The notions of group action and of 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. I will describe several interesting examples of actegories and of morphisms between them, in particular arising from group actions, and also explain how to see (homotopy) colimits and limits as morphisms between two different 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 machine that produces monads or comonads, like those that are central to the construction of the discrete calculus of Bauer-Johnson-McCarthy.

This is joint work with Kristine Bauer, Brenda Johnson, and Julie Rasmussen.