in path component sheaves, and the diagrams

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Rudolf Hill
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1 Cocycle categories Cocycles J.F. Jardine I will be using the injective model structure on the category s Pre(C) of simplicial presheaves on a small Grothendieck site C. You can think in terms of simplicial sheaves if you like both approaches have technical advantages in particular circumstances. I remind you that a cofibration for this theory is a monomorphism, and a weak equivalence is a map f : Y which induces weak equivalences x Y x in all stalks (if there are stalks). Equivalently, f is a weak equivalence if and only if it induces isomorphisms π 0 = π 0 Y in path component sheaves, and the diagrams π n π n Y 0 Y 0 induce pullback diagrams of associated sheaves for all n 1. These are the local weak equivalences. The injective fibrations are those maps which have the right lifting property with respect to all trivial cofibrations. 1

2 The function complex hom(, Y ) is the simplicial set with n-simplices consisting of all maps n Y. Theorem 1. With these definitions, the category s Pre(C) has the structure of a proper closed simplicial model category. This model structure is cofibrantly generated, and weak equivalences are closed under finite products. Suppose that, Y are simplicial presheaves. h(, Y ) = category whose objects are all pairs of maps (f, g) f Z g Y where f is a weak equivalence. A morphism α : (f, g) (f, g ) of h(, Y ) is a commutative diagram f Z g α f Z g h(, Y ) is the category of cocycles from to Y. Example 2. Suppose that k is a field and L/k is a finite Galois extension with Galois group G. Suppose that H is an algebraic group over k. Then Y 2

3 Sp(L) Sp(k) is an étale covering map and represents a sheaf epi Sp(L) on the étale site et k. There is an isomorphism of simplicial sheaves C(L) = EG G Sp(L) where C(L) is the Čech resolution for the covering Sp(L). Then the picture of simplicial sheaf maps = EG G Sp(L) BH is a cocycle on the Galois group G in the algebraic group H in the traditional sense. Example 3. Suppose that H is a sheaf of groups an F is a sheaf with an action H F F. Then F is an H-torsor if and only if the canonical map EH H F is a local weak equivalence. The picture EH H F BH is the canonical cocycle associated to the torsor F. Of course H is an H-torsor since and the standard map EH H H = EH, EH BH 3

4 is the canonical cocycle associated to the trivial H-torsor H. π 0 h(, Y ) = class of path components of h(, Y ). There is a function φ : π 0 h(, Y ) [, Y ] Theorem 4. The canonical map φ : π 0 h(, Y ) [, Y ] (f, g) g f 1 is a bijection for all simplicial presheaves and Y. Cocycles can be defined for all model categories M and the theorem holds if 1) M is right proper, 2) weak equivs. are closed under finite products in M. Examples: spaces, simplicial sets, presheaves of simplicial sets, spectra, presheaves of spectra, any good localizations including motivic homotopy theories. Remark 5. The cocycle category h(, Y ) has appeared before, in the context of the Dwyer-Kan theory of hammock localizations, but all identifications in that theory involve the assumption that Y is fibrant. The interesting applications of the Theorem involve objects Y which are not fibrant. 4

5 Lest you think that I ve done away with the homotopy theory in this statement, suppose that f g : Y. Then there is a picture 1 f d 0 pr I h Y d 1 1 g where h is the homotopy. Then (1, f) (pr, h) (1, g) Thus f [(1, f)] defines a function ψ : π(, Y ) π 0 h(, Y ) If Y is fibrant, then the function ψ is inverse to φ. More generally, there are a couple of things to prove. The following result reduces the proof Theorem 4 to the case where Y is fibrant: Lemma 6. Suppose that, Y Y are weak equivalences. Then the functor h(, Y ) h(, Y ) induces a weak equivalence Bh(, Y ) = Bh(, Y ). 5

6 Proof. (f, g) h(, Y ) is a map (f, g) : Z Y s.t. f is a weak equivalence. There is a functorial factorization Z j (f,g) W (p,p Y ) Y s.t. j is a triv. cofibration and (p, p Y ) is a fibration. p is a weak equivalence. Form the pullback (p,p Y ) W (α β) W (p,p Y ) Y α β Y (p, p Y ) is a fibration and (α β) is a weak equivalence (since α β is a weak equivalence, and by right properness). p is also a weak equivalence. We have found a functor h(, Y ) h(, Y ) which is inverse to the original functor h(, Y ) h(, Y ) up to homotopy. Lemma 7. Suppose that Y is fibrant. Then the canonical map φ : π 0 h(, Y ) [, Y ] 6

7 is a bijection. Proof. π(, Y ) = naive homotopy classes. π(, Y ) [, Y ] is a bijection since Y is fibrant. We have seen that the assignment f [(1, f)] defines a function and there is a diagram ψ : π(, Y ) π 0 h(, Y ) π(, Y ) ψ π 0 h(, Y ) = φ [, Y ] It suffices to show that ψ is surjective, or that any object f Z g Y is in the path component of some a pair 1 k Y for some map k. Form the diagram f p Z V j g where j is a trivial cofibration and p is a fibration; θ exists because Y is fibrant. θ Y 7

8 is cofibrant, so the trivial fibration p has a section σ, and so there is a commutative diagram 1 θσ σ Y p V θ The composite θσ is the required map k. Homotopy classification of torsors: The canonical cocycle functor EH H? : H tors h(, BH) has a left adjoint, and so it induces a weak equivalence B(H tors) = Bh(, BH). This implies the following: Theorem 8. There are bijections [, BH] = π 0 h(, BH) = π 0 (H tors) =: H 1 (C, H). Remark 9. There is a smallness issue here. In principle, the cocycle category h(, Y ) might not be small, but it s possible to assume that all cocycles U Y involve objects U of bounded size (depending on the size of, Y and C). We ll just assume this. 8

9 Relation with the function complex Write we/ for the category whose objects are all weak equivalences U, and whose morphisms are the commutative diagrams U U Suppose that Y is fibrant and consider the functor hom(, Y ) : (we/) op sset which is defined by U hom(u, Y ). There is a canonical map holim U hom(u, Y ) B(we/)op. Then we/ has a terminal object, namely 1 so B(we/) op is contractible, while the diagram hom(, Y ) is a diagram of equivalences since Y is fibrant. It follows (Quillen s Theorem B) that the canonical map hom(, Y ) holim U hom(u, Y ) is a weak equivalence. At the same time, the horizontal simplicial set holim U hom(u, Y ) n 9

10 is the nerve of the cocycle category h(, Y n ) and is therefore weakly equivalent to h(, Y ), for all n. This means that the canonical map Bh(, Y ) holim U hom(u, Y ) is a weak equivalence. We have proved the following: Theorem 10. Suppose that Y is fibrant and that is cofibrant. Then the canonical maps Bh(, Y ) holim U hom(u, Y ) hom(, Y ) are weak equivalences. Theorem 11. Suppose that Y is locally fibrant and that j : Y Z is an injective fibrant model in simplicial presheaves. Then the maps Bh(, Y ) are weak equivalences. holim U hom(u, Y ) holim U hom(u, Z) hom(, Z) A local fibration is a map Y such that x Y x is a Kan fibration in all stalks. Alternatively, the presheaf maps hom( n, ) hom(λ n k, ) hom(λ n k,y )hom( n, Y ) 10

11 are local epimorphisms. Every presheaf of Kan complexes (eg. BG for presheaves of groups G) is locally fibrant but not necessarily injective fibrant. Every injective fibrant simplicial presheaf is locally fibrant. The Verdier hypercovering theorem Write h π (, Y ) for the full subcategory of h(, Y ) whose objects are the cocycles p U f Y with p a hypercover (aka. local fibration plus local weak equivalence). Here s the modern version of the Verdier hypercovering theorem: Theorem 12. Suppose that Y is locally fibrant. Then the inclusion h π (, Y ) h(, Y ) induces a weak equivalence Bh π (, Y ) Bh(, Y ). Proof. Suppose that (g, f) : U Y is a cocycle (g is a weak equivalence), and find a (functorial) factorization U j (g,f) Z (p,q) Y 11

12 with j a trivial cofibration and (p, q) an injective fibration. Then p, or the composite Z (p,q) Y pr is a weak equivalence, and is a local fibration since Y is locally fibrant. The assignment (g, f) (p, q) defines a functor h(, Y ) h π (, Y ) which is inverse to the inclusion h π (, Y ) h(, Y ) up to homotopy specified by the cofibrations j. Write π/ for the category whose objects are the hypercovers p : U, and whose morphisms are the commutative diagrams U p U p Write π(u, Y ) for simplicial homotopy classes of maps U Y, and let [f] be the simplicial homotopy class of a map f. Corollary 13 (Verdier hypercovering theorem). Suppose that Y is locally fibrant. Then the canonical map lim [p]:u is a bijection. π(u, Y ) [, Y ] 12

13 Proof. The displayed colimit is the set of path components of the category h V (, Y ) whose morphisms have the form U [p] [θ] [p ] U [f] [f ] There is an obvious functor h π (, Y ) h V (V, Y ) which is surjective on objects. The composite of the functions π 0 h f (, Y ) π 0 h V (, Y ) [, Y ] is the canonical bijection. There are variants of h V (, Y ), and here s one of them: h V (, Y ) : Y U p [θ] p U [f] [f ] where [θ] denotes a fibre homotopy class of maps. Corollary 14. The canonical function π 0 h V (, Y ) [, Y ] is a bijection if Y is locally fibrant. Y 13

14 Remark 15. It has, up to now, been a big deal that there is no sort of fibrancy assumption on in these results. Here s the function complex version of the Verdier hypercovering theorem (Theorem 12): Corollary 16. Suppose that Y is locally fibrant and that j : Y Z is an injective fibrant model of Y. Then there are weak equivalences holim U p hom(u, Y ) holim U p hom(, Z) hom(u, Z) where the homotopy colimits are indexed by the hypercovers p : U. Proof. All functors h π (, Y n ) h π (, Z n ) are weak equivalences by the hypercovering theorem, and the category π/ has a terminal object 1 14

Local higher category theory

Local higher category theory February 27, 2017 Path category The nerve functor B : cat sset is def. by BC n = hom(n, C), where n is the poset 0 1 n. The path category functor P : sset cat is the left adjoint of the nerve: P(X ) =