#
Descent data and absolute Kan extensions

##
Fernando Lucatelli Nunes

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.

http://www.tac.mta.ca/tac/volumes/37/18/37-18.pdf

TAC Home