### Algebraic Geometry over model categories (a general approach to derived algebraic geometry), by Bertrand Toen and Gabriele Vezzosi

For a (semi-)model category M, we define a notion of a
''homotopy'' Grothendieck topology on M, as well as its
associated model category of stacks. We use this to define a
notion of geometric stack over a symmetric monoidal base model
category; geometric stacks are the fundamental objects to "do
algebraic geometry over model categories". We give two examples of
applications of this formalism. The first one is the
interpretation of DG-schemes as geometric stacks over the model
category of complexes and the second one is a definition of
etale K-theory of E_{\infty}-ring spectra.

This first version is very preliminary and might be considered as
a detailed research announcement. Some proofs, more details and
more examples will be added in a forthcoming version.

Bertrand Toen <toen@math.unice.fr>

Gabriele Vezzosi <vezzosi@dm.unibo.it>