nLab CW-spectrum

Idea

The notion of CW-spectrum is the analogue for sequential spectra in Top of the concept of CW-complex for topological spaces.

Just like CW-complexes are cofibrant objects in the classical model structure on topological spaces, so CW-spectra are cofibrant objects in the stable model structure on topological sequential spectra, see below.

Definition

In components

Definition

A CW-spectrum $X_\bullet\in SeqSpec(Top)$ is a sequential spectrum in Top such that

1. all component spaces $X_n$ are CW-complexes,

2. all structure maps $\Sigma X_n \longrightarrow X_{n+1}$ are inclusions of subcomplexes

e.g. (Adams 74, p. 139) Beware that for instance (Switzer 75, def. 8.1) says just “spectrum” for “CW-spectrum”.

For a CW-spectrum $X$ there is a concept of “cell of a spectrum”:

Definition

A cell of a CW-spectrum, def. is a cell of one of the components CW-complexes $X_n$, together with all its suspensions in all the higher component spaces $X_{\gt n}$, subject to the condition that the first cell itself is not itself the suspension of a cell in $X_{n-1}$.

This way every CW-spectrum is the union of all its cells in the sense of def. . (e.g. Switzer 75, 8.4).

Definition

A CW-spectrum, def. , is called a finite spectrum (or countable spectrum, etc.) if it has finitely many cells (countably many cells) according to def. .

Via spectrum attaching maps

Definition

For $n \in \mathbb{Z}$ (possibly negative) define $\mathbb{S}^n$ to be the sequential prespectrum with component spaces

$$(\mathbb{S}^{n})_k \coloneqq \left\{ \array{ S^{k+n} & if \; k + n \geq 0 \\ \ast & otherwise } \right.$$

and with structure maps the canonical isomorphisms.

(Lewis-May-Steinberger 86, def. 4.3)

Remark

In def.

• for $n = 0$ then $\mathbb{S}^0 = \mathbb{S} = \Sigma^\infty S^0$ is standard sequential incarnation of the sphere spectrum;

• for $n \geq n$ then $\mathbb{S}^n \simeq \Sigma^\infty S^n$ is the suspension spectrum on the n-sphere;

• for general $n$ then $\mathbb{S}^n \simeq F_{-n} S^0$ is also known as the $(-n)$th free spectrum on $S^0$.

Definition

A cell spectrum is a topological sequential spectrum $X$ realized as the colimit over a sequence of spectra $\ast = X_0 \to X_1 \to X_2 \to X_3 \to \cdots$ such that there are morphisms

$$j_n \;\colon\; \left( \underset{i \in I_n}{\sqcup} \mathbb{S}^{q_n} \right) \longrightarrow X_n$$

with $X_{n+1}= Cone(j_n)$ (the mapping cone).

(rmk.)

A cell spectrum is a CW-spectrum if each attaching map $\Sigma^\infty S^{q_n}\to X_n$ factors through a $X_k \to X_n$ with $k \lt q$.

(e.g. Lewis-May-Steinberger 86, def. 5.1, def. 5.2, Weiss)

Category of CW-spectra

There is an obvious category of CW-spectra given as the full subcategory $CWSpec'$ of $SeqSpec(Top)$ consisting of the CW-spectra, ie. a morphism of CW-spectra $f: X \to Y$ is given by levelwise maps $f_n: X_n \to F_n$ that are compatible with the structure maps. In accordance with Switzer 75, 8.9, we call morphisms in this sense functions.

However, often there is a more useful notion of morphisms between CW-spectra.

Definition

Let $E$ be a CW-spectrum. A subspectrum is a CW spectrum $F$ such that each $F_n$ is a subcomplex of $E_n$. A subspectrum is cofinal if for each cell $e_n \in E_n$, there is some $m$ such that $\Sigma^m e_n \in F_{n + m}$.

We can then construct the category $CWSpec$ as the localization of $CWSpec'$ at the cofinal inclusions. Since the class of cofinal inclusions admit a calculus of right fractions, the cateogry $CWSpec$ has the following concrete description - a morphism $X \to Y$ in $CWSpec$ is given by a function $\tilde{X} \to Y$, where $\tilde{X}$ is some cofinal subspectrum of $X$, quotiented by the relation that two such morphisms $f: \tilde{X} \to Y$ and $f': \tilde{X}' \to Y$ are considered equivalent if there is some further cofinal subspectrum $\tilde{X}'' \subseteq \tilde{X} \cap \tilde{X'}$ such that $f|_{\tilde{X}'} = f'|_{\tilde{X}''}$.

Properties

Cofibrancy

Proposition

A sequential spectrum $X\in SeqSpec(Top)_{stable}$ is cofibrant in the stable model structure on topological sequential spectra in particular if all component spaces are cell complexes and all its structure morphisms $S^1 \wedge X_n \to X_{n+1}$ are relative cell complexes. In particular CW-spectra, def. , are cofibrant in $SeqSpec(Top)_{stable}$.

For the proof see there.

Proposition

For $X\in SeqSpec(Top)_{stable}$ a CW-spectrum, then its standard cylinder spectrum $X \wedge (I_+)$ is a good cylinder object, in that the inclusion

$$X \vee X \longrightarrow X \wedge (I_+)$$

is a cofibration in $SeqSpec(Top)_{stable}$.

See this prop..

CW-approximation

The analog of CW-approximation for topological spaces holds true for topological sequential spectra:

Proposition

For $X \in SeqSpec(Top)$ a topological sequential spectrum, there exists a CW-spectrum $\hat X$ and a stable weak homotopy equivalence

$$\hat X \longrightarrow X \,.$$

(e.g. Elmendorf-Kriz-May 95, theorem 1.5)

Proof 1

First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via this prop.. Then proceed by induction: suppose that for $n \in \mathbb{N}$ a CW-approximation $\phi_{k \leq n} \colon \hat X_{k \leq n} \to X_{k \leq n}$ has been found such that all the structure maps are respected. Consider then the continuous function

$$\Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1} \,.$$

Applying that prop. to this function factors it as

$$\Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1} \,.$$

Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

$$\array{ \Sigma \hat X_n &\overset{\Sigma \phi_n}{\longrightarrow}& \Sigma X_n \\ {}^{incl}\downarrow && \downarrow^{\mathrlap{\sigma_n}} \\ \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1} } \,.$$

Proof 2

A high-powered way to see this is to use the Quillen equivalence between the stable model structure on topological sequential spectra and the stable Bousfield-Friedlander model structure (see there) on sequential spectra in simplicial sets. This implies that a CW-approximation is given by

$${\vert Q Sing X\vert} \overset{\in W_{stable}}{\longrightarrow} X \,,$$

where ${\vert - \vert} \dashv Sing$ is degreewise the adjunction between geometric realization and forming singular simplicial complex, and $Q$ denotes any cofibrant replacement in the BF-model structure.

References

• Frank Adams, section III.2 and III.3 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.

• Anthony Elmendorf, Igor Kriz, Peter May, section 1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology, 1995 Amsterdam: North-Holland, pp. 213–253, (pdf)

• Michael Weiss, around pages 43-46 of Homotopy theory and bordism theory (pdf)

Discussion in the generality of equivariant spectra is in

• L. Gaunce Lewis, Peter May, and M. Steinberger (with contributions by J.E. McClure), section I.5 of Equivariant stable homotopy theory Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)