Semantic Factorization and Descent

TL;DR

This paper establishes a connection between semantic factorization and descent in 2-categories, providing a new proof and a monadicity theorem that characterizes monadicity via descent conditions.

Contribution

It offers a direct proof linking semantic factorization with descent in 2-categories and introduces a monadicity theorem based on descent conditions.

Findings

Semantic factorization matches the descent-based factorization under certain conditions.

Monadicity can be characterized as a 2-dimensional exact condition on morphisms.

Conditions on the codensity monad are trivially satisfied when the morphism has a left adjoint.

Abstract

Let $\mathbb{A}$ be a $2$-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism $p$ exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the higher cokernel of $p$ is up to isomorphism the same as the semantic factorization of $p$, either one existing if the other does. The result can be seen as a counterpart account to the celebrated B\'{e}nabou-Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of $p$ trivially hold whenever $p$ has a left adjoint and, hence, in this case, we find monadicity to be a $2$-dimensional exact condition on $p$, namely, to be an effective faithful morphism of the $2$-category $\mathbb{A} $.