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sheaf of spectra

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Stable Homotopy theory

Contents

Idea

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

(Σ Ω ):HΣ Ω Stab(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 H\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:Ch (R)HRMod(Spectra)USpecta DK \;\colon\; Ch_\bullet(R) \stackrel{\simeq}{\longrightarrow} H R Mod(Spectra) \stackrel{U}{\longrightarrow} Specta

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 (,1)(\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

References

General

The homotopy categories of sheaves of combinatorial spectra were discussed in

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

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)

Revised on May 28, 2014 10:21:33 by Urs Schreiber (89.204.153.176)