Axiomatic characterization of ordinary differential cohomology

The Cheeger–Simons differential characters, the Deligne cohomology in the smooth category, the Hopkins–Singer construction of ordinary differential cohomology, and the recent Harvey–Lawson constructions are each in two distinct ways abelian group extensions of known functors. In one description, these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other, they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short-exact sequences mesh with two interlocking long-exact sequences (the Bockstein sequence and the de Rham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Our first theorem shows that on the category of smooth manifolds and smooth maps, any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as mentioned earlier is unique up to a unique natural equivalence. Our second theorem shows that natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of our first theorem couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.