On the formal theory of pseudomonads and pseudodistributive laws

TL;DR

This paper develops the formal theory of pseudomonads in Gray-categories, establishing a triequivalence with existing frameworks and clarifying coherence conditions for pseudodistributive laws.

Contribution

It proves the existence of a Gray-category of pseudomonads for any Gray-category and establishes a triequivalence with Marmolejo's framework, advancing the formal theory of pseudomonads.

Findings

Constructed Gray-category Psm(K) of pseudomonads for any Gray-category K

Established a triequivalence with Marmolejo's Gray-category of pseudomonads

Provided a clear account of coherence conditions for pseudodistributive laws

Abstract

We contribute to the formal theory of pseudomonads, i.e. the analogue for pseudomonads of the formal theory of monads. In particular, we solve a problem posed by Steve Lack by proving that, for every Gray-category K, there is a Gray-category Psm(K) of pseudomonads, pseudomonad morphisms, pseudomonad transformations and pseudomonad modifications in K. We then establish a triequivalence between Psm(K) and the Gray-category of pseudomonads introduced by Marmolejo. Finally, these results are applied to give a clear account of the coherence conditions for pseudodistributive laws. 41 pages. Comments welcome.