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\begin{document}

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\section*{global equivariant homotopy theory}

\begin{quote}%
under construction (some more harmonization needed)


\end{quote}
\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cohesion}{Cohesion}\dotfill \pageref*{cohesion} \linebreak
\noindent\hyperlink{relation_between_global_and_local_equivariant_homotopy_theory}{Relation between global and local equivariant homotopy theory}\dotfill \pageref*{relation_between_global_and_local_equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{relation_to_topological_stacks_and_orbispaces}{Relation to topological stacks and orbispaces}\dotfill \pageref*{relation_to_topological_stacks_and_orbispaces} \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}

What is called \emph{global equivariant homotopy theory} is a variant of [[equivariant cohomology]] in [[homotopy theory]] where [[pointed object|pointed]] [[topological spaces]]/[[homotopy types]] are equipped with $G$-[[infinity-actions]] ``for all [[compact Lie groups]] $G$ at once'', or more generally for a [[global family]].

Sometimes this is referred to just as ``global homotopy theory'', leaving the equivariance implicit. There is also a [[stabilization|stable]] version involving [[spectra]] equipped with [[infinity-actions]], see at \emph{[[global equivariant stable homotopy theory]]}.

More precisely, the \emph{global equivariant homotopy category} is the [[(∞,1)-category]] (or else its [[homotopy category of an (∞,1)-category|homotopy category]]) of [[(∞,1)-presheaves]] $PSh_\infty(Orb)$ on the [[global orbit category]] $Orb$ (\hyperlink{HenriquesGepner07}{Henriques-Gepner 07, section 1.3}), regarded as an [[(∞,1)-category]].

Here $Orb$ has as [[objects]] [[compact Lie groups]] and the [[(∞,1)-categorical hom-spaces]] $Orb(G,H) \coloneqq \Pi [\mathbf{B}G, \mathbf{B}H]$, where on the right we have the [[geometric realization of cohesive infinity-groupoids|fundamental (∞,1)-groupoid]] of the [[topological groupoid]] of [[group homomorphisms]] and [[conjugation|conjugations]].

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

We follow (\hyperlink{Rezk14}{Rezk 14}). Beware that the terminology there differs slightly but crucially in some places from (\hyperlink{HenriquesGepner07}{Henriques-Gepner 07}). Whatever terminology one uses, the following are the key definitions.

The following is the \emph{[[global equivariant indexing category]]}.

\begin{defn}
\label{GlobalIndexingCategory}\hypertarget{GlobalIndexingCategory}{}
Write $Glo$ for the [[(∞,1)-category]] whose

\begin{itemize}%
\item [[objects]] are [[compact Lie groups]];


\item [[(∞,1)-categorical hom-spaces]] $Glo(G,H)$ are the [[geometric realizations]] of the [[Lie groupoid]] of smooth functors and smooth [[natural transformations]] $Top\infty Grpd(\mathbf{B}G, \mathbf{B}H)$.



\end{itemize}
\end{defn}
(\hyperlink{Rezk14}{Rezk 14, 2.1})

\begin{defn}
\label{}\hypertarget{}{}
Equivalent models for the global indexing category, def. \ref{GlobalIndexingCategory} include the category ``$O_{gl}$'' of (\hyperlink{May90}{May 90}). Another variant is $\mathbf{O}_{gl}$ of (\hyperlink{Schwede13}{Schwede 13}).

\end{defn}
(\hyperlink{Rezk14}{Rezk 14, 2.4, 2.5})

The following is the \emph{[[global orbit category]]}.

\begin{defn}
\label{GlobalOrbitCategory}\hypertarget{GlobalOrbitCategory}{}
Write

\begin{displaymath}
Orb \longrightarrow Glo
\end{displaymath}
for the non-[[full sub-(∞,1)-category|full]] [[sub-(∞,1)-category]] of the global indexing category, def. \ref{GlobalIndexingCategory}, on the [[injection|injective]] group homomorphisms.

\end{defn}
(\hyperlink{Rezk14}{Rezk14, 4.5})

The following defines the \emph{global equivariant homotopy theory} $PSh_\infty(Glo)$.

\begin{defn}
\label{GlobalEquivariantHomotopyTopos}\hypertarget{GlobalEquivariantHomotopyTopos}{}
Write

\begin{displaymath}
Top_{Glo} \coloneqq PSh_\infty(Glo)
\end{displaymath}
for the [[(∞,1)-category of (∞,1)-presheaves]] (an [[(∞,1)-topos]]) on the global indexing category $Glo$ of def. \ref{GlobalIndexingCategory}, and write

\begin{displaymath}
\mathbb{B} \;\colon\; Glo \longrightarrow PSh_\infty(Glo)
\end{displaymath}
for the [[(∞,1)-Yoneda embedding]].

Similarly write

\begin{displaymath}
Top_{Orb} \coloneqq PSh_\infty(Orb)
\end{displaymath}
for the [[(∞,1)-category of (∞,1)-presheaves]] on the [[global orbit category]] $Orb$ of def. \ref{GlobalOrbitCategory}, and write again

\begin{displaymath}
\mathbb{B} \;\colon\; Orb \longrightarrow PSh_\infty(Orb)
\end{displaymath}
for its [[(∞,1)-Yoneda embedding]].

\end{defn}
(\hyperlink{Rezk14}{Rezk 14, 3.1 and 4.5})

The following recovers the ordinary (``local'') [[equivariant homotopy theory]] of a given [[topological group]] $G$ (``of $G$-spaces'').

\begin{defn}
\label{LocalEquivariantHomotopyTheory}\hypertarget{LocalEquivariantHomotopyTheory}{}
For $G$ a [[topological group]], write

\begin{displaymath}
G Top \coloneqq PSh_\infty(Orb)/\mathbb{B}G
\end{displaymath}
for the [[slice (∞,1)-topos]] of $PSh_\infty(Orb)$ over the image of $G$ under the [[(∞,1)-Yoneda embedding]], as in def. \ref{GlobalEquivariantHomotopyTopos}.

\end{defn}
This is (\hyperlink{Rezk14}{Rezk 14, 1.5}). Depending on axiomatization this is either a definition or [[Elmendorf's theorem]], see at \emph{[[equivariant homotopy theory]]} for more on this.

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

\hypertarget{cohesion}{}\subsubsection*{{Cohesion}}\label{cohesion}

\begin{prop}
\label{CohesionOfGlobalEquivariantHomotopyTheory}\hypertarget{CohesionOfGlobalEquivariantHomotopyTheory}{}
The global equivariant homotopy theory $PSh_\infty(Glo)$ of def. \ref{GlobalEquivariantHomotopyTopos} is a [[cohesive (∞,1)-topos]] over the canonical [[base (∞,1)-topos]] [[∞Grpd]]:

the [[global section geometric morphism]]

\begin{displaymath}
(\Delta \dashv \Gamma)
  \;\colon\;
  PSh_\infty(Glo)
  \longrightarrow
  \infty Grpd
\end{displaymath}
is given (as for all (∞,1)-presheaf (∞,1)-toposes) by the [[direct image]]/[[global section]] functor being the [[homotopy limit]] over the opposite [[(∞,1)-site]]

\begin{displaymath}
\Gamma X \simeq \underset{\leftarrow}{\lim}(Glo^{op}\stackrel{X}{\to} \infty Grpd)
\end{displaymath}
and the [[inverse image]]/[[constant ∞-stack]] functor literally assigning constant presheaves:

\begin{displaymath}
\Delta S \colon G \mapsto S
 \,.
\end{displaymath}
This is a [[full and faithful (∞,1)-functor]].

Moreover, $\Delta$ has a further [[left adjoint]] $\Pi$ which preserves [[finite products]], and $\Gamma$ has a further [[right adjoint]] $\nabla$.

\end{prop}
(\hyperlink{Rezk14}{Rezk 14, 5.1})

More in detail, the [[shape modality]], [[flat modality]] and [[sharp modality]] of this [[cohesion]] of the global equivariant homotopy theory has the following description.

\hypertarget{relation_between_global_and_local_equivariant_homotopy_theory}{}\subsubsection*{{Relation between global and local equivariant homotopy theory}}\label{relation_between_global_and_local_equivariant_homotopy_theory}

\begin{defn}
\label{InclusionOfGSpacesInTheGlobalTheory}\hypertarget{InclusionOfGSpacesInTheGlobalTheory}{}
For $G$ a [[compact Lie group]] define an [[(∞,1)-functor]]

\begin{displaymath}
\delta_G \;\colon\; G Top \longrightarrow PSh_\infty(Glo)
\end{displaymath}
sending a [[topological G-space]] to the he presheaf which sends a group $H$ to the [[geometric realization]] of the [[topological groupoid]] of maps from $\mathbf{B}H$ to the [[action groupoid]] $X//G$:

\begin{displaymath}
\delta_G(X)\;\colon\; H \mapsto \Pi( [\mathbf{B}H, X//G] )  
  \,.
\end{displaymath}
Observe that by def. \ref{GlobalEquivariantHomotopyTopos} this gives $\delta_G(\ast) \simeq \mathbb{B}G$ and so $\delta_G$ induces a functor

\begin{displaymath}
\Delta_G \;\colon\; G Top \simeq G Top/\ast \simeq PSh_\infty(Orb)/\mathbb{B}G \stackrel{\delta_G}{\longrightarrow} PSh_\infty(Glo)/\mathbb{B}G
 \,.
\end{displaymath}
\end{defn}
(\hyperlink{Rezk14}{Rezk 14, 3.2})

\begin{prop}
\label{OrdinaryQuotientAndHomotopyQuotientViaCohesion}\hypertarget{OrdinaryQuotientAndHomotopyQuotientViaCohesion}{}
\textbf{(ordinary [[quotient]] and [[homotopy quotient]] via equivariant cohesion)}

On a $G$-space $X \in G Top$ included via def. \ref{InclusionOfGSpacesInTheGlobalTheory} into the global equivariant homotopy theory,

\begin{itemize}%
\item the [[shape modality]] of def. \ref{CohesionOfGlobalEquivariantHomotopyTheory} produces the [[homotopy type]] of the ordinary [[quotient]] of the $G$-[[action]]

\begin{displaymath}
\Pi(\delta_G(X)) \simeq \vert X/G \vert
  \,,
\end{displaymath}

\item the [[flat modality]] of def. \ref{CohesionOfGlobalEquivariantHomotopyTheory} produces the [[homotopy type]] of the [[homotopy quotient]]/[[homotopy coinvariants]] of the $G$-[[action]] ([[∞-action]])

\begin{displaymath}
\Gamma(\delta_G(X)) \simeq \vert X//G \vert
  \,,
\end{displaymath}


\end{itemize}
In particular then the [[points-to-pieces transform]] of general [[cohesion]] yields the comparison map

\begin{displaymath}
\vert X//G \vert
  \longrightarrow
  \vert X/G \vert
   \,.
\end{displaymath}
\end{prop}
(\hyperlink{Rezk14}{Rezk 14, 5.1})

\begin{prop}
\label{CohesionOnLocalSlice}\hypertarget{CohesionOnLocalSlice}{}
For $G$ any [[compact Lie group]], the [[cohesion]] of the global equivariant homotopy theory, prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}, descends to the [[slice (∞,1)-toposes]]

\begin{displaymath}
PSh_\infty(Glo)/\mathbb{B}G
  \longrightarrow
  PSh_\infty(Orb)/\mathbb{B}G \simeq G Top
  \,,
\end{displaymath}
hence to cohesion over the ``local'' $G$-[[equivariant homotopy theory]].

The inclusion $\Delta_G$ is that of def. \ref{InclusionOfGSpacesInTheGlobalTheory}.

\end{prop}
(\hyperlink{Rezk14}{Rezk 14, 5.3})

\hypertarget{relation_to_topological_stacks_and_orbispaces}{}\subsubsection*{{Relation to topological stacks and orbispaces}}\label{relation_to_topological_stacks_and_orbispaces}

\begin{quote}%
under construction


\end{quote}
By the main theorem of (\hyperlink{HenriquesGepner07}{Henriques-Gepner 07}) the [[(∞,1)-presheaves]] on the [[global orbit category]] are equivalently ``cellular'' [[topological stacks]]/[[topological groupoids]] (``[[orbispaces]]''), we might write this as

\begin{displaymath}
ETopGrpd^{cell} = PSh_\infty(Orb)
  \,.
\end{displaymath}
(As such the global equivariant homotopy theory should be similar to [[ETop∞Grpd]]. Observe that this is a [[cohesive (∞,1)-topos]] with $\Pi$ such that it sends a topological [[action groupoid]] of a [[topological group]] $G$ acting on a [[topological space]] $X$ to the [[homotopy quotient]] $\Pi(X)//\Pi(G)$.)

The central theorem of (\hyperlink{Rezk14}{Rezk 14}) (using a slightly different definition than \hyperlink{HenriquesGepner07}{Henriques-Gepner 07}) is that $PSh_\infty(Orb)$ is a [[cohesive (∞,1)-topos]] with $\Gamma$ producing homotopy quotients.

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

\begin{itemize}%
\item [[orbifold cohomology]]


\item [[global equivariant stable homotopy theory]]



\end{itemize}
[[!include equivariant homotopy theory -- table]]

[[!include homotopy type representation theory -- table]]

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

The [[global orbit category]] $Orb$ is considered in

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


\item [[Jacob Lurie]], Section 3 of \emph{Elliptic cohomology III: Tempered Cohomology} (\href{http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf}{pdf})



\end{itemize}
Global unstable equivariant homotopy theory is discussed as a [[localization]] of the category of ``orthogonal spaces'' (the unstable version of [[orthogonal spectra]]) in

\begin{itemize}%
\item [[Stefan Schwede]], chapter I of \emph{[[Global homotopy theory]]}, 2013 (\href{http://www.math.uni-bonn.de/~schwede/global.pdf}{pdf})


\item [[Stefan Schwede]], \emph{Orbispaces, orthogonal spaces, and the universal compact Lie group} (\href{https://arxiv.org/abs/1711.06019}{arXiv:1711.06019})


\item [[Stefan Schwede]], \emph{[[Global homotopy theory]]} (\href{https://arxiv.org/abs/1802.09382}{arXiv:1802.09382})



\end{itemize}
see also

\begin{itemize}%
\item [[Peter May]], \emph{Some remarks on equivariant bundles and classifying spaces}, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

\end{itemize}
Discussion of the global equivariant homotopy theory as a [[cohesive (∞,1)-topos]] is in

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

\end{itemize}
Discussion of a model structure for global equivariance with respect to [[geometrically discrete ∞-groupoid|geometrically discrete]] [[simplicial groups]]/[[∞-group]] (globalizing the [[Borel model structure]] for [[∞-actions]]) is in

\begin{itemize}%
\item [[Yonatan Harpaz]], [[Matan Prasma]], \emph{The Grothendieck construction for model categories} (\href{http://arxiv.org/abs/1404.1852}{arXiv:1404.1852})

\end{itemize}
Discussion from a perspective of [[homotopy type theory]] is in

\begin{itemize}%
\item [[Mike Shulman]], \emph{Univalence for inverse EI diagrams} (\href{http://arxiv.org/abs/1508.02410}{arXiv:1508.02410})

\end{itemize}
[[!redirects global homotopy theory]]

[[!redirects global equivariant homotopy category]]

[[!redirects globally equivariant homotopy theory]]



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