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\begin{document}

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\section*{Elmendorf&#39;s theorem}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ModelCategoryPresentation}{Model category presentation / Quillen equivalence}\dotfill \pageref*{ModelCategoryPresentation} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \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}

\emph{Elmendorf's theorem} states that for $G$ a [[topological group]], then the [[(∞,1)-category of (∞,1)-presheaves]] on the [[orbit category]] $Orb_G$ of $G$, naturally regarded as an [[(∞,1)-site]], is [[equivalence of (∞,1)-categories|equivalent]] to the [[localization of an (∞,1)-category|localization]] of [[topological spaces]] with $G$-[[action]] ([[G-spaces]]) at the \emph{[[weak homotopy equivalences]] on [[fixed point spaces]]} (also presented by [[G-CW complexes]], see at \emph{[[equivariant homotopy theory]]} for more).

More in detail, for $G$ a [[topological group]], write $Top^G$ for the [[category]] of [[compactly generated topological spaces]] which are equipped with a [[continuous function|continuous]] $G$-[[action]]. Say that a [[continuous map]] $f \colon X \longrightarrow Y$ between $G$-spaces is a \textbf{weak $G$-homotopy equivalence} if for any [[closed subspace|closed subgroup]] $H \hookrightarrow G$, the induced function on $H$-[[fixed point spaces]] $f^H \colon X^H \longrightarrow Y^H$ is an ordinary [[weak homotopy equivalence]]. Write

\begin{displaymath}
Top^G[\{weak\,G-homotopy\;equivalences\}^{-1}]
  \in (\infty,1)Cat
\end{displaymath}
for the corresponding [[simplicial localization]].

Next, write $Orb_G$ for the [[full subcategory]] of $Top^G$ on the $G$-[[homogeneous spaces]] of the form $G/H$, but regarded as an [[(∞,1)-category]] by regarding each [[compact-open topology|hom-space]] as its [[homotopy type]]. Write moreover $Top^{Orb_G}$ for the category of [[continuous function|continuous]] functors $Orb_G^{op} \longrightarrow Top$. Write finally

\begin{displaymath}
PSh_\infty(Orb_G)
   \in 
 (\infty,1)Cat
  \,.
\end{displaymath}
Then Elmendorf's theorem asserts that there is an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Top^G[\{weak\;G-homotopy\;equivalences\}^{-1}]
  \simeq
  PSh_\infty(Orb_G)
  \,.
\end{displaymath}
\begin{remark}
\label{}\hypertarget{}{}
In particular the theorem hence asserts that the $G$-[[equivariant homotopy theory]] is an [[(∞,1)-topos]].

\end{remark}
\hypertarget{ModelCategoryPresentation}{}\subsection*{{Model category presentation / Quillen equivalence}}\label{ModelCategoryPresentation}

A version of the theorem that applies fairly generally for ([[discrete group|discrete]]) [[group objects]] in suitable [[cofibrantly generated model categories]] is in (\hyperlink{Guillou}{Guillou}, \hyperlink{Stephan10}{Stephan 10}, \hyperlink{Stephan13}{Stephan 13}).

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[cofibrantly generated model category]] and for $G$ a [[discrete group]] (canonically regarded as a [[group object]] of $\mathcal{C}$ via its [[tensoring]] over [[Set]]) write $G \mathcal{C}$ for the category of $G$-[[actions]] in $\mathcal{C}$.

\end{defn}
\begin{defn}
\label{CellularFixedPointFunctor}\hypertarget{CellularFixedPointFunctor}{}
A \emph{cellular fixed point functor} on $\mathcal{C}$ is \ldots{}

\end{defn}
(\hyperlink{Guillou}{Guillou, def. 3.7})

\begin{example}
\label{}\hypertarget{}{}
The [[fixed point spaces]]-functors on the following kinds of model categories are cellular

\begin{itemize}%
\item the standard [[model structure on topological spaces]];


\item the standard [[model structure on simplicial sets]];


\item the global [[model structure on simplicial presheaves]] over any small category;


\item (\ldots{})



\end{itemize}
\end{example}
(\hyperlink{Guillou}{Guillou, section 4})

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[cofibrantly generated model category]] with cellular fixed point functor, def. \ref{CellularFixedPointFunctor}, then the category $G \mathcal{C}$ of $G$-actions in $\mathcal{C}$ carries a [[cofibrantly generated model category]] structure $G \mathcal{C}_{fine}$ whose weak equivalences and fibrations are those maps which induce weak equivalences or fibrations in $\mathcal{C}$, respectively, on objects of $H$-[[fixed points]], for all [[subgroups]] $H$ of $G$.

\end{defn}
(\hyperlink{Guillou}{Guillou, theorem 3.12})

Write $Orb_G$ for the [[orbit category]] of $G$.

Write $(\mathcal{C}^{Orb_G^{op}})_{proj}$ for the projective global [[model structure on functors]] from the $G$-[[orbit category]] to $\mathcal{C}$.

\begin{defn}
\label{}\hypertarget{}{}
There is a pair of [[adjoint functors]]

\begin{displaymath}
(\Theta, \Phi)
    \;\colon\;
  G\mathcal{C}
   \stackrel{\overset{\Theta}{\longleftarrow}}{\underset{\Phi}{\longrightarrow}}
  \mathcal{C}^{Orb_G^{op}}
\end{displaymath}
where $\Phi X \colon G/H \mapsto X^H$ assigns fixed-point objects and where $\Theta S$ has as underlying object $S(G/1)$.

This constitutes a [[Quillen equivalence]] between the above model structures

\begin{displaymath}
(\Theta, \Phi)
    \;\colon\;
  G\mathcal{C}_{fine}
    \underset{Quillen}{\simeq}
  \mathcal{C}^{Orb_G^{op}}_{proj}
 \,.
\end{displaymath}
\end{defn}
(\hyperlink{Guillou}{Guillou, prop. 3.15})

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\begin{enumerate}%
\item Elmendorf's theorem may be generalized to the case where only a sub-family $\mathcal{H}$ of the closed subgroups of $G$ is considered (\hyperlink{Stephan10}{Stephan 10}, also \hyperlink{May96}{May 96}).


\item There is an evident generalization of the [[orbit category]] of a fixed group $G$ to the \emph{[[global orbit category]]}. Under this generalization an analog of Elmendorf's theorem plays a central role in [[global equivariant homotopy theory]] (\hyperlink{Rezk14}{Rezk 14}).


\item The orbit category for $G$ can also be generalized to the orbit category generated by any small category, $I$, where the $I$-orbits are $I$-diagrams in $Top$ whose strict colimit is equal to a point. If the orbits are either small or complete, then the $I$-equivariant model structure is Quillen-equivalent to a presheaf category (see pt. 11 of p. 8 of \href{Parametrized+Higher+Category+Theory+and+Higher+Algebra#PHCTIntro}{PHCTIntro} and \hyperlink{Chorny13}{Chorny13}).



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

\begin{itemize}%
\item [[equivariant Whitehead theorem]]


\item [[tom Dieck splitting]]


\item [[global equivariant homotopy theory]]


\item [[Bredon cohomology]]



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

Lecture notes include

\begin{itemize}%
\item [[Andrew Blumberg]], \emph{Equivariant homotopy theory}, 2017 (\href{https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf}{pdf}, \href{https://github.com/adebray/equivariant_homotopy_theory}{GitHub})

\end{itemize}
The ``fine'' homotopical structure on [[G-spaces]] (with fixed-point-wise weak equivalences) is originally due to

\begin{itemize}%
\item [[Glen Bredon]], \emph{[[Equivariant cohomology theories]]}, Springer Lecture Notes in Mathematics Vol. 34. 1967.

\end{itemize}
The equivalence of the homotopy theory ([[homotopy category]]) of that to presheaves over the orbit category is then due to

\begin{itemize}%
\item [[Anthony Elmendorf]], \emph{Systems of fixed point sets}, Trans. Amer. Math. Soc., 277(1):275--284, 1983 (\href{https://www.jstor.org/stable/1999356}{jstor:1999356})

\end{itemize}
which considered all closed subgroups of $G$. Also

\begin{itemize}%
\item [[Robert Piacenza]], section 6 of \emph{Homotopy theory of diagrams and CW-complexes over a category}, Can. J. Math. Vol 43 (4), 1991 ([[Piazenza91.pdf:file]])

also chapter VI of [[Peter May]] et al, \emph{Equivariant homotopy and cohomology theory}, 1996 (\href{http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf}{pdf})



\end{itemize}
The generalization of the proof to other choices of families of subgroups is due to

\begin{itemize}%
\item [[Peter May]], \emph{Equivariant homotopy and cohomology theory} With contributions by M. Cole, G. Comezaa, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. Number 91 in CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996

\end{itemize}
Discussion in terms of [[Quillen equivalence]] of [[model categories]] is due to

\begin{itemize}%
\item [[Bert Guillou]], \emph{A short note on models for equivariant homotopy theory} (\href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf}{pdf})


\item [[Marc Stephan]], \emph{Elmendorf’s Theorem via Model Categories}, 2010 (\href{http://web.math.ku.dk/~jg/students/stephan.msproject.2010.pdf}{pdf}) -- \emph{Elmendorf’s Theorem for Cofibrantly Generated Model Categories} MS thesis, Zurich 2010 (\href{http://web.math.ku.dk/~jg/students/stephan.msthesis.2010.pdf}{pdf})


\item [[Marc Stephan]], \emph{On equivariant homotopy theory for model categories}, Homology Homotopy Appl. 18(2) (2016) 183-208 (\href{http://arxiv.org/abs/1308.0856}{arXiv:1308.0856})



\end{itemize}
A generalization to [[orbispaces]] is discussed in

\begin{itemize}%
\item [[André Henriques]], [[David Gepner]], \emph{Homotopy Theory of Orbispaces} (\href{http://arxiv.org/abs/math/0701916}{arXiv:math/0701916})

\end{itemize}
Discussion in the broader context of [[global equivariant homotopy theory]] is in

\begin{itemize}%
\item [[Charles Rezk]], \emph{[[Global Homotopy Theory and Cohesion]]} (2014)

\end{itemize}
Some of the categorical aspects of Elmendorf's theorem are examined in

\begin{itemize}%
\item [[Jean-Marc Cordier]], [[Timothy Porter]], \emph{Categorical Aspects of Equivariant Homotopy}, Applied Cat.Structures, \textbf{4} (1996) 195 - 212. doi:\href{https://doi.org/10.1007/BF00122252}{10.1007/BF00122252} (Proceedings of the European Colloquium of Category Theory, 1994)

\end{itemize}
A recent n-cat café discussion initiated by [[John Huerta]] and probing some of its uses in Mathematical Physics, can be found \href{https://golem.ph.utexas.edu/category/2018/06/elmendorfs_theorem.html#more}{here}.

Generalization to $I$-orbits for a small category $I$ is in

\begin{itemize}%
\item [[Boris Chorny]], \emph{Homotopy theory of relative simplicial presheaves}, Israel J. Math. 205 (2015), no. 1, 471–484, (\href{https://arxiv.org/abs/1310.2932}{arXiv:1310.2932})

\end{itemize}
[[!redirects Elmendorf theorem]]



\end{document}