Butterflies II: Torsors for 2-group stacks

TL;DR

This paper explores the structure of torsors over 2-groups, introduces butterfly diagrams for morphisms, and provides a geometric interpretation of non-abelian cohomology changes, advancing the understanding of higher categorical cohomology theories.

Contribution

It introduces butterfly diagrams to encode 2-group morphisms and reinterprets non-abelian cohomology in terms of gerbes, offering explicit cocycle-level maps and geometric insights.

Findings

Butterfly diagrams effectively encode 2-group morphisms.

Gerbes bound by crossed modules represent non-abelian cohomology.

Explicit cocycle maps are derived using butterflies.

Abstract

We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the "fraction" (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level. In addition, we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions.