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\begin{document}

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\section*{orbit category}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_spaces_and_elmendorfs_theorem}{Relation to $G$-spaces and Elmendorf's theorem}\dotfill \pageref*{relation_to_spaces_and_elmendorfs_theorem} \linebreak
\noindent\hyperlink{relation_to_equivariant_homotopy_theory}{Relation to equivariant homotopy theory}\dotfill \pageref*{relation_to_equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{relation_to_mackey_functors}{Relation to Mackey functors}\dotfill \pageref*{relation_to_mackey_functors} \linebreak
\noindent\hyperlink{relation_to_bredon_equivariant_cohomology}{Relation to Bredon equivariant cohomology}\dotfill \pageref*{relation_to_bredon_equivariant_cohomology} \linebreak
\noindent\hyperlink{relation_to_the_category_of_groups_homomorphisms_and_conjugations}{Relation to the category of groups, homomorphisms and conjugations}\dotfill \pageref*{relation_to_the_category_of_groups_homomorphisms_and_conjugations} \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{orbit category} of a [[group]] $G$ is the category of ``all kinds'' of [[orbits]] of $G$, namely of all suitable [[coset spaces]] regarded as [[G-spaces]].

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

\begin{defn}
\label{TheOrbitCategory}\hypertarget{TheOrbitCategory}{}
Given a [[topological group]] $G$ the \textbf{orbit category} $\operatorname{Orb}_G$ (denoted also $\mathcal{O}_G$) is the [[category]] whose

\begin{itemize}%
\item objects are the [[homogeneous space]]s ($G$-orbit types) $G/H$, where $H$ is a closed [[subgroup]] of $G$,


\item and whose morphisms are $G$-equivariant maps.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For suitable continuous [[actions]] of $G$ on a [[topological space]] $X$, every [[orbit]] of the action is [[isomorphism|isomorphic]] to one of the [[homogeneous spaces]] $G/H$ (the [[stabilizer group]] of any point in the orbit is conjugate to $H$). This is the sense in which def. \ref{TheOrbitCategory} gives ``the category of all $G$-orbits''.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Def. \ref{TheOrbitCategory} yields a [[small category|small]] [[enriched category|topologically enriched]] category (though of course if $G$ is a [[discrete group]], the enrichment of $\operatorname{Orb}_G$ is likewise discrete).

Of course, like any category, it has a [[skeleton]], but as usually defined it is not itself skeletal, since there can exist distinct subgroups $H$ and $K$ such that $G/H\cong G/K$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Warning: This should not be confused with the situation where a group $G$ acts on a groupoid $\Gamma$ so that one obtains the [[orbit groupoid]].

\end{remark}
More generally, given a family $F$ of subgroups of $G$ which is closed under conjugation and taking subgroups one looks at the full subcategory $\mathrm{Orb}_F\,G \subset \operatorname{Orb}_G$ whose objects are those $G/H$ for which $H\in F$.

\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

Sometimes a family, $\mathcal{W}$, of subgroups is specified, and then a subcategory of $\operatorname{Orb}_G$ consisting of the $G/H$ where $H\in \mathcal{W}$ will be considered. If the trivial subgroup is in $\mathcal{W}$ then many of the considerations of results such as [[Elmendorf's theorem]] will go across to the restricted setting.

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

\hypertarget{relation_to_spaces_and_elmendorfs_theorem}{}\subsubsection*{{Relation to $G$-spaces and Elmendorf's theorem}}\label{relation_to_spaces_and_elmendorfs_theorem}

[[Elmendorf's theorem]] (see there for details) states that the [[(∞,1)-category of (∞,1)-presheaves]] on the orbit category $Orb_G$ are [[equivalence of (∞,1)-categories|equivalent]] to the [[localization of an (∞,1)-category|localization]] of [[topological spaces]] with $G$-[[action]] at the \emph{[[weak homotopy equivalences]] on [[fixed point spaces]]}.

\begin{displaymath}
L_{we} G Top \simeq PSh_\infty(Orb_G)
  \,.
\end{displaymath}
\hypertarget{relation_to_equivariant_homotopy_theory}{}\subsubsection*{{Relation to equivariant homotopy theory}}\label{relation_to_equivariant_homotopy_theory}

The $G$-orbit category is the [[slice (∞,1)-category]] of the [[global orbit category]] $Orb$ over the [[delooping]] $\mathbf{B}G$:

\begin{displaymath}
Orb_G \simeq Orb_{/\mathbf{B}G}
  \,.
\end{displaymath}
This means that in the general context of [[global equivariant homotopy theory]], the orbit category appears as follows.

[[!include equivariant homotopy theory -- table]]

\hypertarget{relation_to_mackey_functors}{}\subsubsection*{{Relation to Mackey functors}}\label{relation_to_mackey_functors}

Orbit categories are used often in the treatment of [[Mackey functor]]s from the theory of [[locally compact group]]s and in the definition of [[Bredon cohomology]].

\hypertarget{relation_to_bredon_equivariant_cohomology}{}\subsubsection*{{Relation to Bredon equivariant cohomology}}\label{relation_to_bredon_equivariant_cohomology}

It appears in [[equivariant stable homotopy theory]], where the $H$-fixed [[homotopy groups]] of a space form a [[presheaf]] on the [[homotopy category]] of the orbit category (e.g. \href{http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf#page=8}{page 8, 9 here}).

\hypertarget{relation_to_the_category_of_groups_homomorphisms_and_conjugations}{}\subsubsection*{{Relation to the category of groups, homomorphisms and conjugations}}\label{relation_to_the_category_of_groups_homomorphisms_and_conjugations}

See at \emph{[[global equivariant homotopy theory]]}.

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

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


\item [[Bredon cohomology]]



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

Lecture notes include

\begin{itemize}%
\item [[Tammo tom Dieck]], section 1.3 of \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})


\item [[Peter May]], section I.4 of \emph{Equivariant homotopy and cohomology theory} CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf}{pdf})



\end{itemize}
A very general setting for the use of orbit categories is described in

\begin{itemize}%
\item [[W. G. Dwyer]] and [[D. M. Kan]], \emph{Singular functors and realization functors} , Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 -- 153.

\end{itemize}
For more on the relation to [[global equivariant homotopy theory]] see

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

\end{itemize}
[[!redirects orbit categories]]



\end{document}