Pseudoalgebras and non-canonical isomorphisms

TL;DR

The paper characterizes when lax morphisms between pseudoalgebras are pseudomorphisms, linking this to the existence of invertible transformations, and unifies various non-canonical isomorphism results across different 2-categories.

Contribution

It provides a general criterion for identifying pseudomorphisms among lax morphisms in the context of pseudomonads, extending known results to broader 2-categories.

Findings

Lax T-morphisms are T-pseudomorphisms iff there exists an invertible T-transformation.

Unifies non-canonical isomorphism results in monoidal categories and other 2-categories.

Applicable to categories like monoidal, cocomplete, and pseudofunctor categories.

Abstract

Given a pseudomonad $\mathcal{T} $, we prove that a lax $\mathcal{T} $-morphism between pseudoalgebras is a $\mathcal{T} $-pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible $\mathcal{T} $-transformation. This result encompasses several results on \textit{non-canonical isomorphisms}, including Lack's result on normal monoidal functors between braided monoidal categories, since it is applicable in any $2$-category of pseudoalgebras, such as the $2$-categories of monoidal categories, cocomplete categories, pseudofunctors and so on.