Rigidified torsor cocycles, hypercoverings and bundle gerbes

TL;DR

This paper provides a geometric interpretation of higher-degree sheaf cohomology using torsors on hypercoverings with added rigidification data, generalizing known results for degrees one and two.

Contribution

It introduces a new geometric framework for understanding higher-degree sheaf cohomology via rigidified torsors on hypercoverings, extending classical cases.

Findings

Generalizes the interpretation of degree one cohomology as torsors

Expresses degree two cohomology in terms of bundle gerbes

Provides a unified geometric perspective for higher degrees

Abstract

We give a geometric interpretation of sheaf cohomology for higher degrees n in terms of torsors on the member of degree d=n-1 in hypercoverings of type r=n-2, endowed with an additional data, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.