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\begin{document}

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\section*{Spanier-Whitehead category}

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

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

[[!include stable homotopy theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Motivated by the [[Freudenthal suspension theorem]], the \emph{suspension category} of (\hyperlink{SpanierWhitehead53}{Spanier-Whitehead 53}) has as [[objects]] [[pointed topological space|pointed]] [[CW-complexes]], and as [[hom-sets]] it has [[colimits]]

\begin{displaymath}
Hom(X,Y) \coloneqq \underset{\longrightarrow}{\lim}_q [\Sigma^q X , \Sigma^q Y]
\end{displaymath}
over [[homotopy classes]] of [[continuous functions]] between their arbitrary high [[suspensions]].

More generally one considers the category whose objects are pairs $(X,n)$ of a [[pointed topological space|pointed]] [[CW-complex]] $X$ and an [[integer]] $n$ and whose [[hom-sets]] are

\begin{displaymath}
Hom((X_1,n_1), (X_2,n_2)) \coloneqq \underset{\longrightarrow}{\lim}_{q \gt max({\vert n_1\vert},{\vert n_2\vert})} [\Sigma^{q+n_1} X_1 , \Sigma^{q+n_2} X_2]
  \,.
\end{displaymath}
This is a [[triangulated category]].

But the Spanier-Whitehead category lacks other desirable properties, for instance it does not have all [[coproducts]] and the canonical functor from the homotopy category of [[pointed topological spaces]] does not preserve the coproducts that already exist. As a consequence, in particular a [[Brown representability theorem]] does not hold in the SW-category.

Later it was realized (see e.g. \hyperlink{Whitehead62}{Whitehead 62}) that this all this is fixed by regarding the SW-category for [[finite CW complexes]] as a [[full subcategory]] on the (shifted) [[suspension spectra]] inside the larger category of [[spectra]]: the [[stable homotopy category]] (e.g. \hyperlink{Schwede12}{Schwede 12, chapter II theorem 7.2}). As such it is the full subcategory on the [[finite spectra]] (e.g. \hyperlink{Schwede12}{Schwede 12, chapter II theorem 7.4}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Spanier-Whitehead duality]]


\item [[prestable (∞,1)-category]]



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

The definition is due to

\begin{itemize}%
\item [[Edwin Spanier]], [[George Whitehead]], \emph{A first approximation to homotopy theory} Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 655-660.

\end{itemize}
Survey includes

\begin{itemize}%
\item [[George Whitehead]], \emph{Some aspects of stable homotopy theory}, International Confress of Mathematics 1962 (\href{http://www.mathunion.org/ICM/ICM1962.1/Main/icm1962.1.0502.0506.ocr.pdf}{pdf})


\item H. R. Margolis, \emph{Spectra and the Steenrod algebra}, volume 29 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1983


\item [[Stefan Schwede]], chapter II, section 7 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})



\end{itemize}
Discussion in the abstract generality of categories equipped with an abstract suspension-like functor is in

\begin{itemize}%
\item [[Alex Heller]], \emph{Stable homotopy categories}, Bull. Amer. Math. Soc. Volume 74, Number 1 (1968), 28-63. (\href{https://projecteuclid.org/euclid.bams/1183529378}{Euclid})


\item [[Ivo Dell'Ambrogio]], \emph{The Spanier-Whitehead category is always triangulated} (\href{http://math.univ-lille1.fr/~dellambr/diploma.pdf}{pdf})



\end{itemize}
[[!redirects suspension category]]



\end{document}