Simplicial presheaf

In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of 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 formula_1. Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).
Example: Let "G" be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf formula_2. For example, one might set formula_3. These types of examples appear in K-theory.
If formula_4 is a local weak equivalence of simplicial presheaves, then the induced map formula_5 is also a local weak equivalence.
Homotopy sheaves of a simplicial presheaf.
Let "F" be a simplicial presheaf on a site. The homotopy sheaves formula_6 of "F" is defined as follows. For any formula_7 in the site and a 0-simplex "s" in "F"("X"), set formula_8 and formula_9. We then set formula_10 to be the sheaf associated with the pre-sheaf formula_11.
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 (at least) 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 formula_17 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 formula_18.
If "A" is a sheaf of abelian group (on the same site), then we define formula_19 by doing classifying space construction levelwise (the notion comes from the obstruction theory) and set formula_20. One can show (by induction): for any "X" in the site,
where the left denotes a sheaf cohomology and the right the homotopy class of maps.