Descent Data and Absolute Kan Extensions

TL;DR

This paper explores descent theory in 2-categories, establishing conditions under which forgetful functors create Kan extensions and characterizing monadicity in various categorical contexts, including classical descent and indexed categories.

Contribution

It proves that forgetful morphisms in lax descent categories create Kan extensions and characterizes monadicity of functors without relying on the Beck-Chevalley condition.

Findings

Forgetful morphisms create all Kan extensions preserved by certain morphisms.

A right adjoint functor is monadic if it is, up to equivalence, a forgetful functor from descent data.

In indexed categories, effective descent morphisms inducing right adjoints are monadic.

Abstract

The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms create all Kan extensions that are preserved by certain morphisms. As a consequence, in the case $\mathfrak{A} = \mathsf{Cat} $, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, (naturally isomorphic to) a functor that forgets descent data. In particular, within the classical context of descent theory, we show that, in a fibred category, the forgetful functor between the category of internal actions of a precategory $a$ and the category of internal actions of the underlying discrete precategory is monadic if and only if it has a left adjoint. More particularly, this shows that one of the implications of the celebrated Benabou-Roubaud theorem does not depend on the so called Beck-Chevalley condition. Namely, we prove that, in indexed categories, whenever an effective descent morphism induces a right adjoint functor, the induced functor is monadic.