nLab
hypercover

Idea

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.

Definition

Let

$$(L⊣i):\mathrm{Sh}(C)\stackrel{\stackrel{L}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C)$$(L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)

be the geometric embedding defining a sheaf topos $\mathrm{Sh}(C)$ into a presheaf topos $\mathrm{PSh}(C)$.

Definition

A morphism

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

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

$$Y_n\to ({\mathrm{cosk}}_{n-1}Y{)}_n{\times }_{({\mathrm{cosk}}_{n-1}X{)}_n}{X}_n$$Y_{n} \to (\mathbf{cosk}_{n-1} Y)_n \times_{(\mathbf{cosk}_{n-1} X)_n} X_n

in $\mathrm{PSh}(C)$ are local epimorphisms (in other words, $f$ is a “Reedy local-epimorphism”).

Here ${\mathrm{cosk}}_n:\mathrm{PSh}(C{)}^{{\Delta }^{\mathrm{op}}}\to \mathrm{PSh}(C{)}^{{\Delta }^{\mathrm{op}}}$ is the coskeleton functor in degree $n$.

A hypercover is called bounded by $n\in \mathbb{N}$ if for all $k\ge n$ the morphisms $Y_k\to ({\mathrm{cosk}}_{k-1}Y{)}_k{\times }_{({\mathrm{cosk}}_{k-1}X{)}_k}{X}_k$ are isomorphisms.

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

A hypercover that also satisfies a cofibrancy condition in the projective local model structure on simplicial presheaves (being locally a coproduct of representables with degenerate cells splitting off as a direct summand) is called a split hypercover

Remark

This is equivalent to saying that $f:Y\to X$ is a local acyclic fibration: for all $U\in C$ and $n\in \mathbb{N}$ every lifting problem

$$\left(\begin{array}{ccc}\partial \Delta [n]\cdot U& \to & Y\\ \downarrow & & {\downarrow }^f\\ \Delta [n]\cdot U& \to & X\end{array}\right)\simeq \left(\begin{array}{ccc}\partial \Delta [n]& \to & Y(U)\\ \downarrow & & {\downarrow }^{f(U)}\\ \Delta [n]& \to & X(U)\end{array}\right)$$\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 $({\sigma }_i)$ after refining to some covering family $\{U_i\to U\}$ of $U$

$$\forall i:\left(\begin{array}{ccc}\partial \Delta [n]& \to & Y(U_i)\\ \downarrow & {}^{\exists {\sigma }_i}\nearrow & {\downarrow }^{f(U_i)}\\ \Delta [n]& \to & X(U_i)\end{array}\right),$$\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) \,,

Remark

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

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

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

Examples

Example

Consider the case that $X=\mathrm{const}{X}_0$ is simplicially constant. Then the conditions on a morphism $Y\to X$ to be a hypercover is as follows.

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

• in degree 1: The commuting diagram in question is

  $$\begin{array}{ccc}{Y}_1& \to & X_0\\ \downarrow & & {\downarrow }^{\mathrm{diag}}\\ Y_0\times Y_0& \to & X_0\times X_0\end{array}.$$\array{ Y_1 &\to& X_0 \\ \downarrow && \downarrow^{\mathrlap{diag}} \\ Y_0 \times Y_0 &\to& X_0 \times X_0 } \,.

  Its pullback is $(Y_0\times Y_0{)}_{X_0\times X_0}{X}_0\simeq Y_0{\times }_{X_0}{Y}_0$, Hence the condition is that

  $Y_1\to Y_0{\times }_{X_0}{Y}_0$ is a local epimorphism.

• in degree 2: The commuting diagram in question is

  $$\begin{array}{ccc}{Y}_2& \to & X_0\\ \downarrow & & {\downarrow }^{\mathrm{Id}}\\ (Y_1{\times }_{Y_0}{Y}_1{\times }_{Y_0}{Y}_1{)}_{{\times }_{Y_0\times Y_0}}{Y}_0& \to & X_0\end{array}.$$\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 $n\ge 2$ the condition is that

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

  is a local epimorphism.

Properties

Proposition

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

Hypercover homology

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

Forming in the sheaf topos the free abelian group on $f_n$ for each $n\in \mathbb{N}$, we obtain a simplicial abelian group object $\overline{f}\in \mathrm{Ab}(\mathrm{Sh}(C/X){)}^{\Delta }$. As such this has a normalized chain complex $N_{•}(\overline{f})$.

Proposition

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

$$H_p(N(f))\simeq \{\begin{array}{cc}0& \mathrm{for}p\ge 1\\ \mathbb{Z}& \mathrm{for}p=0\end{array}.$$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.

Theorem

In the local model structure on simplicial presheaves $\mathrm{PSh}(C{)}^{{\Delta }^{\mathrm{op}}}$ 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 (DuggerHollanderIsaksen).

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

Theorem

(Verdier’s hypercovering theorem)

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

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

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

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, section 3).

Over Verdier sites

The following definitions and statements capture the fact that over certain nice sites it is sufficient to consider certain nice hypercovers. This is due to (DuggerHollanderIsaksen, section 8).

Definition

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

$$U_i\to U_i{\times }_X{U}_i$$U_i \to U_i \times_X U_i

is also a generating covering. We say that $U_i\to X$ is basal.

Example

It is sufficient that all the $U_i\to X$ are monomorphisms.

Examples include the standard open cover-topology on Top.

Definition

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

Theorem

Over a Verdier site, every hypercover may be refined by a split and basal hypercover.

This is (DuggerHollanderIsaksen, theorem 8.6).

Verdier hypercovering theorem

Reference

An early standard reference 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

• Rick Jardine, Fields Lectures: Simplicial presheaves (pdf)

A discussion of hypercovers of topological spaces is in

• Dan Dugger and Daniel Isaksen, Hypercovers in topology (pdf)

A discussion in a topos with enough points in in

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology

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

• Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves (web)

On the Verdier hypercovering theorem see

• Kenneth Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

• Rick Jardine, The Verdier hypercovering theorem (pdf)