# Čech covers

## Idea

A Čech cover is a Čech nerve $C\left(U\right)$ that comes from a cover $U\to X$.

## Definition

Let $C$ be a site and $\left\{{U}_{i}\to X\right\}$ a covering sieve. Write $U={\bigsqcup }_{i}Y\left({U}_{i}\right)$ for the coproduct of the patches in the presheaf category $\mathrm{PSh}\left(C\right)$ ($Y$ is the Yoneda embedding).

Write $X$ for $Y\left(X\right)$, for short.

Then the Čech nerve $C\left(U\right)$ of $U\to X$ in $\mathrm{PSh}\left(C\right)$, i.e. the simplicial presheaf

$\cdots U{×}_{X}U{×}_{X}U\stackrel{\stackrel{\to }{\to }}{\to }U{×}_{X}U\stackrel{\to }{\to }U$\cdots U \times_X U \times_X U \stackrel{\stackrel{\to}{\to}}{\to} U \times_X U \stackrel{\to}{\to} U

is called a Čech cover.

## Properties

Consider the local model structure on simplicial presheaves on $C$. If the sheaf topos $\mathrm{Sh}\left(C\right)$ has enough points, then the weak equivalences (called local weak equivalences for emphasis) are the stalk-wise weak equivalences of simplicial set (with respect to the standard model structure on simplicial sets).

###### Definition

Write

${\pi }_{0}\left(C\left(U\right)\right)={\mathrm{colim}}_{\left[n\right]\in {\Delta }^{\mathrm{op}}}C\left(U{\right)}_{n}$\pi_0(C(U)) = colim_{[n] \in \Delta^{op}} C(U)_n

for the presheaf of connected components (see simplicial homotopy group) of $C\left(U\right)$. Regard this as a simplicially constant simplicial presheaf.

Remark. By the discussion in the section “Interpretation in terms of descent and codescent” at sieve this ${\pi }_{0}\left(C\left(U\right)\right)$, regarded as an ordinary presheaf, is precisely the subfunctor of $Y\left(X\right)$ that corresponds to the sieve $\left\{{U}_{i}\to X\right\}$.

###### Lemma

For every Čech cover $C\left(U\right)$ the morphism of simplicial presheaves

$C\left(U\right)\to {\pi }_{0}\left(C\left(U\right)\right)$C(U) \to \pi_0(C(U))

is a local weak equivalence.

###### Proof

Being a simplicially discrete simplicial sheaf, for every test object $V$ ${\pi }_{0}\left(C\left(U\right)\right)\left(V\right)$ has all simplicial homotopy groups trivial except possibly the set of connected components. But by the very definition of ${\pi }_{0}\left(C\left(U\right)\right)$ the morphism $C\left(U\right)\to {\pi }_{0}\left(C\left(U\right)\right)$ is a bijection on ${\pi }_{0}$.

Over each test domain $V$ the simplicial set $C\left(U\right)\left(V\right)$ is just the nerve of the Čech groupoid

$\left(C\left(V,U\right){×}_{C\left(V,U{×}_{X}U\right)}\stackrel{\to }{\to }C\left(V,U\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\left( C(V,U)\times_{C(V,U \times_X U)} \stackrel{\to}{\to} C(V,U) \right) \,.

The nerve of that groupoid is readily seen to have vanishing first simplicial homotopy group. Being the nerve of a 1-groupoid, also all higher simplicial homotopy groups vanish.

So $C\left(U\right)\to {\pi }_{0}\left(C\left(U\right)\right)$ induces for each object $V\in C$ an isomorphism of simplicial homotopy groups. It therefore is an objectwise weak equivalence of simplicial sets.

• Daniel Dugger, Sheaves and homotopy theory (web, pdf)
###### Proposition

Every Čech cover

$C\left(U\right)\to X$C(U) \to X

is a stalkwise weak equivalence.

###### Proof

From the above we know that $C\left(U\right)\to X$ factors as

$C\left(U\right)\to {\pi }_{0}\left(C\left(U\right)\right)\to X$C(U) \to \pi_0(C(U)) \to X

and that the first morphism is an objectwise, hence also a stalkwise weak equivalence. It therefore suffices to show that ${\pi }_{0}\left(C\left(U\right)\right)\to X$ is a stalkwise weak equivalence.

But by the remark above, ${\pi }_{0}\left(C\left(U\right)\right)\to X$ is actually the local isomorphism corresponding to the cover $U$. It is therefore even a stalkwise isomorphism.