Bicategories of algebras for relative pseudomonads

TL;DR

This paper develops the theory of pseudoalgebras for relative pseudomonads, establishing bicategory structures, coherence results, and applications to colimit classes, extending pseudomonad theory.

Contribution

It introduces the concept of pseudoalgebras for relative pseudomonads and constructs bicategory frameworks, extending existing pseudomonad theory to a broader context.

Findings

Bicategory of T-pseudoalgebras is terminal among resolutions of T.

The Kleisli bicategory embeds into the bicategory of pseudoalgebras.

A correspondence between relative monads and cocontinuous monads is established.

Abstract

We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $\Phi$, there is a correspondence between monads relative to free $\Phi$-cocompletions, and $\Phi$-cocontinuous monads on free $\Phi$-cocompletions.