nLab
stable homotopy theory

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Context

Homotopy theory

Background

Variations

Definitions

Paths and cylinders

Homotopy groups

Theorems

Stable Homotopy theory

stable homotopy theory

Ingredients

Contents

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 Ho(Top).

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

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 Sp(Top).

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)
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