Quantum Bianchi identities and characteristic classes via DG categories

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.