Descent data and absolute Kan extensions

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 A with lax descent objects, the forgetful morphisms create all Kan extensions that are preserved by certain morphisms. As a consequence, in the case A = Cat, we get a monadicity theorem which says that a right adjoint functor is monadic 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 Bénabou-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.