient stack for the action of G on the sheaf N.

Write E_{G}N for the translation groupoid which is associated to the action of G on N. The sheaf of objects of the category E_{G}N is N, and the sheaf of morphisms is the product G × N, with composeability and composition of morphisms defined in the standard way. The classifying object B(E_{G}N) coincides with the Borel construction EG ×_{G} N.

The path components of the groupoid Tors(C, G)/N can be characterized up to isomorphism as follows:

Theorem 16. Suppose that G is a sheaf of groups on a site C, and that N is a sheaf carrying a G-action. Then there is a bijection

π_{0}(Tors(C, G)/N) ≅ [∗, EG ×_{G} N].

Proof. Suppose that U → ∗ is a sheaf epi, and let T (U) be the associated trivial groupoid with objects U and morphisms U × U. It suffices to show that the set π(T (U), E_{G}N) of homotopy classes of functors T (U) → E_{G}N is isomorphic to the set of isomorphism classes of G-torsors P → N which trivialize over U.

If the object p : P → N trivializes over U, there is a section σ : U → P, and the composite

U ⟶^{σ} P ⟶^{p} N

is the object level map for a functor T (U) → E_{G}N; the map of morphisms is the map U × U → G × N which is defined in sections by the assignment (u_{1}, u_{2}) ↦ (τ(u_{1}, u_{2}), pσ(u_{1})). The element τ(u_{1}, u_{2}) is the unique element such that τ(u_{1}, u_{2})σ(u_{1}) = σ(u_{2}) — τ is, in other words, the usual cocycle defined by the section σ. If σ′ : U → P is a second such section, then σ(u) = h(u)σ′(u) for some unique element h(u) of G, and the associated cocyles τ and τ′ are conjugate by h in the usual way. The map h : U → G × N given by u ↦ (h(u), pσ(u)) defines a homotopy of the functors T (U) → E_{G}N associated to σ and σ′. It follows that isomorphic objects of Tors(C, G)/N which trivialize over U determine homotopic functors.

If f : T (U) → E_{G}N is a functor, then the composite

T (U) ⟶^{f} E_{G}N → BG

determines a G-torsor P which trivializes over U in the usual way, via the Gequivariant coequalizer

G × U × U ⇉ G × U → P,

where the parallel arrows are defined, respectively, by

(g, u_{1}, u_{2}) ↦ (gτ(u_{1}, u_{2}), u_{1}) and
(g, u_{1}, u_{2}) ↦ (g, u_{2}).

Here τ is induced by the functor f in the sense that

f(u_{1}, u_{2}) = (τ(u_{1}, u_{2}), f(u_{1})).

Composing with the G-equivariant map f_{∗} : G×U → N defined by (g, u) ↦ g ·f(u) has the same effect on the parallel pair defining P, and so f_{∗} factors through a unique G-equivariant map p : P → N. If the functors

f, f′ : T (U) → E_{G}N

are homotopic via some map h : U → G × N, with h(u) = (h_{1}(u), f(u)), (so that h is a homotopy f → f′) then the morphism G × U × U → G × U × U defined by

(g, u_{1}, u_{2}) ↦ (gh_{1}(u_{2}), u_{1}, u_{2})

and h_{∗} : G × U → G × U defined by

(g, u) ↦ (gh_{1}(u), u)

together determine a G-equivariant isomorphism h_{∗} : P′ → P on the coequalizer level. At the same time the G-equivariant diagram

\xymatrix{G × U \ar[dd]_{h_{∗}} \ar@{-->}[dr]^{f′_{∗}} \\  & N \\ G × U \ar@{-->}[ur]_{f_{∗}}}

commutes, so that the induced diagram

\xymatrix{P′ \ar[dd]_{h_{∗}} \ar[dr]^{f′_{∗}} \\  & N \\ P \ar[ur]_{f_{∗}}}

commutes as well.

Suppose again that G is a sheaf of groups. Recall [2] that a G-bitorsor consists of a G-torsor P and a G-equivariant map P → Aut(G), where G acts on the sheaf of automorphisms of G by conjugation. A morphism of G-bitorsors is the obvious thing, namely a commutative diagram

\xymatrix{P \ar[dr] \ar[dd]_{f} \\  & Aut(G) \\ P′ \ar[ur]}

where f is G-equivariant. More generally, given sheaves of groups G and H which are locally isomorphic, a (G, H)-bitorsor consists of a G-torsor P and a G-equivariant map P → Iso(H, G) where G acts on the sheaf of isomorphisms Iso(H, G) by composition with conjugation. A morphism of (G, H)-torsors is a G-equivariant commutative diagram, by analogy with the above.

Corollary 17. 1) The set [∗, EG ×_{G} Iso(H, G)] is isomorphic to the set of isomorphism classes of (G, H)-bitorsors.

2) The set [∗, EG ×_{G} Aut(G)] is isomorphic to the set of isomorphism classes of G-bitorsors.

Proof. Statement 2) is the case G = H of statement 1), and 1) follows from Theorem 16.

Example 18. Suppose that S is a Noetherian scheme which is of finite dimension, and is integral and normal. Let Sch|_{S} denote the category of schemes which are of finite type over S.

Following [15], suppose that π : X → S is a finite surjective morphism such that X is integral, and let f : Y → S be the normalization of X in a normal extension of k(S) which contains k(X). Then the natural map

hom_{S}(Y, X) → hom_{k(S)}(k(X), k(Y))

is bijective since Y is normal, and the map f : Y → S is a pseudo-Galois covering in the sense that k(Y)/k(S) is normal and the natural homomorphism

Aut_{S}(Y) → Gal(k(Y)/k(S))

is an isomorphism. Write G for the Galois group of the extension k(Y)/k(S).

Suppose that F is a presheaf of spectra on Sch|_{S} which is globally fibrant for some topology. There is a universal pairing

t_{π} : π_{∗}(F |_{X}) ∧ (EG ×_{G} Y)_{+} → F  (6)

in the stable category of presheaves of spectra on Sch|_{S}, and global sections σ : P → Y of the quotient stack St(E_{G}Y) (if they exist — they may not) determine transfer maps

t_{π,σ} : F (S′ ×_{S} X) → F (S′),

Of course, the meaning of this statement varies with the topology on the site Sch|_{S}. This is really a separable transfer, in contrast with the transfer maps constructed in [15], as only the separable degree of the extension k(X)/k(S) appears in calculations. This construction specializes to a well defined transfer for all finite étale maps π, for both the étale and the qfh topologies. The pairing (6) is the subject of [10].

References

[1] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math., 304, Springer-Verlag, Berlin-Heidelberg-New York (1972).

[2] L. Breen, Bitorseurs et cohomologie non-abélienne, The Grothendieck Festschrift Volume I, Progress in Mathematics, 86, Birkhäuser, Boston-Basel-Berlin (1990), 401–476.

[3] M. Bunge and R. Paré, Stacks and equivalence of indexed categories, Cahiers de Topologie XX(4) (1979), 373–399.

[4] J. Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin-Heidelberg-New York (1971).

[5] A. Grothendieck. Revetements étale et groupe fondamental (SGA1), Lecture Notes in Math. 224, Springer, Heidelberg (1971).

[6] J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Math. 55 (1986), 193–239.

[7] J.F. Jardine, Simplicial presheaves, J. Pure Applied Algebra 47 (1987), 35–87.

[8] J.F. Jardine, Universal Hasse-Witt classes, Contemporary Math. 83 (1989), 83-100.

[9] J.F. Jardine, Generalized Etale Cohomology Theories, Progress in Math. 146, Birkhäuser, Basel-Boston-Berlin (1997).

[10] J.F. Jardine, The separable transfer, Preprint (2001).

[11] A. Joyal, Letter to A. Grothendieck, (1984).

[12] A. Joyal and M. Tierney, Strong stacks and classifying spaces, Category Theory (Como, 1990), Springer Lecture Notes in Math. 1488 (1991), 213–236.

[13] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, New York-Heidelberg-Berlin (1971).

[14] S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic, Springer-Verlag, New York (1992).

[15] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. math. 123 (1996), 61–103.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2001/n2a5/v3n2a5.(dvi,ps,pdf)

J.F. Jardine jardine@uwo.ca

Dept. of Mathematics Univ. of Western Ontario London, ON N6A 5B7 Canada