nLab
Čech cover

Čech covers

Idea

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

Definition

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

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

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

U× XU× XUU× XUU \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 CC. If the sheaf topos Sh(C)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

π 0(C(U))=colim [n]Δ opC(U) 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(U)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 π 0(C(U))\pi_0(C(U)), regarded as an ordinary presheaf, is precisely the subfunctor of Y(X)Y(X) that corresponds to the sieve {U iX}\{U_i \to X\}.

Lemma

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

C(U)π 0(C(U)) C(U) \to \pi_0(C(U))

is a local weak equivalence.

Proof

Being a simplicially discrete simplicial sheaf, for every test object VV π 0(C(U))(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 π 0(C(U))\pi_0(C(U)) the morphism C(U)π 0(C(U))C(U) \to \pi_0(C(U)) is a bijection on π 0\pi_0.

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

(C(V,U)× C(V,U× XU)C(V,U)). \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)π 0(C(U))C(U) \to \pi_0(C(U)) induces for each object VCV \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)X C(U) \to X

is a stalkwise weak equivalence.

Proof

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

C(U)π 0(C(U))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 π 0(C(U))X\pi_0(C(U)) \to X is a stalkwise weak equivalence.

But by the remark above, π 0(C(U))X\pi_0(C(U)) \to X is actually the local isomorphism corresponding to the cover UU. 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.

Revised on March 9, 2010 20:39:04 by Toby Bartels (173.190.149.136)