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\section*{stable homotopy theory}

\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{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{higher_algebra}{Higher algebra}\dotfill \pageref*{higher_algebra} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{History}{History}\dotfill \pageref*{History} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Stable homotopy theory} is [[homotopy theory]] in the case that the operations of [[looping and delooping]] are \emph{[[equivalence of (infinity,1)-categories|equivalences]]}.

As [[homotopy theory]] is the study of [[homotopy types]], so \emph{stable homotopy theory} is the study of [[stable homotopy types]]. As [[homotopy theory]] in generality is [[(∞,1)-category theory]] (or maybe [[(∞,1)-topos theory]]), so stable homotopy theory in generality is the theory of [[stable (∞,1)-categories]].

More specifically, if one thinks of classical [[homotopy theory]] as the study of (just) the [[(∞,1)-category]] $L_{whe}$[[Top]] $\simeq$ [[∞Grpd]] of [[topological spaces]] modulo [[weak homotopy equivalence]] ([[∞-groupoids]]), or rather of its [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(Top)$, then stable homotopy theory is the study of the corresponding [[stabilization]]: To every suitable [[(∞,1)-category]] is associated its corresponding [[stable (∞,1)-category]] of [[spectrum objects]]. For $L_{whe}$[[Top]] this is the [[stable (∞,1)-category of spectra]], $Sp(L_{whe}Top)$. \textbf{Stable homotopy theory} is the study of $Sp(Top)$, or rather of its [[homotopy category of an (infinity,1)-category|homotopy category]], the [[stable homotopy category]] $Ho(Sp(L_{whe}Top))$.

For detailed introduction to the stable homotopy theory of spectra see at \emph{[[Introduction to Stable homotopy theory]]}.

By definition a [[stable homotopy type]] is one on which [[suspension]] and hence [[looping and delooping]] act as an [[equivalence]]. Historically people considered in plain [[homotopy theory]] statements that became true after sufficiently many [[suspensions]], hence once the process of taking [[suspensions]] ``stabilizes''. Whence the name.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{higher_algebra}{}\subsubsection*{{Higher algebra}}\label{higher_algebra}

The study of [[monoid object in a monoidal (infinity,1)-category]] in a [[stable (∞,1)-category]] is the [[homotopy theory|homotopy-theoretic]] version of [[commutative algebra]], hence \emph{[[higher algebra]]} and [[higher linear algebra]].

A tool of central importance in stable homotopy theory and its application to [[higher algebra]] is the [[symmetric monoidal smash product of spectra]] which allows us to describe [[A-∞ rings]] and [[E-∞ rings]] as ordinary [[monoid]] objects in a [[model category]] that presents $Sp(Top)$. (``[[brave new algebra]]'').

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

\begin{itemize}%
\item [[homotopy theory]], [[homotopy type theory]]


\item [[stable homotopy category]], [[(infinity,1)-category of spectra]]


\item [[Spanier-Whitehead duality]], [[Anderson duality]]

\begin{itemize}%
\item [[S-theory]]

\end{itemize}

\item [[stable (infinity,1)-category]]


\item [[parameterized homotopy theory]], [[parameterized stable homotopy theory]]


\item [[rational stable homotopy theory]]


\item [[cohesive stable homotopy theory]]


\item [[noncommutative stable homotopy theory]]


\item [[chromatic homotopy theory]]


\item [[K(n)-local stable homotopy theory]]


\item [[generalized homology theory]], [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]


\item [[stable homotopy homology theory]]



\end{itemize}
When the spaces and spectra in question carry an [[infinity-action]] of a [[group]] $G$ the theory refines to

\begin{itemize}%
\item [[equivariant stable homotopy theory]], [[global equivariant stable homotopy theory]].

\end{itemize}
\hypertarget{History}{}\subsection*{{History}}\label{History}

\begin{quote}%
Stable homotopy theory began around 1937 with the [[Freudenthal suspension theorem]]. $[$\ldots{}$]$ Stable phenomena had of course appeared earlier, at least implicitly: [[reduced cohomology|reduced]] [[generalized homology|homology]] and [[generalized (Eilenberg-Steenrod) cohomology|cohomology]] are examples of functors that are invariant under suspension without limitation on dimension.

Stable homotopy theory emerged as a distinct branch of [[algebraic topology]] with Adams' introduction of [[Adams spectral sequence|his eponymous spectral sequence]] and his spectacular conceptual use of the notion of stable phenomena in his solution to the [[Hopf invariant one]] problem.

Its centrality was reinforced by two related developments that occurred at very nearly the same time, in the late 1950's. One was the introduction of [[generalized (Eilenberg-Steenrod) cohomology|generalized homology and cohomology theories]] and especially [[K-theory]], by Atiyah and Hirzebruch. The other was the work of Thom which showed how to reduce the problem of classifying manifolds up to cobordism to a problem, more importantly, a solvable problem, in stable homotopy theory $[$ [[Thom spectrum]] $]$.

The reduction of geometric phenomena to solvable problems in stable homotopy theory has remained an important mathematical theme, the most recent major success being Stolz's use of [[spin bordism|Spin cobordism]] to study the classication of manifolds with positive scalar curvature.

In an entirely different direction, the early 1970's saw Quillen's introduction of higher [[algebraic K-theory]] and the recognition by Segal and others that it could be viewed as a construction in stable homotopy theory. With algebraic K-theory as an intermediary, there has been a growing volume of work that relates algebraic geometry to stable homotopy theory. With Waldhausen's introduction of the algebraic K-theory of spaces in the late 1970's, stable homotopy became a bridge between algebraic K-theory and the study of diffeomorphisms of manifolds.

Within algebraic topology, the study of stable homotopy theory has been and remains the focus of much of the best work in the subject.

(\hyperlink{ElmendorfKrizMay95}{Elmendorf-Kriz-May 95, p. 2})


\end{quote}
\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}
Quick surveys include

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


\item [[Dylan Wilson]], \emph{Introduction to stable homotopy theory} (\href{http://math.berkeley.edu/~aaron/wcatss/sht1.pdf}{pdf})


\item [[Aaron Mazel-Gee]], \emph{An introduction to spectra} (\href{https://math.berkeley.edu/~aaron/writing/an-introduction-to-spectra.pdf}{pdf})



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[Introduction to Stable homotopy theory]]}


\item [[David Barnes]], [[Constanze Roitzheim]], \emph{Foundations of Stable Homotopy Theory} (\href{https://www.kent.ac.uk/smsas/personal/csrr/stablemodelcatsCUP.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Glossary for stable and chromatic honotopy theory ([[StableChromaticGlossary.pdf:file]])

\end{itemize}
Comprehensive discussion of model with [[symmetric monoidal smash product of spectra]] are

for [[S-modules]]:

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], \emph{[[Modern foundations for stable homotopy theory]]}, in [[Ioan Mackenzie James]], \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})

\end{itemize}
for [[symmetric spectra]]

\begin{itemize}%
\item [[Stefan Schwede]], \emph{Symmetric spectra} (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})

\end{itemize}
for [[orthogonal spectra]]:

\begin{itemize}%
\item [[Stefan Schwede]], \emph{[[Global homotopy theory]]} (take $\mathcal{F} = \{1\}$, on p. 4, to be the trivial collection of groups, in order to specialize from [[global equivariant stable homotopy theory]] to plain stable homotopy theory).

\end{itemize}
A survey of formalisms used in stable homotopy theory -- tools to present the [[triangulated category|triangulated]] [[homotopy category]] of a [[stable (infinity,1)-category]] -- is in

\begin{itemize}%
\item [[Neil Strickland]], \emph{Axiomatic stable homotopy - a survey} (\href{http://front.math.ucdavis.edu/0307.5143}{arXiv:math.AT/0307143})


\item [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]], \emph{Axiomatic stable homotopy theory} (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf}{pdf})



\end{itemize}
An account in terms of [[(∞,1)-category theory]] is in section 1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}

\end{itemize}
Brief indications of open questions and future directions (as of 2013) of [[algebraic topology]] and [[stable homotopy theory]] are in

\begin{itemize}%
\item [[Tyler Lawson]], \emph{The future}, Talbot lectures 2013 (\href{http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf}{pdf})

\end{itemize}
[[!redirects stable homotopy theories]]



\end{document}