What do DG categories form?

We introduce a homotopy 2-category structure on the category of 2-categories

Proof of Swiss Cheese Version of Deligne's Conjecture

For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain operad of Voronov's Swiss Cheese operad. This statement is the Swiss Cheese version of the Deligne conjecture formulated by M. Kontsevich in arXiv:math/9904055.Comment: The article was rewritten. Multiple suggestions of referees were addressed. The paper is accepted for publication to IMR