The Homotopy Theory of DG-Categories

Differentially-graded categories are essential to Derived Algebraic Geometry; but they don't behave as well as we would like them to. In 2005, Bertrand Toën proved that all theories of ∞-categories are Quillen equivalent to the category of complete Segal spaces, basically meaning that we can see any ∞-category as a presheaf from the simplex category to the simplicial sets, satisfying certain conditions. In this talk, we will try and apply a similar logic to dg-categories: we will define a certain "linearized version of the simplex category" and sketch the proof that every dg-category is a simplicial presheaf from that linear simplex category to the category of simplicial sets. We will finish with a few possible applications of the result.