A Representation Result for Free Cocompletions

Abstract

Given a class F of weights, one can consider the construction that
takes a small category C to the free cocompletion of C under weighted colimits, for which the weight lies in F. Provided these free Fcocompletions are small, this construction generates a 2-monad on Cat, or more generally on V-Cat for monoidal biclosed complete and cocomplete V. We develop the notion of a dense 2-monad on V-Cat and characterise free F-cocompletions by dense KZ-monads on V-Cat. We prove various corollaries about the structure of such 2-monads and their Kleisli 2-categories, as needed for the use of open maps in giving an axiomatic study of bisimulation in concurrency. This requires the introduction of the concept of a pseudo-commutativity for a strong 2-monad on a symmetric monoidal 2-category, and a characterisation of it in terms of structure on the Kleisli 2-category.