The Homotopy Theory of A∞categories

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