The Homotopy Theory of A∞categories

In this paper we describe the homotopy category of the (unital and strictly unital) A∞-categories. To do that we introduce the notion of semi-free A∞-category, which plays the role of standard cofibration. Moreover, we define the (non-unital) (resp. DG) A∞-categories with cofibrant morphisms and we prove that any (non-unital) A∞-(resp. DG)category has a resolution of this kind.