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.
Keywords: descent theory, effective descent morphisms, internal actions, indexed categories, creation of absolute Kan extensions, Bénabou-Roubaud theorem, monadicity theorem
2020 MSC: 18N10, 18C15, 18C20, 18F20, 18A22, 18A30, 18A40
Theory and Applications of Categories, Vol. 37, 2021, No. 18, pp 530-561.
Published 2021-05-20.
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