# Second nonabelian group cohomology: cocycles vs. gerbes

In 1965 Jean Giraud published two Comptes Rendus notes titled "Cohomologie non abélienne", and in 1971 he published a book with the same title. In 1966 Tonny A. Springer's paper "Nonabelian $H^2$ in Galois cohomology" appeared, where he, in particular, constructs nonabelian $H^2$ of a group in terms of group extensions and in terms of cocycles. Springer writes that his definition for group cohomology "seems to be essentially equivalent to to that of Dedecker and Giraud". Giraud in his book (page 452) writes that "la définition de $H^2$ en termes de gerbes ... redonne, dans ce cas, la théorie de Springer".

I do not understand the latter assertion. Let $\Gamma$ be a group, and let $G$ be a group together with a "$\Gamma$-kernel": a homomorphism $\kappa\colon \Gamma\to {\rm Out}(G)$, where ${\rm Out}(G):= {\rm Aut}(G)/{\rm Inn}(G)$ is the group of outer automorphisms of $G$. Springer defines $H^2(\Gamma, G,\kappa)$ in terms of group extensions $$1\to G\to E\to \Gamma\to 1$$ inducing the "kernel" $\kappa$. He also describes $H^2(\Gamma, G,\kappa)$ in terms of 2-cocycles coming from the group extension. (A 2-cocycle is a pair of maps $(f,g)$ of maps $f\colon \Gamma\to {\rm Aut}(G)$, $g\colon \Gamma\times \Gamma\to G$ satisfying certain conditions.) Giraud defines $H^2$ in terms of gerbes (on the category of $\Gamma$-sets?).

Question: How can I get a gerbe $\mathcal{G}$ (i.e., a stack over the category of $\Gamma$-sets) from a group extension? In other words, for any $\Gamma$-set $S$, I want to get a groupoid $\mathcal{G}_S$ defined in terms of the given group extension. Conversely, I would like get a group extension from a gerbe.

## 3 Answers

The cocycle data which you review together is a map of 2-groupoids $B \Gamma \to B Aut(B G)$ to the delooping of the automorphism 2-group "of $G$" (really: of $BG$). As for any cocycle with coefficients in an automorphism group, there is the corresponding associatived 2-bundle, hence a $BG$-fiber bundle. That's the corresponding Giraud G-gerbe and that's essentially the group extension.

There are some details on this spelled out in the nLab entry nonabelian group cohomology, though maybe that needs another touch.

The general abstract story of nonabelian cocycle, infinity-gerbes and associated infinity-bundles is in section 4 of

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal infinity-bundles - General theory (arXiv:1207.0248, web)
• @Urs: Thank you! I will try to read your preprint. Still, I would appreciate if you could write a more explicit answer... – Mikhail Borovoi May 11 '13 at 6:05

I am not sure if this is the most general definition, but my proposal for 2nd nonabelian cohomology is combinatorial descent data in a cosimplicial crossed gropoid, modulo gauge equivalence. This is a development of an earlier idea of L. Breen.

This is worked out in my paper

Combinatorial descent data for gerbes, Journal of Noncommutative Geometry Volume 8, Issue 4, 2014, pp. 1083–1099 arXiv:1109.1919

For a geometric application see the paper

Twisted Deformation Quantization of Algebraic Varieties, Advances in Mathamatics, Volume 268, 2 January 2015, Pages 241–305 arXiv:0905.0488

Nonabelian $$H^2$$ in Galois cohomology can be defined in terms of: (1) cocycles, (2) extensions, (3) gerbes. The relations between these three definitions are described in Section 2.2 of Le principe de Hasse pour les espaces homogènes : réduction au cas des stabilisateurs finis by Cyril Demarche and Giancarlo Lucchini Arteche.