What is the universal property of the 2-category of monads?

TL;DR

This paper investigates the universal property of the 2-category of monads within a 2-category, revealing it as a free completion related to enriched category theory and establishing foundational results in 2-category theory.

Contribution

It characterizes the 2-functor from Mnd(𝓚) to EM(𝓚) as a free completion of the identity-on-objects-and-1-cells 2-functor, using enriched category theory over a cartesian closed category.

Findings

The 2-functor Mnd(𝓚) → EM(𝓚) is a free completion.

Develops theory of categories enriched over a cartesian closed category.

Provides a new perspective on the universal property of the 2-category of monads.

Abstract

For a 2-category $\mathcal{K}$, we consider Street's 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street's 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$) $\to$ EM($\mathcal{K}$) between them. We show that this 2-functor can be obtained as a "free completion" of the 2-functor $1\colon \mathcal{K} \to \mathcal{K}$. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category $\mathbf{BO}$ whose objects are identity-on-objects functors. We also develop some of the theory of $\mathbf{BO}$-enriched categories.