Functors Preserving Effective Descent Morphisms

TL;DR

This paper introduces a new approach to studying effective descent morphisms by establishing conditions under which certain functors preserve these morphisms, with applications to Grothendieck fibrations and topological functors.

Contribution

The paper presents general results on the preservation of effective descent morphisms by functors, expanding the methods used beyond reflection properties.

Findings

Grothendieck (op)fibrations satisfying mild conditions preserve effective descent morphisms

Topological functors and other forgetful functors are shown to preserve effective descent morphisms

New framework broadens understanding of functorial preservation of descent properties

Abstract

Effective descent morphisms, originally defined in Grothendieck descent theory, form a class of special morphisms within a category. Essentially, an effective descent morphism enables bundles over its codomain to be fully described as bundles over its domain endowed with additional algebraic structure, called descent data. Like the study of epimorphisms, studying effective descent morphisms is interesting in its own right, providing deeper insights into the category under consideration. Moreover, studying these morphisms is part of the foundations of several applications of descent theory, notably including Janelidze-Galois theory, also known as categorical Galois theory. Traditionally, the study of effective descent morphisms has focused on investigating and exploiting the reflection properties of certain functors. In contrast, we introduce a novel approach by establishing general results on the preservation of effective descent morphisms. We demonstrate that these preservation results enhance the toolkit for studying such morphisms, by observing that all Grothendieck (op)fibrations satisfying mild conditions fit our framework. To illustrate these findings, we provide several examples of Grothendieck (op)fibrations that preserve effective descent morphisms, including topological functors and other forgetful functors of significant interest in the literature.