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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 inftΩ Stab(H) (\Sigma^\infty \dashv \Omega^\infty) : \mathbf{H} \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\inft}{\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.

Examples

References

General

The homotopy categories of sheaves of combinatorial spectra are discussed in

part II of

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] (http://cms.math.ca/cjm/v39/cjm1987v39.0733-0747.pdf))

Plenty of further discussion is in

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

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 February 27, 2014 12:46:09 by Urs Schreiber (89.204.137.80)