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.

Keywords: Connection, curvature, groupoid, first neighbourhood of the diagonal.

1991 MSC: 58A03, 53C05, 18F15 .

Theory and Applications of Categories, Vol. 2, 1996, No. 7, pp 69-89.

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