Nonconilpotent Koszul Duality

Koszul duality is a phenomenon appearing in many areas of mathematics, such as rational homotopy theory and deformation theory and representation theory. For differential graded (dg) algebras, it is often formulated as a Quillen equivalence between model categories of augmented dg algebras and conilpotent dg coalgebras, and their corresponding dg modules and comodules. In this talk we consider what happens when the conilpotence condition is removed; the result is an exotic model structure on dg modules that is "of second kind", i.e. weak equivalences are finer than quasi-isomorphisms. This is joint work with Andrey Lazarev.