Comodule-Contramodule Correspondence, Higher Geometry Viewpoint

Both comodules and contramodules were introduced in 1960-s by Eilenberg and Moore. While the comodules spread widely, the contramodules remained a poor cousin. The numbers of MathSciNet hits for them are 1327 and 14 correspondingly. In fact, the contramodules do not appear in the literature at all between 1970 and 2007, when Positselski resurrected interest to them by discovering what is now known as Comodule-Contramodule Correspondence but should probably be called Positselski Duality. It seems to underline some seemingly unrelated duality phenomena: Matlis Duality and Riemann-Hilbert Correspondence are examples. In the first half of the talk we will review Comodule-Contramodule Correspondence on a cakewalk example of a DG-coalgebra. In the second half we will discuss an approach to it from the viewpoint of model categories, It is a work in progress, joint with K. Hristova and J. D, S, Jones.