Moduli of objects in dg-categories

To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the triangulated category $[T]$ associated to $T$. The main result of this work states that under certain finiteness conditions on $T$ (e.g. if it is saturated) the $D^{-}$-stack $\mathcal{M}_{T}$ is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of a saturated dg-category. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as well as complexes of representations of a finite quiver.Comment: 64 pages. Minor corrections. Section 3.4 including some corollaries has been added. Sections 1 and 2.5 added, as well as some remarks. To appear in Annales de l'EN

Toward a Galoisian interpretation of homotopy theory

Given any pointed CW complex (X,x), it is well known that the fondamental group of X pointed at x is naturally isomorphic to the automorphism group of the functor which associates to a locally constant sheaf on X its fibre at x. The purpose of this work is to generalize this fact to higher homotopy. For this we introduce the (infinite) category of locally constant stacks on X, and we prove that the loop-space of endomorphisms of its fibre functor at x is naturally equivalent to the loop space of X based at x.Comment: French, 38 pages. To appear in "Cahiers de topologie et geometrie differentielle categoriques