Seminar

Adjunctions describe a “duality” between mathematical objects of different types. Monads encode algebraic structure. Descent refers to classical recognition or gluing problems in algebra, algebraic geometry, and homotopy theory that have been connected by the language of adjunctions, monads, and comonads. In this talk, we explain how adjunctions and monads should be thought of as diagrams (“resolutions”) and how algebras, coalgebras, and thus descent data are then described by particular “weighted” limits of these diagrams. Maps between the weights, which describe the shapes of each limit notion, recover the expected maps between these objects, allowing us to prove basic theorems completely independently of the mathematical context in which we are working.