nLab sheaf of spectra

Idea

The stabilization of an (∞,1)-topos $\mathbf{H}$

$$(\Sigma^\infty \dashv \Omega^\infty) : \mathbf{H} \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}} Stab(\mathbf{H})$$

consist of spectrum objects in $\mathbf{H}$. By the “stable Giraud theorem” this is the localization of an (∞,1)-category of (∞,1)-functors with values in the stable (∞,1)-category of spectra: $\infty$-sheaves of spectra.

This may be presented by a model structure on presheaves of spectra.

Properties

Relation to sheaves of chain complexes and abelian sheaf cohomology

The stable Dold-Kan correspondence

$$DK \;\colon\; Ch_\bullet(R) \stackrel{\simeq}{\longrightarrow} H R Mod(Spectra) \stackrel{U}{\longrightarrow} Spectra$$

is an (∞,1)-limits-preserving (∞,1)-functor from the (∞,1)-category of chain complexes (over a given commutative ring) to the (∞,1)-category of spectra. Hence every (∞,1)-sheaf/∞-stack of chain complexes (as it appears (maybe implicitly) in abelian sheaf cohomology/hypercohomology canonically incarnates as an $(\infty,1)$-sheaf of spectra).

Symmetric monoidal structure

The smash tensor product of the symmetric monoidal (∞,1)-category of spectra induced also a smash tensor product on any (∞,1)-category of sheaves of spectra

(Lurie, "Spectral Schemes", prop 1.5).

Examples

• generalized étale cohomology

• tangent cohesion

  • smooth spectrum

  • differential cohomology,

• differential algebraic K-theory

Related concepts

• sheaf of L-∞ algebras

• parameterized spectrum

References

General

The homotopy categories of sheaves of combinatorial spectra were discussed in

• Kenneth Brown, part II of Abstract homotopy theory and generalized sheaf cohomology (1973)

A model category structure of presheaves of spectra akin to the model structure on simplicial presheaves is discussed in

• Rick Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39(1987), 733-747 (pdf)

Plenty of further discussion in terms of model category theory is in

• Rick Jardine, Generalized Étale cohomology theories, 1997 Progress in mathematics volume 146

Discussion in terms of (∞,1)-category/(∞,1)-topos-theory is in

• Jacob Lurie, Sheaves of spectra, section 1 of Spectral Schemes

Application to K-theory

• Bertrand Toën, section 1.2 of K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems (arXiv:math/9908097)

• Michael Paluch, Algebraic K-theory and topological spaces (pdf)

Application to differential cohomology

Discussion of sheaves of spectra on the $\infty$-cohesive site of SmoothManifolds (or, equivalently, its dense subsite of Cartesian spaces) as a model for Whitehead-generalized differential cohomology

• Urs Schreiber, Section 4.1.2 in: Differential cohomology in a cohesive $\infty$-topos (arXiv:1310.7930)

and observing that the differential cohomology hexagon emerges in this context:

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra, J. Homotopy Relat. Struct. 11, 1–66 (2016) (arXiv:1311.3188)

Discussion within the broader context of differential non-abelian cohomology:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 4.3 in: The character map in (twisted differential) non-abelian cohomology (arXiv:2009.11909)

A book-length overview:

• Araminta Amabel, Arun Debray, Peter J. Haine (eds.), Differential Cohomology: Categories, Characteristic Classes, and Connections. Based on Fall 2019 talks at MIT’s Juvitop seminar by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. (arXiv:2109.12250)