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\begin{document}

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\section*{stable homotopy 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{via_eventually_defined_maps}{Via eventually defined maps}\dotfill \pageref*{via_eventually_defined_maps} \linebreak
\noindent\hyperlink{ViaSpectrificationofSequentialPreSpectra}{Via left homotopy}\dotfill \pageref*{ViaSpectrificationofSequentialPreSpectra} \linebreak
\noindent\hyperlink{via_model_structures}{Via model structures}\dotfill \pageref*{via_model_structures} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{symmetric_monoidal_structure}{Symmetric monoidal structure}\dotfill \pageref*{symmetric_monoidal_structure} \linebreak
\noindent\hyperlink{triangulated_structure}{Triangulated structure}\dotfill \pageref*{triangulated_structure} \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}

The \emph{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 [[equivalence of categories|equivalences]] on the stable homotopy category.

\begin{displaymath}
\itexarray{
     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)
  }
  \,.
\end{displaymath}
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]] [[category of chain complexes|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 monoidal category|symmetric]] [[closed monoidal category]].

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

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

\begin{displaymath}
E_\bullet \coloneqq \pi_0((\Sigma^\bullet E) \wedge(-)) \;\colon\; Ho(Spectra) \longrightarrow Ab
\end{displaymath}
is a [[generalized homology theory]], while the functor

\begin{displaymath}
E^\bullet \coloneqq \pi_0[-,\Sigma^\bullet E] \;\colon\; Ho(Spectra)^\op \longrightarrow Ab
\end{displaymath}
is a [[generalized cohomology theory]].

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

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

\hypertarget{via_eventually_defined_maps}{}\subsubsection*{{Via eventually defined maps}}\label{via_eventually_defined_maps}

One of the first constructions of the stable homotopy category is due to (\hyperlink{Adams74}{Adams 74, part III, sections 2 and 3}), following (\hyperlink{Boardman65}{Boardman 65}). This \emph{[[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.

Hostorically this was advertized as being a construction free of tools of [[category theory]]. See (\hyperlink{LewismaySteinberger86}{Lewis-May-Steinberger 86, pages 1-3}) for review and critical assessment

\hypertarget{ViaSpectrificationofSequentialPreSpectra}{}\subsubsection*{{Via left homotopy}}\label{ViaSpectrificationofSequentialPreSpectra}

(\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, ``preamble'' pages 1-7}, \hyperlink{ElmendorfKrizMay95}{Elmendorf-Kriz-May 95, p. 8}, \hyperlink{Malkiewich14}{Malkiewich 14})

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

\begin{displaymath}
(E \wedge X)_n \coloneqq E_n \wedge X
\end{displaymath}
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 \emph{[[cylinder spectrum]]}.

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

\begin{displaymath}
\phi \;\colon\; E_1\wedge I_+ \longrightarrow E_2
\end{displaymath}
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

\begin{displaymath}
[E_1,E_2]
  \coloneqq
  Hom(\Gamma E_1, L E_2)_{\sim_{left\;homotopy}}
\end{displaymath}
for the corresponding [[homotopy classes]] of maps.

\hypertarget{via_model_structures}{}\subsubsection*{{Via model structures}}\label{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 spectrum|sequential]] [[pre-spectra]] in [[simplicial sets]] (\hyperlink{BousfieldFriedlander78}{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 \emph{[[Model categories of diagram spectra]]}.

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

\hypertarget{symmetric_monoidal_structure}{}\subsubsection*{{Symmetric monoidal structure}}\label{symmetric_monoidal_structure}

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

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

\hypertarget{triangulated_structure}{}\subsubsection*{{Triangulated structure}}\label{triangulated_structure}

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

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

\begin{itemize}%
\item [[Ho(Top)]]


\item [[Whitehead theorem]]



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

The original direct definitions of the stable homotopy category (for precursors see at \emph{[[Spanier-Whitehead category]]}) is due to

\begin{itemize}%
\item [[Michael Boardman]], \emph{Stable homotopy theory}, mimeographed notes, University of Warwick, 1965 onward

\end{itemize}
Early accounts include

\begin{itemize}%
\item [[Rainer Vogt]], \emph{Boardman's stable homotopy category}, lectures, spring 1969


\item J. M. Cohen, \emph{Stable Homotopy}, Springer Lecture Notes in Math., No. 165, Springer-Verlag, Berlin, 1970.


\item [[Dieter Puppe]], \emph{On the stable homotopy category}, Topology and its application (1973) ([[PuppeStableHomotopyCategory.pdf:file]])


\item [[Frank Adams]], Part III, section 2 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.



\end{itemize}
A fun scan of the (pre-)history of the stable homotopy category is in

\begin{itemize}%
\item [[Peter May]], \emph{The hare and the hedgehog}, speech at [[Michael Boardman]]`s birthday meeting (\href{http://hopf.math.purdue.edu/May/boardman.txt}{txt})

\end{itemize}
See also the references at \emph{[[stable homotopy theory]]}.

Original articles realizing the stabel homotopy category as the [[homotopy category]] of a [[model category]] include

\begin{itemize}%
\item [[Aldridge Bousfield]], [[Eric Friedlander]], \emph{Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets}, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf}{pdf})

\end{itemize}
Original articles in the context of [[highly structured spectra]] include

\begin{itemize}%
\item [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics, 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf})


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


\item [[John Greenlees]], [[Peter May]], \emph{Equivariant stable homotopy theory}, in I.M. James (ed.), \emph{[[Handbook of Algebraic Topology]]} , pp. 279-325. 1995. (\href{http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf}{pdf})



\end{itemize}
A textbook account in the context of [[symmetric spectra]] is

\begin{itemize}%
\item [[Stefan Schwede]], chapter II, section 1\_[[Symmetric spectra]]\_ (2012)

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Cary Malkiewich]], \emph{The stable homotopy category}, 2014 (\href{http://math.stanford.edu/~carym/stable.pdf}{pdf})

\end{itemize}
[[!redirects HoSpectra]]

[[!redirects Ho(Spectra)]]



\end{document}