Lax distributive laws for topology, II

Abstract

For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf 2-monad ℙ and the copresheaf 2-monad ℙ†, as well as by their two composite 2-monads, and establish that they all laxly distribute over ℙ. These four 2-monads therefore admit lax extensions to the category Q-Dist of Q-categories and their distributors. We characterize the corresponding (T,Q)-categories in each of the four cases, leading us to both known and novel categorical structures.