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stable homotopy theory

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stable homotopy theory

Ingredients

Contents

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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

Revised on January 21, 2010 17:22:26 by Urs Schreiber (92.237.184.59)
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