From HAG to DAG: derived moduli spaces

These are expanded notes from some talks given during the fall 2002, about homotopical algebraic geometry'' (HAG) with special emphasis on its applications to derived algebraic geometry’’ (DAG) and derived deformation theory''. We use the general framework developed in Toen, Vezzosi, Homotopical Algebraic Geometry I: Topos theory’‘, and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of the etale topology to the category of non-positively graded commutative differential algebras, and we show that its homotopy category contains interesting objects, such as schemes, algebraic stacks, higher algebraic stacks, dg-schemes … . We define the notion of ``geometric D-stack’’ and present some related geometric constructions ($\mathcal{O}$-modules, perfect complexes, K-theory, derived tangent stacks, cotangent complexes, various notion of smoothness … .). Finally, we define and study the derived moduli problems classifying local systems on a topological space, vector bundles on a smooth projective variety, and $A_{\mathrm{\infty }}$-categorical structures. We state geometricity and smoothness results for all of these examples.