Simplicial presheaf


In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site taking values in simplicial sets. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.
Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf. Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf.
Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf. For example, one might set. These types of examples appear in K-theory.
If is a local weak equivalence of simplicial presheaves, then the induced map is also a local weak equivalence.

Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves of F is defined as follows. For any in the site and a 0-simplex s in F, set and. We then set to be the sheaf associated with the pre-sheaf.

Model structures

The category of simplicial presheaves on a site admits many different model structures.
Some of them are obtained by viewing simplicial presheaves as functors
The category of such functors is endowed with three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps
such that
is a weak equivalence / fibration of simplicial sets, for all U in the site S. The injective model structure is similar, but with weak equivalences and cofibrations instead.

Stack

A simplicial presheaf F on a site is called a stack if, for any X and any hypercovering H →X, the canonical map
is a weak equivalence as simplicial sets, where the right is the homotopy limit of
Any sheaf F on the site can be considered as a stack by viewing as a constant simplicial set; this way, the category of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on the site. The inclusion functor has a left adjoint and that is exactly.
If A is a sheaf of abelian group, then we define by doing classifying space construction levelwise and set. One can show : for any X in the site,
where the left denotes a sheaf cohomology and the right the homotopy class of maps.