Géométrie algébrique dérivée, n-catégorie et théorie de Hodge – HODAG

The purpose of this proposal is to settle down the foundations of non-abelian Hodge theory. For this we propose to rework and update the previous works of C. Simpson and of its collaborators with the help of derived algebraic geometry (Toën-Vezzosi, Lurie). We will propose a notion of "derived non-abelian Hodge structures" that we think is better behaved than the notions of non-abelian Hodge structures that have been introduced previously. We will use this notion to provide solid basis for non-abelian Hodge theory, as well as to compare the different existing approaches to this subject.