Descent data and absolute Kan extensions
Lucatelli Nunes, Fernando
(2021) Theory and Applications of Categories, volume 37, pp. 530 - 561
(Article)
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
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= 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.
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Keywords: Bénabou-Roubaud theorem, Creation of absolute Kan extensions, Descent theory, Effective descent morphisms, Indexed cate-gories, Internal actions, Monadicity theorem, Mathematics (miscellaneous)
ISSN: 1201-561X
Publisher: Mount Allison University
Note: 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.
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