In this example Pαβ is simply the restriction of the trivial bundle gerbe over ∐Uij to Uαβ . A trivialisation is given by Tαβ .
Constructions with bundle gerbes
The bundle 2-gerbe product is deﬁned by a D-trivialisation of this bundle gerbe, Jij k .
Constructions with bundle gerbes
This is because the T ’s have zero connections and ﬂat trivialisations so their D-obstructions are zero.
Constructions with bundle gerbes
With the presence of sections rather than trivialisations the D-obstruction forms used to deﬁne the full Deligne class may be replaced by the pullbacks of the relevant connections and curvings by the sections.
Constructions with bundle gerbes
Note that the global version is not explicitly independent of the choice of trivialisation, h, however we may deduce this from the explicit independence of the local version (5.7).
Constructions with bundle gerbes
To see that this is well deﬁned consider that when we pull back the bundle gerbe P to Σ using ψ the resulting bundle gerbe has an induced curving which we denote ψ ∗η and for dimensional reasons has a trivialisation L.
Constructions with bundle gerbes
Denote the curvature of this trivialisation (given some connection which is compatible with the bundle gerbe connection) by FL .
Constructions with bundle gerbes
This is independent of the choice of trivialisation since a different choice just changes FL by a closed 2-form which descends to Σ.
Constructions with bundle gerbes
For the case of a bundle 2-gerbe we must ﬁrst establish the notation associated with the ﬂat holonomy and with trivialisations.
Constructions with bundle gerbes
We choose a trivialisation with connection and curving.
Constructions with bundle gerbes
The 3-form deﬁned by the difference between the 3-curving induced by the pullback and the 3-curvature of the trivialisation may be integrated over X to deﬁne the holonomy.
Constructions with bundle gerbes
It is important to remember that in this case χ is no longer a D-obstruction form so we cannot be sure that the construction is independent of the choice of the trivialisation h.
Constructions with bundle gerbes
This expression is independent of the choice of ρ, but the dependence on a choice of trivialisation causes diﬃculties.
Constructions with bundle gerbes
We would prefer to have a ρ dependence instead, this could be achieved by restricting to a particular ρ and then choosing a trivialisation, however this is rather technical.
Constructions with bundle gerbes
It is easy enough to choose a trivialisation over each element of the triangulation using the canonical trivialisation over an element of the open cover which was described in §3.2.
Constructions with bundle gerbes
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