# Differential cohomology theories as sheaves of spectra

@article{Bunke2013DifferentialCT,
title={Differential cohomology theories as sheaves of spectra},
author={Ulrich Bunke and Thomas Nickelsen Nikolaus and Michael V{\"o}lkl},
journal={Journal of Homotopy and Related Structures},
year={2013},
volume={11},
pages={1-66}
}
• Published 13 November 2013
• Mathematics
• Journal of Homotopy and Related Structures
We show that every sheaf on the site of smooth manifolds with values in a stable $$(\infty ,1)$$(∞,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…
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## References

SHOWING 1-10 OF 27 REFERENCES
Uniqueness of smooth extensions of generalized cohomology theories
• Mathematics
• 2010
We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqueness of smooth extensions, and the identification of the
Index theorem and equivariant cohomology on the loop space
In this paper, extending ideas of Witten and Atiyah, we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the Dirac
Quadratic functions in geometry, topology, and M-theory
• Mathematics
• 2002
We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results
Axiomatic characterization of ordinary differential cohomology
• Mathematics
• 2008
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
SMOOTH K-THEORY
• Mathematics
• 2009
In this paper we consider smooth extensions of cohomology theo- ries. In particular we construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth
Loop Differential K-theory
• Mathematics
• 2012
In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an
Differential cohomology in a cohesive infinity-topos
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we
Loop Spaces, Characteristic Classes and Geometric Quantization
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical
Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic 𝐾-theory
• Mathematics
• 2013
We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic