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Combinatorics of curvature, and the Bianchi identity. (English) Zbl 0877.58002

The author’s abstract: “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”.

MSC:

58A03 Topos-theoretic approach to differentiable manifolds
53C05 Connections (general theory)
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
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