nLab Spectrum

Idea

By $Spectrum$ or $Spectra$ one may denote the stable (∞,1)-category of spectra. It is also denoted $Sp$, or sometimes $Spec$ although that can be confusing.

Its homotopy category is the classical stable homotopy category.

For more see the entry stable (∞,1)-category of spectra.

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).

Other abstract characterizations

$Spectra$ is equivalently

• the free stable locally presentable (∞,1)-category on one compact generator, namely the sphere spectrum.

• the stable (infinity,1)-category of quasicoherent infinity-stacks on Spec(S).

Related concepts

• model structure on spectra

• PermCat

category: category