Koszul duality in exact categories

In this paper we establish Koszul duality type results in the setting of chain complexes in exact categories. In particular we prove generalisations of Vallette's cooperadic Koszul duality theorem, and operadic Koszul duality along the lines of Lurie. We also prove a connective version. We conclude with some applications, including our main example, the category of complete bornological spaces over a Banach field.Comment: 65 Pages; Preliminary, feedback welcom

Level-rank duality via tensor categories

We give a new way to derive branching rules for the conformal embedding (\asl_n)_m\oplus(\asl_m)_n\subset(\asl_{nm})_1. In addition, we show that the category \Cc(\asl_n)_m^0 of degree zero integrable highest weight (\asl_n)_m-representations is braided equivalent to \Cc(\asl_m)_n^0 with the reversed braiding.Comment: 16 pages, to appear in Communications in Mathematical Physics. Version 2 changes: Proof of main theorem made explicit, example 4.11 removed, references update