The unreasonable effectiveness of actegories

The notion of a module over a ring can be categorified to that of an actegory, consisting essentially of a functorial action of a monoidal category on a (2-)category.  Though not always explicitly identified as such, actegories show up all over pure mathematics, mathematical physics, and theoretical computer science. I will describe several interesting examples of actegories and of morphisms between them and explain in particular how to see (homotopy) colimits and limits as morphisms between two distinct actegory structures on the 2-category of categories. I’ll conclude by explaining how this type of actegory morphism can be used to build a natural machine that produces monads or comonads, like those that are central to the construction of the discrete calculus of Bauer-Johnson-McCarthy, and present a broad range of examples of such machines.

Joint work with Kristine Bauer, Brenda Johnson, and Julie Rasmussen