Generalized principal bundles and quotient stacks

TL;DR

This paper generalizes the concept of principal bundles within a broad categorical framework, constructing quotient prestacks and proving their descent properties under certain conditions, advancing the theoretical understanding of stacks.

Contribution

It introduces a generalized notion of principal bundles in sites with pullbacks, and constructs quotient prestacks with descent properties, extending classical theory.

Findings

Quotient prestacks satisfy descent in subcanonical sites.

Explicit construction of quotient prestacks via presheaves of categories.

Generalization of principal bundles to abstract categorical contexts.

Abstract

We consider the internalization of the usual notion of principal bundle in a site that has all pullbacks and a terminal object. We use this notion to consider the explicit construction of quotient prestacks via presheaves of categories of principal bundles equipped with equivariant morphisms in this abstract context. We then prove that, if the site is subcanonical and the underlying category satisfies some mild conditions, these quotient prestacks satisfy descent in the sense of stacks.