Combinatorics of curvature, and the Bianchi identity
Anders Kock
We analyze the Bianchi Identity as an instance of a
basic fact of combinatorial groupoid theory, related to
the Homotopy Addition Lemma. Here it becomes formulated
in terms of 2-forms with values in the gauge group bundle
of a groupoid, and leads in particular to the
(Chern-Weil) construction of characteristic classes.
The method is that of synthetic differential geometry,
using "the first neighbourhood of the diagonal" of a
manifold as its basic combinatorial structure. We introduce
as a tool a new and simple description of wedge (= exterior)
products of differential forms in this context.