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stable (infinity,1)-category of spectra

Contents

Context

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

Stable Homotopy theory

Contents

Idea

The collection of spectra forms an (∞,1)-category Sp(Grpd)=Sp(\infty Grpd) = Spectra, 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

Finite homotopy (co)limits of spectra

Proposition

A sequence of morphisms of spectra EFGE \longrightarrow F \longrightarrow G is a homotopy fiber sequence if and only if it is a homotopy cofiber sequence:

A proof is spelled out at Introduction to Stable homotopy theory (this Prop., following Lewis-May-Steinberger 86, chapter III, theorem 2.4 )

In fact:

Proposition

A homotopy-commuting square in Spectra is a homotopy pullback if and only it is a homotopy pushout.

This follows from Prop. by the fact that Spectra is additive (this Prop.).

See also arXiv:1906.04773, Prop. 6.2.11, MO:q/132347.

Remark

This property of Spectra (Prop. , Prop. ) reflects one of the standard defining axioms on stable (∞,1)-categories (see there) and on stable derivators (see there).

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.

Model category presentation

There are several presentations of the (,1)(\infty,1)-category of spectra by model categories of spectra. In particular there are symmetric monoidal model categories where the smash product of spectra is presented by an ordinary tensor product, so that A-∞ rings, E-∞ rings and ∞-modules are presented by 1-categorical monoid objects and module objects, respectively (“brave new algebra”). See at:

model structure on spectra, symmetric monoidal smash product of spectra

References

The stable (,1)(\infty,1)-category of spectra is described in chapter 1 of

or 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

Last revised on January 16, 2021 at 09:59:15. See the history of this page for a list of all contributions to it.