### Homotopical Algebraic Geometry I Topos theory, by Bertrand Toen and Gabriele Vezzosi

This is the first of a series of papers devoted to the foundations of
Algebraic Geometry in homotopical and higher categorical contexts. In this
paper we investigate a notion of higher topos. For this, we use S-categories
(i.e. simplicially enriched categories) as models for certain kind of
$\infty$-categories, and we develop the notions of S-topologies, S-sites and
stacks over them. We prove that for an S-site T, there is a model category of
stacks over T, generalizing Joyal-Jardine structure on simplicial presheaves
on a Grothendieck site. We also shows, as an analog of the relation between
topologies and localizing subcategories of the categories of presheaves, that
there is a bijection between S-topologies on an S-category T, and certain
left exact Bousfield localizations of the model category of pre-stacks on
T. Then we study the notion of model topos due to C. Rezk, and 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
Dwyer-Kan simplicial localization provides a canonical way to pass from the
theory of stacks over model sites to the theory of stacks over S-sites. As an
application, we propose a definition of \'etale K-theory of ring spectra. An
appendix gives an alternative approach to the theory which uses Segal
categories. We define Segal topologies, Segal sites, stacks over Segal sites
and Segal topoi. The existence of internal Hom's in this context allows us to
define the Segal category of geometric morphisms between Segal topoi. An
application to the reconstuction of a space via its Segal category of stacks
is given.

Bertrand Toen <toen@math.unice.fr>

Gabriele Vezzosi <vezzosi@dm.unibo.it>