A model structure on the category of A${}_{\mathrm{\infty }}$-categories with strict morphisms

We prove that the category of (strictly unital) A${}_{\mathrm{\infty }}$-categories, linear over a commutative ring $R$, with strict A${}_{\mathrm{\infty }}$-morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the cofibrant objects have cofibrant morphisms. As a consequence we prove that the semi-free A${}_{\mathrm{\infty }}$-categories (resp. resolutions) are cofibrant objects (resp. resolution) in this model structure.