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\begin{document}

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\section*{CW-spectrum}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{via_spectrum_attaching_maps}{Via spectrum attaching maps}\dotfill \pageref*{via_spectrum_attaching_maps} \linebreak
\noindent\hyperlink{category_of_cwspectra}{Category of CW-spectra}\dotfill \pageref*{category_of_cwspectra} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Cofibrancy}{Cofibrancy}\dotfill \pageref*{Cofibrancy} \linebreak
\noindent\hyperlink{CWApproximation}{CW-approximation}\dotfill \pageref*{CWApproximation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{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 category|stable]] [[model structure on topological sequential spectra]], see \hyperlink{Cofibrancy}{below}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

\begin{defn}
\label{CWSpectrum}\hypertarget{CWSpectrum}{}
A \textbf{CW-spectrum} $X_\bullet\in SeqSpec(Top)$ is a [[sequential spectrum]] in [[Top]] such that

\begin{enumerate}%
\item all component spaces $X_n$ are [[CW-complexes]],


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



\end{enumerate}
\end{defn}
e.g. (\hyperlink{Adams74}{Adams 74, p. 139}) Beware that for instance (\hyperlink{Switzer75}{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'':

\begin{defn}
\label{CellOfACWSpectrum}\hypertarget{CellOfACWSpectrum}{}
A \textbf{cell} of a CW-spectrum, def. \ref{CWSpectrum} 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 $X_{n-1}$.

\end{defn}
This way every CW-spectrum is the union of all its cells in the sense of def. \ref{CellOfACWSpectrum}. (e.g. \href{Switzer75}{Switzer 75, 8.4}).

\begin{defn}
\label{}\hypertarget{}{}
A CW-spectrum, def. \ref{CWSpectrum}, is called a \emph{[[finite spectrum]]} (or countable spectrum, etc.) if it has [[finite number|finitely many]] cells (countably many cells) according to def. \ref{CellOfACWSpectrum}.

\end{defn}
\hypertarget{via_spectrum_attaching_maps}{}\subsubsection*{{Via spectrum attaching maps}}\label{via_spectrum_attaching_maps}

\begin{defn}
\label{SphereSpectrumOfIntegerDimension}\hypertarget{SphereSpectrumOfIntegerDimension}{}
For $n \in \mathbb{Z}$ (possibly negative) define $\mathbb{S}^n$ to be the [[sequential prespectrum]] with component spaces

\begin{displaymath}
(\mathbb{S}^{n})_k
   \coloneqq
 \left\{
   \itexarray{
     S^{k+n} & if \; k + n \geq 0
     \\
     \ast & otherwise
   }
 \right.
\end{displaymath}
and with structure maps the canonical isomorphisms.

\end{defn}
(\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, def. 4.3})

\begin{defn}
\label{}\hypertarget{}{}
In def. \ref{SphereSpectrumOfIntegerDimension}

\begin{itemize}%
\item for $n = 0$ then $\mathbb{S}^0 = \mathbb{S} = \Sigma^\infty S^0$ is standard sequential incarnation of the [[sphere spectrum]];


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


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



\end{itemize}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{[[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

\begin{displaymath}
j_n 
   \;\colon\; 
   \left(
      \underset{i \in I_n}{\sqcup} \mathbb{S}^{q_n}
   \right)
   \longrightarrow
   X_n
\end{displaymath}
with $X_{n+1}= Cone(j_n)$ (the [[mapping cone]]).

(\href{Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra}{rmk.})

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

\end{defn}
(e.g. \hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, def. 5.1, def. 5.2}, \hyperlink{Weiss}{Weiss})

\hypertarget{category_of_cwspectra}{}\subsubsection*{{Category of CW-spectra}}\label{category_of_cwspectra}

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 \href{Switzer75}{Switzer 75, 8.9}, we call morphisms in this sense \textbf{functions}.

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

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

\end{defn}
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 $f|_{\tilde{X}'} = f'|_{\tilde{X}''}$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Cofibrancy}{}\subsubsection*{{Cofibrancy}}\label{Cofibrancy}

\begin{prop}
\label{}\hypertarget{}{}
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. \ref{CellOfACWSpectrum}, are cofibrant in $SeqSpec(Top)_{stable}$.

\end{prop}
For the proof see \href{model+structure+on+topological+sequential+spectra#CellSpectraAreCofibrantInModelStructureOnTopologicalSequentialSpectra}{there}.

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

\begin{displaymath}
X \vee X \longrightarrow X \wedge (I_+)
\end{displaymath}
is a cofibration in $SeqSpec(Top)_{stable}$.

\end{prop}
See \href{model+structure+on+topological+sequential+spectra#CylinderSpectrumOverCWSpectrumIsGood}{this prop.}.

\hypertarget{CWApproximation}{}\subsubsection*{{CW-approximation}}\label{CWApproximation}

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

\begin{prop}
\label{}\hypertarget{}{}
For $X \in SeqSpec(Top)$ a topological [[sequential spectrum]], there exists a CW-spectrum $\hat X$ and a [[stable weak homotopy equivalence]]

\begin{displaymath}
\hat X \longrightarrow X
 \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{ElmendorfKrizMay95}{Elmendorf-Kriz-May 95, theorem 1.5})

\begin{proof}
First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via \href{CW+approximation#CWApproximationForContinuousFunctions}{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

\begin{displaymath}
\Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1}
  \,.
\end{displaymath}
Applying \href{CW+approximation#CWApproximationForContinuousFunctions}{that prop.} to this function factors it as

\begin{displaymath}
\Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1}
  \,.
\end{displaymath}
Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

\begin{displaymath}
\itexarray{  
    \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}
  }
  \,.
\end{displaymath}
\end{proof}
\begin{proof}
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

\begin{displaymath}
{\vert Q Sing X\vert}
  \overset{\in W_{stable}}{\longrightarrow}
  X
  \,,
\end{displaymath}
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.

\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Frank Adams]], section III.2 and III.3 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Robert Switzer]], \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.


\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 1 of \emph{[[Modern foundations for stable homotopy theory]]}, in [[Ioan Mackenzie James]] (ed.), \emph{[[Handbook of Algebraic Topology]]}, 1995 Amsterdam: North-Holland, pp. 213--253, (\href{http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf}{pdf})


\item [[Michael Weiss]], around pages 43-46 of \emph{Homotopy theory and bordism theory} (\href{http://www.maths.ed.ac.uk/~aar/papers/abdnbordism.pdf}{pdf})



\end{itemize}
Discussion in the generality of [[equivariant spectra]] is in

\begin{itemize}%
\item [[L. Gaunce Lewis]], [[Peter May]], and M. Steinberger (with contributions by J.E. McClure), section I.5 of \emph{Equivariant stable homotopy theory} Springer Lecture Notes in Mathematics Vol.1213. 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf})

\end{itemize}
[[!redirects CW spectra]]

[[!redirects CW-spectrum]] [[!redirects CW-spectra]]

[[!redirects CW spectrum]]



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