Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures

TL;DR

This paper introduces the concept of relative pseudomonads, extends the Kleisli bicategory construction, and provides a uniform method to define pseudomonads on bicategories of profunctors, impacting operad theory, logic, and computer science.

Contribution

It generalizes pseudomonads to relative pseudomonads and offers a systematic way to define pseudomonads on Kleisli bicategories of these structures.

Findings

Defined the notion of a relative pseudomonad.

Developed an efficient method for pseudomonad construction.

Applied to bicategories of profunctors for uniform definitions.

Abstract

We introduce the notion of a relative pseudomonad, which generalises the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonas on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.