Simplicial presheaves and simplicial sheaves seem to play an important role in homotopy theory. For an introduction see Jardine’s Fields lecture notes, or his longer notes
Toen Barcelona notes.
nLab on model structures, see also local model structure on simplicial presheaves
http://ncatlab.org/nlab/show/descent+for+simplicial+presheaves
Brillian and brief intro in Toen Essen talk, to stacks, simplicial presheaves with various model structures, and the conceptual difference between simplicial sheaves and presheaves. Also the key ideas on simplicial presheaves on a model site, i.e. taking into account a model structure on the base category.
Toen: Thm de RR pour les champs de DM. File Toen web publ rrchamp.pdf. Discusses K-theory and RR for DM stacks, but also various notions of descent, including basic stuff on simplicial presheaves, homological descent, and etale descent.
Lectures of Vaquie in HTAG09: Some notes: For any category, presheaves on it can be endowed with the (global) projective model structure. Also have the local proj model str, and we can view this as a Bousfield loc of the global one. Same result for injective instead of projective. The class S needed for the Bousfield loc is defined in terms of the notion of hypercover, and descent wrt a hypercover. Also local model str on simplicial presheaves on a model site. Universality statement for and functors from to a model cat, works for the projective (global) model structure only.
http://mathoverflow.net/questions/9490/what-are-the-fibrant-objects-in-the-injective-model-structure
In http://www.math.uiuc.edu/K-theory/0129, Jardine proves that the categories of simplicial presheaves and simplicial sheaves on an arbitrary Grothendieck site have proper closed simplicial model structures.
In http://www.math.uiuc.edu/K-theory/0175 there is something about localization theories for simplicial presheaves.
Blander: In this note we introduce the local projective model structure on simplicial presheaves and as an application give a simplified proof of Voevodsky’s comparison theorem, which asserts that the motivic stable homotopy categories in the Nisnevich and cdh topologies are equivalent.
There might be many interesting articles by Dugger and Isaksen, for example http://www.math.uiuc.edu/K-theory/0563
Flasque model structures for simplicial presheaves , by Daniel C. Isaksen http://www.math.uiuc.edu/K-theory/0676. This is an alternative to the projective and the injective model structures.
Consider the general setting of a category of functors from a category to . We can define the projective model structure on this functor category as follows: WEs and fibrations are defined pointwise, while a cofibration is a “retract of a cellular inclusion” (see Dundas, p.40). Can define another model structure (page 41) in which WEs and cofibrations are defined pointwise. This is Quillen equivalent to the projective model structure (Heller).
Rezk: Fibrations and homotopy colimits of simplicial sheaves. Abstract: We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest.
C. Simpson. The topological realization of a simplicial presheaf, q-alg/9609004.
Feliu (thesis) reviews the homotopy theory of simplicial sheaves, in particular the Zariski site. Simplicial sheaves on a site form a simplicial model cat. Generalized cohomology theories defined with simplicial sheaves in both variables. Also generalised cohomology of a simplicial sheaf, with coeffs in an infinite loop spectrum.
Feliu, cont: Def of space “constructed from schemes”. Any simplicial scheme gives rise to such a space, but not the other way around, ref to Huber and Wildeshaus.
Feliu, cont: A limit lemma which describes maps from K-theory to another cohomology theory. Maybe this could also be applied to for example maps from l-adic cohomology.
nLab page on Simplicial presheaves