Group cohomology with coefficients in a crossed-module

We compare three different ways of defining group cohomology with coefficients in a crossed-module: 1) explicit approach via cocycles; 2) geometric approach via gerbes; 3) group theoretic approach via butterflies. We discuss the case where the crossed-module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed-modules and also prove the “long” exact cohomology sequence associated to a short exact sequence of crossed-modules and weak morphisms.

💡 Research Summary

The paper investigates three distinct but equivalent ways of defining group cohomology with coefficients in a crossed‑module ( \mathbb{G}=(G_{1}\xrightarrow{\partial}G_{0}) ). The authors compare an explicit cocycle description, a geometric description via gerbes (or more precisely, the classifying 2‑stack ( B\mathcal{G} ) of the associated 2‑group), and a group‑theoretic description using butterflies, which encode weak morphisms between crossed‑modules.

In the cocycle approach, a 0‑cochain is a ( \Gamma )‑invariant element of ( G_{0} ); a 1‑cochain consists of a map ( \Gamma\to G_{1} ) together with a map ( \Gamma\times\Gamma\to G_{0} ) satisfying compatibility conditions that involve the boundary map ( \partial ) and the ( \Gamma )‑action. The usual coboundary operators are defined, and the resulting cohomology groups ( H^{n}(\Gamma,\mathbb{G}) ) are shown to be independent of the chosen normalisation.

The gerbe approach treats the crossed‑module as a strict 2‑group ( \mathcal{G} ). A ( \Gamma )‑action on ( \mathcal{G} ) gives rise to a 2‑stack ( B\mathcal{G} ) equipped with a ( \Gamma )‑equivariant structure. Then ( H^{1}(\Gamma,\mathbb{G}) ) classifies ( \Gamma )‑equivariant ( \mathcal{G} )‑torsors, while ( H^{2}(\Gamma,\mathbb{G}) ) classifies ( \Gamma )‑equivariant gerbes banded by ( \mathcal{G} ). The authors prove that these geometric objects are in bijection with the cocycle classes described above, thereby providing a conceptual interpretation of the cohomology groups as isomorphism classes of higher‑bundles.

Butterflies give a purely algebraic model for weak morphisms between crossed‑modules. A butterfly consists of a middle group ( E ) together with two monomorphisms ( \iota\colon G_{1}\hookrightarrow E ) and ( \kappa\colon H_{1}\hookrightarrow E ) and two epimorphisms ( \pi\colon E\to G_{0} ), ( \rho\colon E\to H_{0} ) satisfying natural compatibility conditions. The paper shows that a butterfly from the trivial crossed‑module to ( \mathbb{G} ) encodes exactly the data of a 1‑cocycle, and that composition of butterflies corresponds to the coboundary operation. Consequently, the butterfly category is equivalent to the 2‑category of cocycles and to the gerbe‑based description.

When the crossed‑module carries a braiding, the cohomology acquires extra symmetry. For a braided crossed‑module the authors prove that ( H^{2}(\Gamma,\mathbb{G}) ) classifies central extensions of ( \Gamma ) by ( \mathbb{G} ) compatible with the braiding, and that the braiding induces a commutative product on cohomology. If the braiding is symmetric, the product is fully commutative and the groups ( H^{n} ) become abelian for ( n\le 3 ). This analysis clarifies how higher‑order commutativity constraints affect the algebraic structure of the cohomology groups.

A major technical achievement of the work is the construction of a long exact sequence associated to a short exact sequence of crossed‑modules and weak morphisms
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