# nLab stable homotopy theory

## Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

One may think of classical homotopy theory as the study of the (∞,1)-category Top of topological spaces, or rather of its homotopy category $\mathrm{Ho}\left(\mathrm{Top}\right)$.

To every (∞,1)-category is associated its corresponding stable (∞,1)-category of spectrum objects. For Top this is the stable (∞,1)-category of spectra, $\mathrm{Sp}\left(\mathrm{Top}\right)$. Stable homotopy theory is the study of $\mathrm{Sp}\left(\mathrm{Top}\right)$, or rather of its homotopy category, the stable homotopy category $\mathrm{StHo}\left(\mathrm{Top}\right):=\mathrm{Ho}\left(\mathrm{Sp}\left(\mathrm{Top}\right)\right)$.

A tool of major importance in stable homotopy theory and its application to higher algebra is the symmetric monoidal smash product of spectra which allows us to describe A-∞ rings and E-∞ rings as ordinary monoid objects in a model category that presents $\mathrm{Sp}\left(\mathrm{Top}\right)$.

## Variations

When the spaces and spectra in question carry an action of a group $G$ the theory refines to

## References

An excellent general survey is

• A. Elmendorf, I. Kriz, P. May, Modern foundations for stable homotopy theory (pdf)

A survey of formalisms used in stable homotopy theory – tools to present the triangulated homotopy category of a stable (infinity,1)-category – is in

An account in terms of (∞,1)-category theory is in section 7 of

Revised on April 24, 2013 19:39:54 by Urs Schreiber (131.174.42.61)