Butterflies II: Torsors for 2-group stacks

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.

💡 Research Summary

The paper “Butterflies II: Torsors for 2‑group stacks” develops a comprehensive framework for studying torsors (principal homogeneous spaces) over 2‑groups, also known as crossed modules, and for understanding how these torsors behave under changes of coefficients. The authors begin by recalling the notion of a 2‑group stack and the associated first non‑abelian cohomology set (H^{1}(X,\mathcal{G})), which classifies (\mathcal{G})‑torsors (or, equivalently, gerbes bound by the crossed module (\mathcal{G})). They point out that while strict morphisms between 2‑groups are well‑understood, many natural transformations are only weak, and existing categorical language does not give a concrete handle on the induced maps on cohomology.

To fill this gap they introduce the “butterfly” diagram, a four‑corner commutative diagram consisting of two crossed modules ((G_{1}\to H_{1})) and ((G_{2}\to H_{2})) together with a middle group (E) and two exact sequences linking (E) to each crossed module. The butterfly encodes a weak morphism (or “fraction”) (\mathcal{B}:\mathcal{G}{1}\dashrightarrow\mathcal{G}{2}). The central insight is that such a fraction can be used to lift a gerbe bound by (\mathcal{G}{1}) to a gerbe bound by (\mathcal{G}{2}). Concretely, given a (\mathcal{G}{1})‑torsor (\mathcal{X}), one first equips (\mathcal{X}) with an action of the middle group (E) via the left exact sequence, then uses the right exact sequence to reinterpret that action as an action of (\mathcal{G}{2}). The resulting object is a (\mathcal{G}_{2})‑torsor, and the construction yields a functorial “change‑of‑coefficients” map \