Differential cohomology theories as sheaves of spectra
Ulrich Bunke, Thomas Nikolaus, Michael Völkl
TL;DR
This work treats differential cohomology as a general, sheaf-theoretic refinement in stable $(\infty,1)$-categories: any ${\mathbf{C}}$-valued sheaf on smooth manifolds yields a differential cohomology diagram with a homotopy formula, obtained by decomposing into a homotopy-invariant part and a cycle-data piece. The authors develop a cohesive framework using adjoints ${\mathcal H}, {\tt const}, {\mathcal S}, {\mathcal G}$ and associated functors to produce the differential data ${\mathcal Z}$, ${\mathcal A}$, $U$, $S$, and a curvature map $R$, together with an integration map $\int$ that encodes the homotopy formula. They instantiate the general theory across a broad spectrum of examples, including classical differential forms, generalized Deligne cohomology, Hopkins–Singer differential K-theory, loop K-theory refinements, and Snaith-type differential K-theory, computing the invariants and offering a scheme to analyze new sheaves by their cycle data and gluing maps. The framework also yields universal differential characteristic classes and provides a unifying viewpoint on secondary invariants via the functor $S$, connecting to Cheeger–Simons theory and loop-space refinements. Overall, the paper offers a principled, model-agnostic method to derive and compare differential refinements, with potential to guide construction of new theories and invariants in differential cohomology.
Abstract
We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-category (like spectra or chain complexes) gives rise to a differential cohomology diagram and a homotopy formula, which are common features of all classical examples of differential cohomology theories. These structures are naturally derived from a canonical decomposition of a sheaf into a homotopy invariant part and a piece which has a trivial evaluation on a point. In the classical examples the latter is the contribution of differential forms. This decomposition suggest a natural scheme to analyse new sheaves by determining these pieces and the gluing data. We perform this analysis for a variety of classical and not so classical examples.