Koszul duality and Calabi Yau strutures

I will talk about two aspects of Koszul duality. Firstly, Koszul duality for dg categories provides a way of modelling dg categories as certain curved coalgebras. This is a linearization of the correspondence of simplicial categories as simplicial sets (quasi-categories). Secondly, Koszul duality exchanges smooth and proper Calabi-Yau structures for dg categories and curved coalgebras. This is a generalization and conceptual explanation of the following phenomen: For a topological space X with the homotopy type of a finite complex having an oriented Poincaré duality structure (with local coefficients) is equivalent to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X.  This is joint work with Andrey Lazarev and with Manuel Rivera, respectively.