TITLE: A-infinity tensor categories
QUESTION [11 upvotes]: My question is rather simple:
What is the correct notion of a monoidal A-infinity category C?
Or is there any reference where such a notion is explained?

REPLY [5 votes]: One way to define a monoidal $A_\infty$-category (which is what Chris Brav suggests in the comments - and of course presumably to Kevin and Scott's) is as an associative algebra object in the monoidal $(\infty,1)$-category of $A_\infty$ categories (which is the same as that of dg categories). In other words, to define a monoidal structure we first need to ask "what do $A_\infty$-categories form?" and then just take the notion of associative multiplication in that world. Now of course to say that sentence I need to know there's a theory of monoidal $(\infty,1)$ -categories, for which we have Lurie's DAG II. But the advantage is that with this definition we can immediately perform all the operations of ordinary algebra as if we were just dealing with an associative ring -- there's automatically a theory of module categories eg, inner homs, bimodules and tensors, we can even talk about Hochschild homology and each module category defines a character object in this Hochschild homology, and of course many many other things. So I strongly believe it's worth the investment.
As to how to define this monoidal $(\infty,1)$-category, there are many ways, but I'll take the cheapest (again modulo the worthwhile investment which is DAG): I would define them as modules for dg-Vect, the monoidal $\infty$-category of chain complexes of $k$-vector spaces. This category itself is given as modules for the commutative ring k, hence its monoidal structure. (Again modulo DAG 2 and now 3 for commutative rings and their tensor products of modules).
[Let me attempt to preempt the obvious complaint with all this: yes it's not very explicit and close to the ground, i.e. to models, and for calculations you very well may want a more concrete objectwise definition in terms of higher homotopies. This approach is however a very powerful one for proving formal properties -- ie if the questions you're asking have the feeling "this would be easy/formal/follow for abstract general reasons if I were living in the toy model world of just plain monoidal categories, or even associative algebras, but seem hard in this homotopical world" then this is the approach for you, otherwise it's not.]