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… CONTINUE READING