A Čech cover is a Čech nerve that comes from a cover .
Let be a site and a covering sieve. Write for the coproduct of the patches in the presheaf category ( is the Yoneda embedding).
Write for , for short.
Then the Čech nerve of in , i.e. the simplicial presheaf
is called a Čech cover.
Consider the local model structure on simplicial presheaves on . If the sheaf topos 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).
Write
for the presheaf of connected components (see simplicial homotopy group) of . 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 , regarded as an ordinary presheaf, is precisely the subfunctor of that corresponds to the sieve .
For every Čech cover the morphism of simplicial presheaves
is a local weak equivalence.
Being a simplicially discrete simplicial sheaf, for every test object has all simplicial homotopy groups trivial except possibly the set of connected components. But by the very definition of the morphism is a bijection on .
Over each test domain the simplicial set is just the nerve of the Čech groupoid
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 induces for each object 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
Every Čech cover
is a stalkwise weak equivalence.
From the above we know that factors as
and that the first morphism is an objectwise, hence also a stalkwise weak equivalence. It therefore suffices to show that is a stalkwise weak equivalence.
But by the remark above, is actually the local isomorphism corresponding to the cover . It is therefore even a stalkwise isomorphism.
See also for instance lemma 3.4.9 in
So this says that every Čech cover is a hypercover. But not conversely. Localization of simplicial presheaves at Čech covers yields Čech cohomology.
Last revised on March 9, 2010 at 20:39:04. See the history of this page for a list of all contributions to it.