Abstract We prove that the category of (strictly unital) A A ∞ -categories, linear over a commutative ring R, with strict A A ∞ -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 A ∞ -categories (resp. resolutions) are cofibrant objects (resp. resolution) in this model structure.
Mattia Ornaghi (Mon,) studied this question.