symmetric monoidal (∞,1)-category of spectra
The construction that sends a smooth manifold $X$ to the algebraic K-theory spectrum $K(C^\infty(X,\mathbb{C}))$ of its ring of smooth functions (with values in the complex numbers) presents (after infinity-stackification) a sheaf of spectra on the site of smooth manifolds, hence a smooth spectrum
i.e. an object of the tangent cohesive (∞,1)-topos of Smooth∞Grpd. (See also this definition at differential cohomology hexagon.)
The shape of this $\mathbf{K}$ is the topological K-theory spectrum $ku$ (Bunke-Nikolaus-Voelkl 13, lemma 6.3, Bunke 14, (48) with def. 2.21):
Hence $\mathbf{K}$ is a differential cohomology refinement of $ku$, a form of differential K-theory.
There is also the more standard differential K-theory refinement $\mathbf{ku}_{conn}$ of $ku$ (Hopkins-Singer 05, Bunke-Nikolaus-Voelkl 13) which is obtained by pulling back suitable sheaves of ($\mathbb{C}$-valued) differential forms $\mathbf{DD}^-$ along the usual Chern character map $ch \colon ku \longrightarrow DD^{per}$. This Chern character lifts through the shape modality to a regulator map (Bunke 14, (50))
Moreover, this induces a differential regulator (BNV 13, p.40 and example 6.9, Bunke 14, def. 2.29):
See also this proposition at differential cohomology diagram.
In (Bunke 14) all this is generalized to the mapping spaces $[X,\mathbf{K}]$ out of a smooth manifold $X$, hence to the moduli stacks (here: moduli spectra) of algebraic K-theory cocycles on $X$.
The regulator from above induces a map
(…)
Ulrich Bunke, Thomas Nikolaus, Michael Völkl, section 6 of Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures October 2014 (arXiv:1311.3188)
Ulrich Bunke, A regulator for smooth manifolds and an index theorem (arXiv:1407.1379)
Ulrich Bunke, Smooth aspects of algebraic K-theory (pdf)
Last revised on November 7, 2015 at 10:07:45. See the history of this page for a list of all contributions to it.