Abstract
We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes due to W. Messing and the author [5], and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition than in [5] of the corresponding coboundary relations. We also show that the diagrammatic proofs of certain local curving and curvature equations may be replaced by computations with differential forms.
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Notes
- 1.
At least in a differential geometric setting, see [9], but the same construction can be carried out within the context of algebraic geometry.
- 2.
The canonical divided power 1∕2[ω,ω] of the 2-form [ω,ω] is also denoted ω∧ω or [ω](2).
- 3.
- 4.
We prefer to emphasize the fact that λ ij is a 1-cochain since this is more consistent with a simplicial definition of the associated cohomology, even though it is more customary to view the pair of equations (46) as a 2-cocycle equation, with (44) an auxiliary condition.
- 5.
See [5] (4.1.28) for a proof of this identity.
- 6.
The chosen orientation of the arrow B i is consistent with that in [5].
- 7.
This is true for diagram (??) since ν i (γ ij 02)=γ ij 02.
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Breen, L. (2011). Differential Geometry of Gerbes and Differential Forms. In: Cattaneo, A., Giaquinto, A., Xu, P. (eds) Higher Structures in Geometry and Physics. Progress in Mathematics, vol 287. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4735-3_4
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