Categories of fibrant objects, and Higher stacks as an example of a category of fibrant objects
by Ezra Getzler
Abstract: First part: There are two approaches to the homotopy theory of simplicial sheaves: one may study the category of simplicial presheaves, or its subcategory consisting of those simplicial sheaves whose stalk at each point is a Kan complex. The former is a closed model category in the sense of Quillen, while the latter is a category of fibrant objects. In a category of fibrant objects, we retain only the fibrations and weak equivalences from Quillen's axioms for a closed model category, but not the cofibrations. Still, the axioms are rich enough to be able to do a lot with them, in particular, to study the simplicial localization of the category, the "category of homotopy types".
In the second part, we show that for each natural number k, the category of k-stacks is a category of fibrant objects whose underlying category is a subcategory of the category of simplicial Banach analytic spaces. (This is analogous to thinking of the category of manifolds as a category of Cech nerves of atlases.) In particular, we make explicit what the weak equivalences are. (These are the analogues of diffeomorphisms of manifolds, defined in terms of Cech nerves of atlases.)