Twisted tensor product of dg categories and its application

If one redefines the natural transformations between dg categories in a homotopically correct way, one gets a 2-quiver C, whose objects are small dg categories over a given field, whose 1-arrows are dg functors, and whose 2-arrows are redefined natural transformations. This 2-quiver fails to have a strict 2-category structure, and the question “What do dg categories form”, raised by Drinfeld, is a question on existence of a weak 2-category structure on C.
I'll talk on a solution to this question, based on a construction of a contractible 2-operad, which is defined via the “twisted tensor product”of dg categories (another construction, based on the McClure-Smith approach to Deligne conjecture, was found by Tamarkin). This 2-operad acts on our 2-quiver C. According to Batanin's approach to weak higher categories, an action of a contractible 2-operad is considered as a weak 2-categorical structure on C. In particular, one gets a solution to the Deligne conjecture for Hochschild cochains of a small dg category.
If time permits, I'll talk on a generalisation of these constructions for monoidal dg categories, which potentially leads to a solution of Deligne conjecture on action of the chain operad C(E_3) on the deformation complex of a small monoidal dg category.