nLab stable homotopy category

Idea

The stable homotopy category $Ho(Spectra)$ is the category of spectra and homotopy classes of morphisms between them, the object of study in classical stable homotopy theory. Equivalently this is the homotopy category of an (∞,1)-category of the stable (∞,1)-category of spectra and the latter is the proper context for stable homotopy theory. But with due care exercised, the stable homotopy category itself is useful.

The stable homotopy category may be thought of as the stabilization of the classical homotopy category $Ho(Top)$ under the operation of forming loop space objects $\Omega$ and reduced suspensions $\Sigma$: via forming suspension spectra $\Sigma^\infty$ every pointed object in the classical homotopy category maps to the stable homotopy category, and under this map the loop space- and reduced suspension-functors become inverse equivalences on the stable homotopy category.

$$\array{ Ho(Top)^{\ast/} & \underoverset{\underset{\Omega}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{} & Ho(Top)^{\ast/} \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ Ho(Spectra) & \underoverset{\underset{\Omega}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{\simeq} & Ho(Spectra) } \,.$$

In contrast to the classical homotopy category, the stable homotopy category is a triangulated category (a shadow of the fact that the (∞,1)-category of spectra is a stable (∞,1)-category). As such it may be thought of as a refinement of the derived category of chain complexes (of abelian groups): every chain complex gives rise to a spectrum and every chain map to a map between these spectra (the stable Dold-Kan correspondence), but there are many more spectra and maps between them than arise from chain complexes and chain maps.

Equipped with the smash product of spectra “$\wedge$” and with function spectra $[-,-]$, the stable homotopy category becomes a symmetric closed monoidal category.

A (commutative) monoid object with respect to $\wedge$ is (commutative) ring spectrum.

For $E \in Ho(Spectra)$ any spectrum, then the functor

$$E_\bullet \coloneqq \pi_0((\Sigma^\bullet E) \wedge(-)) \;\colon\; Ho(Spectra) \longrightarrow Ab$$

is a generalized homology theory, while the functor

$$E^\bullet \coloneqq \pi_0[-,\Sigma^\bullet E] \;\colon\; Ho(Spectra)^\op \longrightarrow Ab$$

is a generalized cohomology theory.

Definition

There are various different-looking ways to define the stable homotopy category.

Via eventually defined maps

One of the first constructions of the stable homotopy category is due to (Adams 74, part III, sections 2 and 3), following (Boardman 65). This Adams category is defined to be the category of CW-spectra with homotopy classes (with respect to cylinder spectra) of “eventually defined” functions between them.

Historically this was advertised as being a construction free of tools of category theory. See (Lewis-May-Steinberger 86, pages 1-3) for review and critical assessment

Via left homotopy

(Lewis-May-Steinberger 86, “preamble” pages 1-7, Elmendorf-Kriz-May 95, p. 8, Malkiewich 14)

For $E$ a sequential pre-spectrum and $X$ a pointed topological space, write

$$(E \wedge X)_n \coloneqq E_n \wedge X$$

for the degreewise smash product of pointed topological spaces. Write $I_+$ for the unit interval with a base point adjoined, such that for any spectrum $E$, the smash product $E \wedge I_+$ is its cylinder spectrum.

A left homotopy between morphism of pre-spectra $f,g \colon E_1 \longrightarrow E_2$ is a morphism of spectra

$$\phi \;\colon\; E_1\wedge I_+ \longrightarrow E_2$$

such that $\phi|_0 = f$ and $\phi|_1 = g$.

In order for this to be the homotopy-correct notion, we need to apply it with domain a CW-spectrum and codomain an Omega-spectrum.

Let $PreSpectra \stackrel{\overset{L}{\longrightarrow}}{\underset{\ell}{\longleftarrow}} Spectra$ be the 1-categorical adjunction between Omega-spectra and prespectra, in the sense defined at coordinate-free spectrum, where $\ell$ is the forgetful functor and $L$ is spectrification.

There is also a CW-spectrum-replacement functor $\Gamma$.

Write then

$$[E_1,E_2] \coloneqq Hom(\Gamma E_1, L E_2)_{\sim_{left\;homotopy}}$$

for the corresponding homotopy classes of maps.

Via model structures

There are several model categories which exhibit model structures for spectra, hence whose homotopy category of a model category is equivalent to the stable homotopy category.

The most lightweight of these is the Bousfield-Friedlander model structure of sequential pre-spectra in simplicial sets (Bousfield-Friedlander 78)

Model structures on more highly structured spectra include the model structure on symmetric spectra, the model structure on orthogonal spectra, and ultimately the model structure for excisive functors. It is only with these model structures that the smash product of spectra is represented by a symmetric monoidal smash product of spectra even before passing to the stable homotopy category.

A unified account of all thes model structures is at Model categories of diagram spectra.

Properties

Symmetric monoidal structure

The smash product of spectra makes the stable homotopy category into a symmetric monoidal category.

An (commutative) monoid object with respect to this is a (commutative) ring spectrum. A module object over such is a module spectrum.

Triangulated category structure

The homotopy fiber sequences of spectra gives the stable homotopy category the structure of a triangulated category.

For proof see at Introduction to Stable homotopy theory this Prop.

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).

Related concepts

• Ho(Top)

• Whitehead theorem

• Spectra

References

The original direct definitions of the stable homotopy category (for precursors see at Spanier-Whitehead category) is due to

• Michael Boardman, Stable homotopy theory, mimeographed notes, University of Warwick, 1965 onward

Early accounts include

• Rainer Vogt, Boardman’s stable homotopy category, lectures, spring 1969

• J. M. Cohen, Stable Homotopy, Springer Lecture Notes in Math., No. 165, Springer-Verlag, Berlin, 1970.

• Dieter Puppe, On the stable homotopy category, Topology and its application (1973) (pdf)

• Frank Adams, Part III, section 2 of Stable homotopy and generalised homology, 1974

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

Expository introduction:

• Cary Malkiewich, The stable homotopy category, 2014 (pdf, pdf)

A fun scan of the (pre-)history of the stable homotopy category:

• Peter May, The hare and the tortoise, speech at Michael Boardman‘s birthday meeting (txt)

See also the references at stable homotopy theory.

Original articles realizing the stable homotopy category as the homotopy category of a model category include

• Aldridge Bousfield, Eric Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

Original articles in the context of highly structured spectra include

• L. Gaunce Lewis, Peter May, M. Steinberger, Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics, 1986 (pdf)

• Anthony Elmendorf, Igor Kriz, Peter May, Modern foundations for stable homotopy theory, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology, Amsterdam: North-Holland, 1995 pp. 213–253, (pdf)

• John Greenlees, Peter May, Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

A textbook account in the context of symmetric spectra is

• Stefan Schwede, chapter II, section 1_Symmetric spectra_ (2012)