TY - JOUR

T1 - Descent data and absolute Kan extensions

AU - Lucatelli Nunes, Fernando

N1 - Funding Information:
This research was partially supported by the Institut de Recherche en Math?matique et Physique (IRMP, UCLouvain, Belgium), and by the Centre for Mathematics of the University of Coimbra-UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.
Funding Information:
This research was partially supported by the Institut de Recherche en Mathématique et Physique (IRMP, UCLouvain, Belgium), and by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Received by the editors 2020-06-08 and, in final form, 2021-05-17. Transmitted by Tim van der Linden. Published on 2021-05-20. 2020 Mathematics Subject Classification: 18N10, 18C15, 18C20, 18F20, 18A22, 18A30, 18A40. Key words and phrases: descent theory, effective descent morphisms, internal actions, indexed categories, creation of absolute Kan extensions, Bénabou-Roubaud theorem, monadicity theorem. © Fernando Lucatelli Nunes, 2021. Permission to copy for private use granted.
Publisher Copyright:
© Fernando Lucatelli Nunes, 2021.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Bénabou-Roubaud theorem

KW - Creation of absolute Kan extensions

KW - Descent theory

KW - Effective descent morphisms

KW - Indexed cate-gories

KW - Internal actions

KW - Monadicity theorem

UR - http://www.scopus.com/inward/record.url?scp=85108073567&partnerID=8YFLogxK

M3 - Article

SN - 1201-561X

VL - 37

SP - 530

EP - 561

JO - Theory and Applications of Categories

JF - Theory and Applications of Categories

M1 - 18

ER -