nLab Čech cover

Idea

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

Definition

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

Write $X$ for $Y(X)$, for short.

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

$$\cdots U \times_X U \times_X U \stackrel{\stackrel{\to}{\to}}{\to} U \times_X U \stackrel{\to}{\to} U$$

is called a Čech cover.

Properties

Consider the local model structure on simplicial presheaves on $C$. If the sheaf topos $Sh(C)$ 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

$$\pi_0(C(U)) = colim_{[n] \in \Delta^{op}} C(U)_n$$

for the presheaf of connected components (see simplicial homotopy group) of $C(U)$. 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 $\pi_0(C(U))$, regarded as an ordinary presheaf, is precisely the subfunctor of $Y(X)$ that corresponds to the sieve $\{U_i \to X\}$.

Lemma

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

$$C(U) \to \pi_0(C(U))$$

is a local weak equivalence.

Proof

Being a simplicially discrete simplicial sheaf, for every test object $V$ $\pi_0(C(U))(V)$ has all simplicial homotopy groups trivial except possibly the set of connected components. But by the very definition of $\pi_0(C(U))$ the morphism $C(U) \to \pi_0(C(U))$ is a bijection on $\pi_0$.

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

$$\left( C(V,U)\times_{C(V,U \times_X U)} \stackrel{\to}{\to} C(V,U) \right) \,.$$

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 $C(U) \to \pi_0(C(U))$ induces for each object $V \in C$ an isomorphism of simplicial homotopy groups. It therefore is an objectwise weak equivalence of simplicial sets.

See also for instance lemma 3.3.5 in

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

Proposition

Every Čech cover

$$C(U) \to X$$

is a stalkwise weak equivalence.

Proof

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

$$C(U) \to \pi_0(C(U)) \to X$$

and that the first morphism is an objectwise, hence also a stalkwise weak equivalence. It therefore suffices to show that $\pi_0(C(U)) \to X$ is a stalkwise weak equivalence.

But by the remark above, $\pi_0(C(U)) \to X$ is actually the local isomorphism corresponding to the cover $U$. It is therefore even a stalkwise isomorphism.

See also for instance lemma 3.4.9 in

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

Remark

So this says that every Čech cover is a hypercover. But not conversely. Localization of simplicial presheaves at Čech covers yields Čech cohomology.