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?