Centre for Mathematics, University of Coimbra

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Type:	Preprint
National /International:	International
Title:	Descent data and absolute Kan extensions
Publication Date:	2019-06-01
Authors:	- Fernando Lucatelli Nunes
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 absolute Kan extensions. As a consequence of this result, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, 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. In particular, this proves that one of the implications of the celebrated B´ enabou-Roubaud theorem does not depend on the so called Beck-Chevalley condition. Namely, we show that, in a fibred category with pullbacks, whenever an effective descent morphism induces a right adjoint functor, the functor is monadic.
Institution:	DMUC 19-19
Online version:	http://www.mat.uc.pt...prints/eng_2019.html
Download:	Not available