nLab
hypercover

Idea

Depending a bit on context, a hypercover is a local weak equivalence or local acyclic fibration in the model structure on simplicial presheaves for a given site $C$.

In particular, for $C=*$ the point hypercovers are hypercovers of simplicial sets and hence of topological spaces.

So a hypercover is a morphism $Y\to X$ of simplicial presheaves (thought of a models for ∞-stacks and hence for generalized spaces) that exhibits $Y$ as a possibly big “puffed-up” version of $X$.

The term hypercover is to be read in contrast with the term Čech cover of which it is, in general, a generalization. If all hypercovers are Čech covers one deals with the Čech model structure on simplicial presheaves. The hom-sets in the corresponding homotopy category are then Čech cohomology.

More generally in the standard local model structure on simplicial presheaves hypercovers are the local (stalkwise if the sheaf topos $\mathrm{Sh}(C)$ has enough points) weak equivalences (local acyclic fibrations), of which there are more than Čech covers. One also speaks of the hypercompletion of the Čech model.

Generally, cohomology defined in terms of the (∞,1)-topos presented by the hypercompleted local model structure on simplicial presheaves has been regarded as the “right” notion of cohomology. In particular that coincides with standard abelian sheaf cohomology when restricted to the abelian case.

On the other hand, more rencently in

• Jacob Lurie, Higher Topos Theory

it is argued that the (∞,1)-topos coming from just localizing at Čech covers is in fact better behaved in many respects.

Origin of the term

The term hypercover originates in the fact that for $\pi :Y\to X$ any regular epimorphism, hence an ordinary cover, the simplicial object

$$Y^{•}:=(\cdots \to Y{\times }_{X}Y{\times }_{X}Y\stackrel{\overrightarrow{\to }}{\to }Y{\times }_{X}Y\overrightarrow{\to }Y)$$Y^\bullet := (\cdots \to Y \times_X Y \times_X Y \stackrel{\stackrel{\to}{\to}}{\to} Y \times_X Y \stackrel{\to}{\to} Y)

is an acyclic fibration

$$Y^{•}\to X$$Y^\bullet \to X

over $X$ of simplicial objects, but not all acyclic fibrations arise this way: a generic hypercover of simplicial objects is obtained by starting with a cover $Y\to X$, then choosing a cover of the fiber product $Y{\times }_{X}Y$, and so on.

Characterization by lifting property

Hypercovers are usually (for instance in the model structure on simplicial sets characterized as being those morphisms $\pi :Y\to X$ for which all images $\pi (\partial C^n)$ of boundaries $\partial C^n$ of standard $n$-cells globes $C^n=G^n$ or simplices, $C^n={\Delta }^n$) every $n$-cell filling this boundary in $X$ lifts to $Y$.

Diagrammatically this means: $f:Y\to X$ is a hypercover if for all commuting diagrams

$$\begin{array}{ccc}\partial C^n& \to & Y\\ \downarrow & & {\downarrow }^{\pi }\\ C^n& \to & X\end{array}$$\array{ \partial C^n &\to & Y \\ \downarrow && \downarrow^\pi \\ C^n &\to& X }

there exists a diagonal lift

$$\begin{array}{ccc}\partial C^n& \to & Y\\ \downarrow & {}^{\exists }\nearrow & {\downarrow }^{\pi }\\ C^n& \to & X\end{array}.$$\array{ \partial C^n &\to & Y \\ \downarrow &{}^{\exists}\nearrow& \downarrow^\pi \\ C^n &\to& X } \,.

Hypercovers in different model categories

In the context of the model structure on simplicial sets, these are the hypercovers proper. In the context of the folk model structure on strict ω-categories this are the $\omega$-functors which are $k$-surjective for all $k$.

Relation to fibrations

The condition on hypercovers, being acyclic fibrations is closely related to the condition on fibrations. Usually the lifting property for fibrations is obtained from that for hypercovers by removing in the boundary $\partial C^n$ of the standard $n$-cell one face.

For instance the definition of a hypercover of simplicial sets becomes that of a Kan fibration if the full boundary $\partial \Delta$ is replaced by a horn ${\Lambda }^k[n]$.

In the globular set by replacing the inclusion $\partial G^n\hookrightarrow G^n$ of the boundary of the standard $n$-globe into the $n$-globe with the inclusion $G^{n-1}\hookrightarrow G^n$ of the standard $(n-1)$-globe (which is one-half of the full boundary), the above lifting condition is that of fibrations in the folk model structure on ω-groupoids.

Reference

See the remark at the end of section 2, on p. 6 of

• Jardine, Fields Lectures: Simplicial presheaves (pdf)

For a thorough discussion see

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