Abstract:
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.