Definition 1. An m-linear function f which maps the m-fold cartesian product V m of a vector space V into some other vector space W is called alternating if f(v1, . . . , vm) = 0 whenever v1, . . . , vm ∈ V and vi = vj for i 6= j. We let ∧ (V ,W ) be the vector space of mlinear alternating functions mapping V m intoW . We then define ∧m V by the property that if f ∈ ∧m(V,W ) then there exists a...

Remark. It is also possible to define tensors over a complex vector space (with no essential modifications in the definitions and results below, except that the special Euclidean case no longer applies). This is important in many other applications (including complex manifolds), but will not be considered here. 1.1. The dual space. Let V ∗ = hom(V ;R) be the dual of V , i.e. the vector space of...

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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 curvatu...