Four Equivalent Versions of Non-Abelian Gerbes

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth non-abelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.

💡 Research Summary

The paper establishes a comprehensive equivalence between four prevailing models of smooth non‑abelian gerbes: Čech cocycles, classifying maps, bundle gerbes, and principal 2‑bundles. Beginning with a systematic review, the authors recast each model in a differential‑geometric setting, emphasizing the role of a crossed module (G,H) as the underlying non‑abelian 2‑group. In the Čech approach, they replace the usual continuous transition data with smooth maps g_{ij}:U_{ij}→G and h_{ijk}:U_{ijk}→H, thereby enabling the definition of connections and curvatures. The classifying‑map perspective constructs the classifying space B𝔾 for the 2‑group and shows that smooth maps f:X→B𝔾 classify gerbes up to isomorphism. A key technical achievement is the proof of a bijection between continuous non‑abelian cohomology H¹_{cont}(X,𝔾) and its smooth counterpart H¹_{∞}(X,𝔾), achieved by a careful model‑category analysis of mapping spaces.

In the bundle‑gerbe section, the authors generalize Murray’s U(1) gerbe to a 𝔾‑bundle gerbe, encoding the multiplication and associativity data as a 2‑monoid object in the 2‑category of smooth manifolds. They explicitly construct the associator and unitors required for the non‑abelian coherence conditions. The principal 2‑bundle part follows Baez‑Schreiber’s framework, defining a 2‑bundle with a smooth 2‑group action and describing its local trivializations via 2‑cocycles. By translating the gerbe data into a global 2‑section, they demonstrate a precise correspondence with principal 2‑bundles, using 2‑natural transformations and 2‑equivalences.

The core of the work is the construction of explicit 2‑stack equivalences among the four models. The authors build 2‑functors that send a Čech cocycle to a classifying map, a classifying map to a bundle gerbe, and a bundle gerbe to a principal 2‑bundle, and they prove that each functor admits a quasi‑inverse up to coherent 2‑isomorphism. The proof relies on the completeness of the model structures on smooth mapping spaces, the uniqueness of connections on crossed modules, and the preservation of the higher‑categorical coherence laws. Consequently, all four descriptions represent the same 2‑stack of smooth non‑abelian gerbes.

Beyond the equivalence theorem, the paper provides new tools for applications: the smooth‑continuous cohomology bijection facilitates the passage between topological and differential data, while the explicit gerbe‑to‑2‑bundle equivalence offers a concrete bridge for physicists working with higher gauge fields and non‑abelian surface holonomies. The authors conclude with a discussion of potential extensions to higher (n‑)gerbes and to twisted or equivariant settings, suggesting that the unified framework will streamline future research across geometry, topology, and quantum field theory.