Codescent and bicolimits of pseudo-algebras

TL;DR

This paper extends cocompleteness results from monad theory to pseudomonads within 2-categories, establishing conditions for bicocompleteness of pseudo-algebras through codescent objects and bicoequalizers.

Contribution

It introduces a general framework for constructing weighted bicolimits from oplax bicolimits and bicoequalizers, and proves a reduction theorem for bicocompleteness of pseudo-algebras.

Findings

Bicolimits can be built from oplax bicolimits and bicoequalizers of codescent objects.

A reduction theorem links bicocompleteness of pseudo-algebras to existence of bicoequalizers.

Bicocompleteness is guaranteed for bifinitary pseudomonads.

Abstract

We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of codescent objects. After prerequisites on pseudomonads and their pseudo-algebras, we give a 2-dimensional Linton theorem reducing bicocompleteness of 2-categories of pseudo-algebras to existence of bicoequalizers of codescent objects. Finally we prove this condition to be fulfilled in the case of a bifinitary pseudomonad, ensuring bicocompleteness.