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.