Colax adjunctions and lax-idempotent pseudomonads

TL;DR

This paper generalizes a theorem on colax adjunctions via relative Kan extensions and explores their application to lax-idempotent pseudomonads, revealing new structural insights in 2-category theory.

Contribution

It extends classical results by establishing weak completeness and change-of-base adjunctions for Kleisli 2-categories of lax-idempotent pseudomonads.

Findings

Weak completeness of Kleisli 2-category

Existence of colax change-of-base adjunctions

Lax analogues of classical monad theory results

Abstract

We prove a generalization of a theorem of Bunge and Gray about forming colax adjunctions out of relative Kan extensions and apply it to the study of the Kleisli 2-category for a lax-idempotent pseudomonad. For instance, we establish the weak completeness of the Kleisli 2-category and describe colax change-of-base adjunctions between Kleisli 2-categories. Our approach covers such examples as the bicategory of small profunctors and the 2-category of lax triangles in a 2-category. The duals of our results provide lax analogues of classical results in two-dimensional monad theory: for instance, establishing the weak cocompleteness of the 2-category of strict algebras and lax morphisms and the existence of colax change-of-base adjunctions.