Department of MathematicsThe algebraic analogue of the loop space construction of topological spaces is Adams’ cobar construction, it induces a Koszul duality between algebras and coalgebras, providing an equivalence of suitable homotopy theories of augmented differential graded algebras and differential graded conilpotent coalgebras.
I will talk about far-reaching generalisation of this result to categorical Koszul duality, introducing a category of coalgebras Quillen equivalent to differential graded categories.
This equivalence is moreover quasi-monoidal and by constructing internal homs of certain coalgebras we can construct a concrete closed monoidal model for dg categories. In particular this gives natural descriptions of mapping spaces and internal homs between dg categories.
This is joint work with A. Lazarev.