The theory of derived stacks has been developed by Toën and Vezzosi and by Lurie, and the definitions have been clarified by Pridham. In these talks, we will present a down-to-earth account of the subject, using the language of categories of fibrant objects, which are closer to the way in which geometers think about higher Lie groupoids than the closed model categories used by the above authors.

Our starting point is the observation that the category of (smooth Artin) stacks is a localization of the category of Lie groupoids, in which certain morphisms, called hypercovers, are formally inverted. It turns out that Lie groupoids form a category of fibrant objects, and that the localization is best understood in this context. One of the advantages of this approach (developed in collaboration with Kai Behrend) is that it applies without modification to the Banach analytic setting.

To generalize this construction to derived stacks, we must introduce notions of fibrancy on simplicial and cosimplicial objects. We will develop all of this machinery, and construct examples of derived stacks associated to differential graded algebras. For algebras, this construction specializes to the group of invertible elements. In particular, for algebras of matrices, we obtain the general linear groups. Our construction yields in the case of graded matrices (where the basis vectors have integer degree) the derived stack Perf of perfect complexes.

We will then develop the theory of the de Rham complex on derived stacks: in other words, the de Rham complex is invariant under hypercovers. Using cyclic homology and our explicit construction of Perf, we will give a more direct proof of a theorem of Toën and Vezzosi, the existence of the Chern character in the de Rham complex of Perf.