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?
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?
Since no one else has posted an answer I'll take the opportunity to plug a recent paper with Scott Morrison, arxiv.org/abs/1009.5025. Section 6 of that paper gives a definition of "$A_\infty$ $n$-category", where for us "$n$-category" usually means an $n$-category with lots of duality, like a pivotal 2-category, and the "$A_\infty$" part means that at level $n$ things (associativity, various compositions and dualities) only hold up to (coherent) higher homotopies.
(Based on feedback from others we will soon revise the terminology of the paper, e.g. replace "$A_\infty$" with "infinity".)
The case you ask about, a monoidal $A_\infty$ category, would correspond to one of our $A_\infty$ 2-categories with only one object. (The case of $A_\infty$ 2-category with only one object and only one 1-morphism would correspond to an $E_2$ algebra.)
Quoth Bugs Bunny in a comment on the original question: "Are you after higher homotopy on compositions or tensor products or, God forbid, both?" For us, the answer is both, and moreover the pivotal structure is only up to higher homotopy. The way to make this manageable is to impose more coherence conditions, rather than fewer. Roughly speaking, our $n$-category axioms are parameterized by all ways of gluing $n$-balls together, and by all (families of) homeomorphisms of $n$-balls. This is a very long list of axioms/coherence conditions, but it is relatively easy to show that the basic examples we have in mind satisfy them all. Presumably this long list of axioms is implied by a much shorter and more combinatorial list (finite or at least finitely generated in some sense), but coming up with such a list is a difficult problem which we need not and do not try to solve.
If you're not interested in pivotal monoidal $A_\infty$ categories, our definition can be adapted to not require so much duality, though we don't develop this in the paper.
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.]