# 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 $\mathbf{H}$

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

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

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

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 $(\infty,1)$-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

## 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)