The 2-category of categories, controls, and flows


David Spivak


January 29, 2024

For any monoidal category C, there is an associated 2-category Comod(C) of comonoids in C, homomorphisms, and what I’ll call natural cotransformations between homomorphisms; colax monoidal functors C→D induce 2-functors Comod(C)→Comod(D). I’ll briefly discuss the 2-category Comod(Rel) associated to Rel as a monoidal category, but we’ll focus on the 2-category Cat^# associated to the monoidal category of polynomial functors under substitution. I’ll also briefly discuss a certain colax monoidal functor Poly→Rel relating them. An object in Cat^# is a category c, whose objects and morphisms we imagine as the points and paths in c as a kind of space. A morphism c→d is a cofunctor, which we imagine as a d-control on c; we refer to a natural cotransformation between these controllers as a flow, which looks something like a homotopy. This imagery is especially vivid in the case of 𝓎^ℝ-control on the fundamental groupoid of a space. We’ll explore how various flavors of polynomial monoids m represent various monoidal structures on m-controlled categories.