# nLab sheaf of spectra

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

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

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

consist of spectrum objects in $\mathbf{H}$. 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.

## 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)