# nLab stable (infinity,1)-category of spectra

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

## Models

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

The collection of spectra form an (∞,1)-category $Sp(\infty Grpd)$ which is in fact a stable (∞,1)-category. Indeed, it is the universal property stabilization of the $(\infty,1)$-category ∞Grpd, equivalently of the simplicial localization of the category Top at the weak homotopy equivalences.

$Sp(\infty Grpd)$ plays a role in stable homotopy theory analogous to the role played by the 1-category Ab of abelian groups in homological algebra, or rather of the category of chain complexes $Ch_\bullet(Ab)$ of abelian groups.

## Definition

In the context of (∞,1)-categories a spectrum is a spectrum object in the (∞,1)-category $L_{whe} Top_*$ of pointed topological spaces.

Recall that spectrum objects in the (infinity,1)-category $C$ form a stable (∞,1)-category $Sp(C)$.

The stable (∞,1)-category of spectrum objects in $L_{whe} Top_*$ is the stable $(\infty,1)$-category of spectra

$Stab(L_{whe}Top) := Sp(L_{whe}Top_*) \,.$

## Properties

### Prime spectrum and Morava K-theory

The prime spectrum of a monoidal stable (∞,1)-category for p-local and finite spectra is labeled by the Morava K-theories. This is the content of the thick subcategory theorem.

## References

the stable $(\infty,1)$-category of spectra is described in section 9 of

Its monoidal structure is described in section 4.2

That this is a symmetric monoidal structure is described in section 6 of

Revised on March 24, 2014 07:14:35 by Urs Schreiber (89.204.138.56)