# Model structure on the category of small $A_\infty$ categories, hocolims.

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?

• Here's a guess: Take weak equivalences as stated, and have fibrations be functors surjective (degreewise) on Hom complexes and induce an isofibration on homotopy categories (i.e. RLP with respect to the inclusion 0 --> J where J is the groupoid with two objects 0, and 1 with a unique isomorphism between them). This gives the correct model structure on dg-categories over a ring, for example... – Dylan Wilson Apr 14 '13 at 0:31
• Another idea is to ramp up Lurie's nerve construction for dg-categories to work for A_infty categories (he says this can be done), and then maybe transfer the model structure from the Joyal model structure? Dunno if this would work though... – Dylan Wilson Apr 14 '13 at 0:32

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.)