Quantum Bianchi identities and characteristic classes via DG categories

Abstract: We show how DG categories arise naturally in noncommutative differential geometry and use them to derive noncommutative analogues of the Bianchi identities for the curvature of a connection. We also give a derivation of formulae for characteristic classes in noncommutative geometry following Chern's original derivation, rather than using cyclic cohomology. We show that a related DG category for extendable bimodule connections is a monoidal tensor category and in the metric compatible case give an analogue of a classical antisymmetry of the Riemann tensor. The monoidal structure implies the existence of a cup product on noncommutative sheaf cohomology. Another application is to prove that the curvature of a line module reduces to a 2-form on the base algebra. We also extend our geometric approach to Dirac operators. We illustrate the theory on the q-sphere, the permutation group S_3 and the bicrossproduct model quantum spacetime with algebra [r,t]=\lambda r.