nLab stable (infinity,1)-category of spectra

Idea

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

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

Definition

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

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

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

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

Properties

Finite homotopy (co)limits of spectra

Proposition

A sequence of morphisms of spectra $E \longrightarrow F \longrightarrow G$ 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

• With the smash product of spectra $Sp(L_{whe}Top_*)$ becomes a symmetric monoidal (∞,1)-category.

  • an algebra object in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is an associative ring spectrum;

  • a commutative algebra object in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is a commutative ring spectrum;

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 $(\infty,1)$-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

• symmetric spectrum, model structure on symmetric spectra

• orthogonal spectrum, model structure on orthogonal spectra

• S-module, model structure on S-modules

References

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

• Jacob Lurie, Higher Algebra

or section 9 of

• Jacob Lurie, Stable Infinity-Categories .

Its monoidal structure is described in section 4.2

• Jacob Lurie, Noncommutative algebra.

That this is a symmetric monoidal structure is described in section 6 of

• Jacob Lurie, Commutative algebra.