The second descent obstruction and gerbes

TL;DR

This paper generalizes the concept of rational points and obstructions on varieties to categories fibred in groupoids, introducing new obstructions like the derived obstruction and relating second cohomology to gerbes.

Contribution

It extends descent theory and cohomological obstructions to a categorical framework, proposing new obstructions including the derived obstruction with favorable properties.

Findings

Established the generalized theory of descent by torsors.

Interpreted second cohomology obstructions via gerbes.

Constructed new obstructions that are not larger than the descent obstruction.

Abstract

We extend the notion of rational points and cohomological obstructions on varieties to categories fibred in groupoids. We also establish the generalized theory of descent by torsors. Then we interpret the obstruction given by the second cohomology of abelian sheaves in terms of categorical points of gerbes, analogue to descent by torsors. As an application, we construct some composite obstructions not larger than the descent obstruction. We also propose some new kinds of obstructions including the derived obstruction, which has good behavior under a product and is also not larger than the descent obstruction.