I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A\_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in the case underlying chain complexes are unbounded, ungraded. 


Related question, let $M$ denote the category of small $A\_\infty$ categories, $M^D$ category of $D$ shaped diagrams with $D$ Reedy with fibrant constants. Think of $M^D$ as a homotopical category with weak equivalences objectwise quasi-equivalences, similarly with $M$. Is the functor
$$colim M^D \to M$$ homotopical? It is not homotopical in the dg world, but it seems it might be in the $A\_\infty$ world. If it is not homotopical and the answer to question one is "none" does the derived functor $hocolim: ho M^D \to ho M$ exist? 



Model structure on the category of small $A_\infty$ categories, hocolims.
Look at Lefevre-Hasegawa PhD Thesis (unfortunately in French) at <http://arxiv.org/abs/math/0310337>, where this is done. For a reference in English, you can look there: <http://math.unice.fr/~brunov/HomotopyTheory.pdf> , where you have to restrict to the the Koszul operad $As$ of associative algebras and then consider "mutliple objects", ie $A\_\infty$-categories. 


Strictly speaking, you cannot have a model category structure on $A\_\infty$-categories since not all the equalizers exist. But "all" the other axioms hold. In this case, weak equivalences are $\infty$-quasi-isomorphisms (does this correspond to your quasi-equivalences?), cofibrations are $\infty$-monomorphisms (the first component map is a monomorphism) and fibrations are $\infty$-epimorphisms (the first component map is an epimorphism). In this model category structure, everybody is fibrant and cofibrant. (I have an extended version of loc. cit. including this, that I can share.) 


