Generalized nerves of monads

TL;DR

This paper introduces generalized nerve constructions for monads that extend the Kleisli construction, providing fully faithful 2-functors from monads to double categories, enriching the categorical understanding of monads.

Contribution

It develops more general nerve constructions on 2-categories of monads, expanding the classical Kleisli-based approach with new fully faithful 2-functors.

Findings

New nerve constructions for monads are fully faithful.

These constructions generalize the Kleisli nerve.

They establish a richer categorical framework for monads.

Abstract

One interpretation of the Kleisli construction (given by Miranda and related to work of Par\'e) is as a nerve sending a monad $P$ to the Kleisli double category of $P$. In this paper we find more general nerve constructions on the 2-categories of monads, which also give fully faithful nerve 2-functors $\mathbf{Mnd}\left(\mathcal{K}\right)\to\mathbf{Dbl}$.