The Homotopy Theory of $A_{\mathrm{\infty }}$Categories

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