Locality and descent

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(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



A hypercover is the generalization of a Cech nerve of a cover: it is a simplicial resolution of an object obtained by iteratively applying covering families.



(Li):Sh(C)LPSh(C) (L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)

be the geometric embedding defining a sheaf topos Sh(C)Sh(C) into a presheaf topos PSh(C)PSh(C).


A morphism

(YfX)PSh(C) Δ op (Y \stackrel{f}{\to} X) \in PSh(C)^{\Delta^{op}}

in the category of simplicial objects in PSh(C)PSh(C), hence the category of simplicial presheaves, is called a hypercover if for all nn \in \mathbb{N} the canonical morphism

Y n(cosk n1Y) n× (cosk n1X) nX n Y_{n} \to (\mathbf{cosk}_{n-1} Y)_n \times_{(\mathbf{cosk}_{n-1} X)_n} X_n

in PSh(C)PSh(C) are local epimorphisms (in other words, ff is a “Reedy local-epimorphism”).

Here this morphism into the fiber product is that induced from the naturality square

Y X cosk n1Y cosk n1Y \array{ Y &\longrightarrow& X \\ \downarrow && \downarrow \\ \mathbf{cosk}_{n-1} Y &\longrightarrow& \mathbf{cosk}_{n-1} Y }

of the unit of the coskeleton functor cosk n:PSh(C) Δ opPSh(C) Δ op\mathbf{cosk}_n : PSh(C)^{\Delta^{op}} \to PSh(C)^{\Delta^{op}}.

A hypercover is called bounded by nn \in \mathbb{N} if for all knk \geq n the morphisms Y k(cosk k1Y) k× (cosk k1X) kX kY_{k} \to (\mathbf{cosk}_{k-1} Y)_k \times_{(\mathbf{cosk}_{k-1} X)_k} X_k are isomorphisms.

The smallest nn for which this holds is called the height of the hypercover.


A hypercover that also satisfies the cofibrancy condition in the projective local model structure on simplicial presheaves in that

  1. it is simplicial-degree wise a coproduct of representables;

  2. degenerate cells split off as a direct summand)

is called a split hypercover.

(Dugger-Hollander-Isaksen 02, def. 4.13) see also (Low 14-05-26, 8.2.15)


Definition 1 is equivalent to saying that f:YXf : Y \to X is a local acyclic fibration: for all UCU \in C and nn \in \mathbb{N} every lifting problem

(Δ[n]U Y f Δ[n]U X)(Δ[n] Y(U) f(U) Δ[n] X(U)) \left( \array{ \partial \Delta[n] \cdot U &\to& Y \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \Delta[n]\cdot U &\to& X } \right) \;\;\simeq \;\; \left( \array{ \partial \Delta[n] &\to& Y(U) \\ \downarrow && \downarrow^{\mathrlap{f(U)}} \\ \Delta[n] &\to& X(U) } \right)

has a solution (σ i)(\sigma_i) after refining to some covering family {U iU}\{U_i \to U\} of UU

i:(Δ[n] Y(U i) σ i f(U i) Δ[n] X(U i)), \forall i : \left( \array{ \partial \Delta[n] &\to& Y(U_i) \\ \downarrow &{}^{\mathllap{\exists \sigma_i}}\nearrow& \downarrow^{\mathrlap{f(U_i)}} \\ \Delta[n] &\to& X(U_i) } \right) \,,

(Dugger-Hollander-Isaksen 02, prop. 7.2


If the topos Sh(C)Sh(C) has enough points a morphism f:YXf : Y \to X in Sh(C) Δ opSh(C)^{\Delta^{op}} is a hypercover if all its stalks are acyclic Kan fibrations.

In this form the notion of hypercover appears for instance in (Brown 73).

In some situations, we may be interested primarily in hypercovers that are built out of data entirely in the site CC. We obtain such hypercovers by restricting XX to be a discrete simplicial object which is representable, and each Y nY_n to be a coproduct of representables. This notion can equivalently be formulated in terms of diagrams (Δ/A)C(\Delta/A) \to C, where AA is some simplicial set and (Δ/A)(\Delta/A) is its category of simplices.



Consider the case that X=constX 0X = const X_0 is simplicially constant. Then the conditions on a morphism YXY \to X to be a hypercover is as follows.

  • in degree 0: Y 0X 0Y_0 \to X_0 is a local epimorphism.

  • in degree 1: The commuting diagram in question is

    Y 1 X 0 diag Y 0×Y 0 X 0×X 0. \array{ Y_1 &\to& X_0 \\ \downarrow && \downarrow^{\mathrlap{diag}} \\ Y_0 \times Y_0 &\to& X_0 \times X_0 } \,.

    Its pullback is (Y 0×Y 0) X 0×X 0X 0Y 0× X 0Y 0(Y_0 \times Y_0)_{X_0 \times X_0} X_0 \simeq Y_0 \times_{X_0} Y_0, Hence the condition is that

    Y 1Y 0× X 0Y 0Y_1 \to Y_0 \times_{X_0} Y_0 is a local epimorphism.

  • in degree 2: The commuting diagram in question is

    Y 2 X 0 Id (Y 1× Y 0Y 1× Y 0Y 1) × Y 0×Y 0Y 0 X 0. \array{ Y_2 &\to& X_0 \\ \downarrow && \downarrow^{Id} \\ (Y_1 \times_{Y_0} Y_1 \times_{Y_0}Y_1)_{\times_{Y_0 \times Y_0}} Y_0 &\to& X_0 } \,.

    So the condition is that the vertical morphism is a local epi.

  • Similarly, in any degree n2n \geq 2 the condition is that

    Y n(cosk n1Y) n Y_n \to (\mathbf{cosk}_{n-1} Y)_n

    is a local epimorphism.


Existence and refinements

Cech nerves


For U={U iX}U = \{U_i \to X\} a cover, the Cech nerve projection C(U)XC(U) \to X is a hypercover of height 0.

Over general sites


Given any site (𝒞,J)(\mathcal{C},J) and given a diagram of simplicial presheaves

Y hcov X X \array{ && Y \\ && \downarrow^{\mathrlap{hcov}} \\ X' &\longrightarrow& X }

where the vertical morphism is a hypercover, then there exists a completion to a commuting diagram

Y Y hcov hcov X X \array{ Y' &\longrightarrow& Y \\ \downarrow^{\mathrlap{hcov}} && \downarrow^{\mathrlap{hcov}} \\ X' &\longrightarrow& X }

where the left vertical morphism is a split hypercover, def. 2.

Moreover, if (𝒟,K)(𝒞,J)(\mathcal{D}, K)\to (\mathcal{C},J) is a dense subsite then YY' as above exists such that it is simplicial-degree wise a coproduct of (representables by) objects of 𝒟\mathcal{D}.

(e.g. Low 14-05-26, lemma 8.2.20)


In particular taking XXX'\to X in prop. 2 to be an identity, the proposition says that every hypercover may be refined by a split hypercover.

(see also Low 14-05-26, lemma 8.2.23)

Over Verdier sites


A Verdier site is a small category with finite pullbacks equipped with a basis for a Grothendieck topology such that the generating covering maps U iXU_i \to X all have the property that their diagonal

U iU i× XU i U_i \to U_i \times_X U_i

is also a generating covering. We say that U iXU_i \to X is basal.


It is sufficient that all the U iXU_i \to X are monomorphisms.

Examples include the standard open cover-topology on Top.


A basal hypercover over a Verdier site is a hypercover UXU \to X such that for all nn \in \mathbb{N} the components of the maps into the matching object U nMU nU_n \to M U_n are basal maps, as above.


Over a Verdier site, every hypercover may be refined by a split (def. 2) and basal hypercover (def. 4).

This is (Dugger-Hollander-Isaksen 02, theorem 8.6).

Hypercover homology

Let f:YXf : Y \to X be a hypercover. We may regard this as an object in the overcategory Sh(C)/XSh(C)/X. By the discussion here this is equivalently Sh(C/X)Sh(C/X). Write Ab(Sh(C/X))Ab(Sh(C/X)) for the category of abelian group objects in the sheaf topos Sh(C/X)Sh(C/X). This is an abelian category.

Forming in the sheaf topos the free abelian group on f nf_n for each nn \in \mathbb{N}, we obtain a simplicial abelian group object f¯Ab(Sh(C/X)) Δ\bar f \in Ab(Sh(C/X))^{\Delta}. As such this has a normalized chain complex N (f¯)N_\bullet(\bar f).


For f:YXf : Y \to X a hypercover, the chain homology of N(f¯)N(\bar f) vanishes in positive degree and is the group of integers in degree 0, as an object in Ab(Sh(C)(X)Ab(Sh(C)(X):

H p(N(f)){0 forp1 forp=0. H_p(N(f)) \simeq \left\{ \array{ 0 & for \; p \geq 1 \\ \mathbb{Z} & for \; p = 0 } \right. \,.

Descent and cohomology

The following theorem characterizes the ∞-stack/(∞,1)-sheaf-condition for the presentation of an (∞,1)-topos by a local model structure on simplicial presheaves in terms of descent along hypercovers.


In the local model structure on simplicial presheaves PSh(C) Δ opPSh(C)^{\Delta^{op}} an object is fibrant precisely if it is fibrant in the global model structure on simplicial presheaves and in addition satisfies descent along all hypercovers over representables that are degreewise coproducts of representables.

This is the central theorem in (Dugger-Hollander-Isaksen 02).

The following theorem is a corollary of this theorem, using the discussion at abelian sheaf cohomology. But historically it predates the above- theorem.


(Verdier’s hypercovering theorem)

For XX a topological space and FF a sheaf of abelian groups on XX, we have that the abelian sheaf cohomology of XX with coefficients in FF is given

H q(X,F)lim YXH q(Hom Sh(Y ,F)) H^q(X, F) \simeq {\lim_{\to}}_{Y \to X} H^q(Hom_{Sh}(Y^\bullet,F))

by computing for each hypercover YXY \to X the cochain cohomology of the Moore complex of the cosimplicial abelian group obtained by evaluating FF degreewise on the hypercover, and then taking the colimit of the result over the poset of all hypercovers over XX.

A proof of this result in terms of the structure of a category of fibrant objects on the category of simplicial presheaves appears in (Brown 73, section 3).


The concept of hypercovers was introduced for abelian sheaf cohomology in

An early standard reference founding étale homotopy theory is

  • Michael Artin, Barry Mazur, Étale Homotopy , Lecture Notes in Mathematics 100, Springer- Verlag, Berline-Heidelberg-New York (1972).

The modern reformulation of their notion of hypercover in terms of simplicial presheaves is mentioned for instance at the end of section 2, on p. 6 of

A discussion of hypercovers of topological spaces and relation to étale homotopy type of smooth schemes and A1-homotopy theory is in

A discussion in a topos with enough points in in

A thorough discussion of hypercovers over representables and their role in descent for simplicial presheaves is in

On the Verdier hypercovering theorem see

Split hypercover refinement over general sites is discussed in

Revised on June 10, 2014 23:47:58 by Urs Schreiber (