# $A_\infty$-categories and their equivalent dg-categories: the case of $\mathcal{RH}om(\mathcal A,\mathcal B)$

Hello everyone,

I'm currently working on dg-categories, in particular I'm looking for some convenient characterization of the dg-category $\mathcal{RH}om(\mathcal A,\mathcal B)$, for two given dg-categories $\mathcal A$ and $\mathcal B$, at least in cases when $\mathcal A$ is "easy". For example, if $\mathcal A$ is the category $\Delta^1$ with two objects $0$ and $1$ and freely generated over $k$ (a fixed ground commutative ring) by one nontrivial morphism $0 \to 1$, then $\mathcal{RH}om(\mathcal A,\mathcal B) = \mathcal{M}or(\mathcal B)$ (equality up to quasi-equivalence, I suppose), where $\mathcal{M}or(\mathcal B)$ is the dg-category of morphisms of $\mathcal B$ as defined in Drinfeld's article "dg quotients of dg categories" (Drinfeld assumes $\mathcal B$ pretriangulated, but it is not really necessary).

In Keller's survey "on differential graded categories" it is stated that $\mathcal{RH}om(\mathcal A,\mathcal B)$ is (quasi equivalent to) the category of strictly unital $A_\infty$ functors from $\mathcal A$ to $\mathcal B$, both viewed as $A_\infty$-categories.

Moreover, it seems a well-known fact that a given $A_\infty$-category is equivalent - in the $A_\infty$ sense - to some dg-category. Drinfeld himself (first paragraph of Appendix IV of "Dg quotients of dg categories") sketches a procedure to associate a dg-functor to a $A_\infty$-functor between dg-categories, upon changing the source dg-category, in a simple case: as far as I can understand, the idea is to formally add morphisms and coboundary relations to the source category, in a smart way.

My question is the following: how do you explain the above procedure, in general? Namely, what morphisms and what differentials I really have to add, keeping Drinfeld's example in mind? I am looking for something as elementary as possible, and living solely in the "world of dg categories". Even an explanation in some simple situations would be of great help. A caveat: I know nearly nothing about $A_\infty$-categories!

Thanks in advance; I hope everything above is clear enough.

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This doesn't stay solely in the dg world, but the proof I know of "every Aoo category is equivalent to a dg category" is more conceptual--it's the Yoneda Lemma. Consider the category of all contravariant Aoo functors from $\mathcal{A}$ to $Ch$, the category of chain complexes. The category of Aoo functors would normally be an Aoo category, but because $Ch$ is a dg category, this functor category is actually a dg category. A version of the usual Yoneda argument shows that $\mathcal{A}$ is in fact equivalent as an Aoo category to its image inside this dg category. –  Hiro Lee Tanaka Dec 6 '12 at 19:31
Let me quote: "The category of Aoo functors would normally be an Aoo category, but because Ch is a dg category, this functor category is actually a dg category" This is, in fact, quite the point. I would like to <i>describe in an elementary way</i> such functor category, and functor categories of this kind. –  Francesco Genovese Dec 7 '12 at 9:32

I don't quite know what kind of properties do you want the DG-category $RHom(A,B)$ to have, but there is a natural nice DG-category which may work. Namely, the category of right quasi-representable h-projective $(A^{op}\otimes B)$-DG-modules.
The dg-category $\mathcal{RH}om(\mathcal A, \mathcal B)$ is the internal hom in the homotopy category of dg-categories. As far as I could see, it is quite difficult to describe in general, but I still hope that some useful characterizations can be obtained when the source dg-category is "simple" enough. What I really seek is a... "hands on" description, something I'm able to do computations with. One example is outlined above: when $\mathcal A = \Delta^1$, then $\mathcal{RH}om(\mathcal A, \mathcal B)$ is the dg-category of morphisms in $\mathcal B$, which is "elementary" enough. –  Francesco Genovese Dec 7 '12 at 9:41