Geodesic
Combinatorics of curvature, and the Bianchi identity
Theory and applications of categories, Tome 2 (1996), pp. 69-89

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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.

Classification : 58A03, 53C05, 18F15 .
Keywords: Connection, curvature, groupoid, first neighbourhood of the diagonal. 
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     author = {Anders Kock},
     title = {Combinatorics of curvature, and the {Bianchi} identity},
     journal = {Theory and applications of categories},
     pages = {69--89},
     publisher = {mathdoc},
     volume = {2},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_1996_2_a6/}
}
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Anders Kock. Combinatorics of curvature, and the Bianchi identity. Theory and applications of categories, Tome 2 (1996), pp. 69-89. http://geodesic.mathdoc.fr/item/TAC_1996_2_a6/