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I am reading a 2007 article of Bressler et al. on deformation quantization of gerbes. In the article, the authors state that a gerbe on a manifold is defined using certain two-cocycles $c_{ijk}$ but how is this cocycle data used to write down a corresponding bundle gerbe in practice?

The theoretical reason that one can do this is that there is an equivalence of the $2$-groupoids for a gerbe and a bundle gerbe when both are expressed in terms of their cocycle data, but how would one write down the connection on the bundle gerbe using this data, or is there some freedom in choosing such a connection?

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A gerbe on a manifold $M$ is a morphism of simplicial presheaves $$\def\tB{{\sf B}}\def\U{{\rm U}}\def\cC{{\sf\check C}}\cC(U)→\tB^2\U(1),$$ where $\{U_i\}_{i∈I}$ is an open cover of $M$, $\cC(U)$ is the Čech nerve of $U$, $\U(1)$ is the representable presheaf of the Lie group $\U(1)$, and $\tB$ denotes the delooping functor.

Unfolding this definition yields precisely the cocycle data for a gerbe on $M$.

Likewise, a gerbe with connection on $M$ is a morphism of simplicial presheaves $$\cC(U)→Γ(Ω^2←Ω^1←\U(1))=\tB^2_∇\U(1),$$ where $Γ$ denotes the Dold–Kan functor, and $Ω^k$ denotes the presheaf of differential $k$-forms.

In general, other models for bundle gerbes are obtained by picking simplicial presheaves weakly equivalent to $M$ respectively $\tB^2\U(1)$ (or $\tB^2_∇\U(1)$), and writing down morphisms between them.

For example, a commonly encountered definition of gerbes uses the local data of a principal $\U(1)$-bundle on a submersion over $M$. This is encoded by replacing $M$ with the Čech nerve of the submersion and replacing $\tB^2\U(1)$ with the delooping of the presheaf of principal $\U(1)$-bundles and their isomorphisms.

Since the corresponding simplicial mapping spaces happen to be derived, this also proves the equivalence of various models of bundle gerbes and bundle gerbes with connection.

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  • $\begingroup$ This discussion is somewhat abstract for me, I am being more pedestrian here. If a gerbe on $M$ is defined by a cocycle $c$ whose class is in $H^2 ( M, \mathcal{O}_M/ 2 \pi i \mathbb{Z})$, how does one write down a connection? $\endgroup$
    – Hollis Williams
    Commented Apr 2, 2023 at 15:26
  • $\begingroup$ See for example page 7 of the lecture notes of Hitchin arxiv.org/pdf/math/9907034.pdf $\endgroup$
    – Hollis Williams
    Commented Apr 2, 2023 at 15:27
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    $\begingroup$ @HollisWilliams: A class in the second cohomology of M with coefficients in $\def\cO{{\cal O}}\def\Z{{\bf Z}}\cO/2πi\Z$ is equivalently a map in the derived category $M→\cO/2πi\Z[2]$, where $[2]$ denotes the shift of a chain complex by 2 degrees. Now replace $\cO/2πi\Z[2]$ with the Deligne complex $\cO/2πi\Z→Ω^1→Ω^2$ (sometimes it is more convenient to take $(2πi)^3$ here, like Deligne does) and take morphisms in the derived category as before (this is also known as the hypercohomology of $M$). $\endgroup$
    – Dmitri Pavlov
    Commented Apr 2, 2023 at 22:50
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    $\begingroup$ @HollisWilliams: To connect with the simple cocycle description mentioned by Hitchin, observe that unfolding the definition of a map of simplicial presheaves (or, equivalently, presheaf of corresponding chain complexes) $\def\C{{\sf\check C}}\C(U)→(Ω^0/2πiZ→Ω^1→Ω^2)$ yields an element in the totalization of the Čech–de Rham (in this case, Čech–Deligne) bicomplex. To match Hitchin's description exactly, take the homotopy cofiber of the curvature map from the presheaf of line bundles with connection to differential 2-forms, and replace the Deligne complex with this cofiber, recovering Hitchin. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 2, 2023 at 22:56

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