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stable (infinity,1)-category of spectra

Context

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

Stable Homotopy theory

Contents

Idea

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

Sp(Grpd)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 (Ab)Ch_\bullet(Ab) of abelian groups.

Definition

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

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

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

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

Properties

Monoidal structure

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 (,1)(\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)