Enriched Kleisli objects for pseudomonads

TL;DR

This paper explores the structure of Kleisli objects for pseudomonads in 2-category theory, providing a new presentation of weighted colimits and analyzing their homotopical properties.

Contribution

It introduces a presentation for the weighted colimit defining Kleisli objects for pseudomonads and studies their universal properties and homotopical aspects.

Findings

The weight for Kleisli objects is cofibrant in the projective model structure.

Comparison 2-functors induced by pseudoadjunctions are bi-fully faithful.

Biessential surjectivity characterizes certain left pseudoadjoints.

Abstract

A pseudomonad on a $2$-category whose underlying endomorphism is a $2$-functor can be seen as a diagram $\mathbf{Psmnd} \rightarrow \mathbf{Gray}$ for which weighted limits and colimits can be considered. The $2$-category of pseudoalgebras, pseudomorphisms and $2$-cells is such a $\mathbf{Gray}$-enriched weighted limit \cite{Coherent Approach to Pseudomonads}, however neither the Kleisli bicategory nor the $2$-category of free pseudoalgebras are the analogous weighted colimit \cite{Formal Theory of Pseudomonads}. In this paper we describe the actual weighted colimit via a presentation, and show that the comparison $2$-functor induced by any other pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a consequence, we see that biessential surjectivity on objects characterises left pseudoadjoints whose codomains have an `up to biequivalence' version of the universal property for Kleisli objects. This motivates a homotopical study of Kleisli objects for pseudomonads, and to this end we show that the weight for Kleisli objects is cofibrant in the projective model structure on $[\mathbf{Psmnd}^\text{op}, \mathbf{Gray}]$.