Homotopical algebraic geometry. I. Topos theory.

This is the rst of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this rst part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of 1-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bouseld localizations of the model category of pre-stacks on T . Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a denition of etale K-theory of ring spectra, extending the etale K-theory of commutative rings.