# nLab sheaf of spectra

### Context

#### $\left(\infty ,1\right)$-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 $H$

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

consist of spectrum objects in $H$. 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)

### Application to K-theory

section 1.2 of

• Bertrand Toën, 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 July 13, 2012 16:18:15 by Urs Schreiber (89.204.130.60)