\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{orbifold cohomology}

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

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

[[!include cohomology - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{deficiency_of_orbifolds_as_geometric_groupoids}{Deficiency of orbifolds as geometric groupoids}\dotfill \pageref*{deficiency_of_orbifolds_as_geometric_groupoids} \linebreak
\noindent\hyperlink{lift_of_orbifolds_to_geometric_global_homotopy_theory}{Lift of orbifolds to geometric global homotopy theory}\dotfill \pageref*{lift_of_orbifolds_to_geometric_global_homotopy_theory} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{GloballyEquivariantCohesiveToposes}{Globally equivariant cohesive toposes}\dotfill \pageref*{GloballyEquivariantCohesiveToposes} \linebreak
\noindent\hyperlink{EquivariantCohesiveToposes}{$G$-Equivariant cohesive toposes}\dotfill \pageref*{EquivariantCohesiveToposes} \linebreak
\noindent\hyperlink{OnOrbifolds}{Orbifolds}\dotfill \pageref*{OnOrbifolds} \linebreak
\noindent\hyperlink{orbifold_cohomology}{Orbifold cohomology}\dotfill \pageref*{orbifold_cohomology} \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}

Since the crucial extra [[structures]] carried by an [[orbifold]] are

\begin{enumerate}%
\item geometric structure (e.g. [[topology|topological]], [[algebraic geometry|algebro-geometric]], [[differential geometry|differential-geometric]], [[supergeometry|super-geometric]], etc.)


\item [[singularities]]



\end{enumerate}
the [[cohomology]] of orbifolds should be such as to provide [[invariants]] which are sensitive not just to the underlying plain [[homotopy type]] of an orbifold (its [[shape modality|shape]]) but also to this extra structure. This means that \emph{orbifold cohomology} should, respectively, unify

\begin{enumerate}%
\item geometric cohomology (e.g. [[sheaf cohomology|sheaf]] [[hypercohomology]], [[differential cohomology]], etc.)


\item [[equivariant cohomology]] in its fine form of [[Bredon cohomology]]



\end{enumerate}
in the sense that [[differential cohomology|geometric cohomology]] is recovered away from the orbifold [[singularities]] and [[equivariant cohomology]] is recovered right at the [[singularities]], while globally orbifold cohomology provides a unification of these two aspects.

Since any concept of [[cohomology]] (as discussed there) is effectively equivalent to the choice of ambient [[(∞,1)-topos]], the question of defining orbifold cohomology is closely related to the question of how exactly to define the [[(∞,1)-category]] of orbifolds (usually a [[(2,1)-category]]) in the first place. This question is notoriously more subtle than the simple intuitive idea of orbifolds might suggest, as witnessed by the convoluted history of the concept (see e.g. \href{orbifold#Lerman08}{Lerman 08, Introduction}).

$\,$

\hypertarget{deficiency_of_orbifolds_as_geometric_groupoids}{}\subsubsection*{{Deficiency of orbifolds as geometric groupoids}}\label{deficiency_of_orbifolds_as_geometric_groupoids}

A proposal popular among [[Lie theory|Lie theorists]] (\href{orbifold#MoerdijkPronk97}{Moerdijk-Pronk 97}) is to regard an orbifold with local charts $U_i \in G_i Actions$ ([[actions]] of some [[group]] on some local model [[space]]) as the [[geometric stack]] obtained by gluing the corresponding [[homotopy quotients]]/[[quotient stacks]] $U_i \!\sslash \!G_i$. If $\mathbf{H}$ is the ambient [[cohesive (∞,1)-topos]] in which this takes place (for instance $\mathbf{H} =$ [[Smooth∞Groupoids]], [[SuperFormalSmooth∞Groupoids]], etc.) and if $G \in Grp \overset{Disc}{\hookrightarrow} Grp(\mathbf{H})$ is a [[discrete group]] in which all the [[isotropy groups]] of the orbifold are contained, this gives an object

\begin{displaymath}
\left(
  \itexarray{
    \mathcal{X}
    \\
    \downarrow^{\mathrlap{faith}}
    \\
    \mathbf{B}G
  }
  \phantom{{}^{faith}}
  \right)
  \;\in\;
  \left( 
    \mathbf{H}_{/\mathbf{B}G}
  \right)_{\leq 0}
  \hookrightarrow
  \mathbf{H}_{/\mathbf{B}G}
\end{displaymath}
in the [[slice (∞,1)-topos]] over the [[delooping]] $\mathbf{B}G = \ast \sslash G$ of $G$, which is still a [[0-truncated object]], reflecting that as a [[functor]] of [[groupoids]] the morphism $\mathcal{X} \to \mathbf{B}G$ is a [[faithful functor]].

Accordingly, if this is -- or were -- the correct formalization of the nature of [[orbifolds]] $\mathcal{X}$, then the corresponding orbifold cohomology has [[coefficients]] given by objects $\mathcal{A} \in \mathbf{H}_{/\mathbf{B}G}$ and cohomology sets being the [[connected components]] of the [[(∞,1)-categorical hom-spaces]]

\begin{displaymath}
H_{\mathbf{B}G}\big( \mathcal{X}, \mathcal{A}\big)
  \;\coloneqq\;
  \pi_0 \mathbf{H}_{/\mathbf{B}G}\big( \mathcal{X}, \mathcal{A}\big)
  \,.
\end{displaymath}
This concept of orbifold cohomology does fully reflect the geometric nature of orbifolds. It also reflects \emph{some} equivariance aspect. For example if $\mathcal{X} = \ast \sslash G$ is the one-point orbifold with singularity given by a finite group $G$, and if $V \in G Representations$ is a [[linear representation]], with $K(V,n)\sslash G \in \infty Groupoids_{/\mathbf{B}G} \overset{Disc}{\hookrightarrow} \mathbf{H}_{/\mathbf{B}G}$ its [[Eilenberg-MacLane space]], then

\begin{displaymath}
H_{\mathbf{B}G}\big( 
    \mathbf{B}G, K(V,n)\sslash G
  \big)
  \;\simeq\;
  H^n_{grp}(G,V)
\end{displaymath}
is the [[group cohomology]] of $G$ with [[coefficients]] in $V$.

However, this definition does \emph{not} reflect [[Bredon cohomology|Bredon]]-[[equivariant cohomology]] around the orbifold singularities. Instead, it really given (geometric/stacky refinement) of \emph{[[cohomology with local coefficients]]}.

$\,$

\hypertarget{lift_of_orbifolds_to_geometric_global_homotopy_theory}{}\subsubsection*{{Lift of orbifolds to geometric global homotopy theory}}\label{lift_of_orbifolds_to_geometric_global_homotopy_theory}

Hence the proposal of \href{orbifold#MoerdijkPronk97}{Moerdijk-Pronk 97}, that an [[orbifold]] should be regarded as a certain [[geometric stack]], is missing something. It was briefly suggested in \hyperlink{Schwede17}{Schwede 17, Introduction}, \hyperlink{Schwede18}{Schwede 18, p. ix-x} that the missing aspect is provided by [[global equivariant homotopy theory]], but details seem to have been left open.

Here we discuss how to define the required orbifold cohomology in detail and in general. We combine the [[differential cohesion]] for the geometric aspect with the cohesion of [[global equivariant homotopy theory]] that was observed and highlighted in \hyperlink{Rezk14}{Rezk 14}.

The following may serve as intuition for the issue with the nature of orbifolds:

Envision the picture of an orbifold singularity and hold a mathemagical magnifying glass over the singular point. Under this magnification you can see resolved the singular point as a fuzzy fattened point, to be called $\mathbb{B}G$.

Removing the magnifying glass, what one sees with the bare eye depends on how one squints:

\begin{itemize}%
\item The physicist says that what he sees is a singular point, but a point after all. This is the plain [[quotient]] $\ast = \ast / G$.


\item The Lie geometer says that what she sees is a point transforming under the $G$-[[action]] that fixes it, hence the [[homotopy quotient]] [[groupoid]] $\mathbf{B}G =\ast \sslash G$.



\end{itemize}
These are two [[adjoint modality|opposite extreme aspects]] of the orbifold singularity $\mathbb{B}G$, but the orbifold singularity itself is more than both of these aspects. The real nature of an orbifold singularity is in fact a point, not a big [[classifying space]] $\mathbf{B} G$ (recall that already $\mathbf{B}\mathbb{Z}_2 = \mathbb{R}P^\infty$), but it is a point that also remembers the group [[action]], for that characterizes how the singularity is being singular.

\begin{displaymath}
\itexarray{
       && 
       { \text{orbifold singularity} \atop {\mathbb{B}G} }
      \\
      & {}^{\mathllap{&#643;_{sing}}}\swarrow & {{\phantom{A}} \atop {  \text{opposite extreme} \atop \text{aspects of orbifold singularity} }} & \searrow^{\mathrlap{ \flat_{sing} }}
     \\
     { \text{plain quotient} \atop {\ast = \ast/G} }
     &&
     &&
     { \text{homotopy quotient} \atop { \mathbf{B}G = \ast \sslash G } }
   }
\end{displaymath}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

The definition of orbifold cohomology below in Def. \ref{OrbifoldCohomology} is the canonical [[cohomology]] in a [[slice (infinity,1)-topos|slice]] of the [[globally equivariant homotopy theory]] $\mathbf{H}_{sing}$ of the given ambient [[cohesive (∞,1)-topos]] $\mathbf{H}$.

For completeness, we first introduce/recall $\mathbf{H}_{sing}$ in Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory} below, as well as its slices to $G$-[[equivariant homotopy theory]] $\mathbf{H}_G$ (Def. \ref{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos} below), following \hyperlink{Rezk14}{Rezk 14}.

The key point in the following is the identification of [[orbifolds]] as the [[n-truncated object in an (∞,1)-category|0-truncated]] ``$Singularities$-[[codiscrete objects]]'' in $\mathbf{H}_{sing}$ (Def. \ref{Orbifold} below), which is consistent in that under passage to [[shape modality|shapes]], i.e. the underlying bare [[globally equivariant homotopy theory|globally equivariant]] [[homotopy types]], it reproduces the standard embedding of [[G-spaces]] into [[globally equivariant homotopy theory]] (Example \ref{GlobalHomotopyQuotientsOfSmoothManifolds} below.)

$\,$

\hypertarget{GloballyEquivariantCohesiveToposes}{}\subsubsection*{{Globally equivariant cohesive toposes}}\label{GloballyEquivariantCohesiveToposes}

We consider here the evident refinement to \emph{geometric cohomology} hence to \emph{[[differential cohomology]]} ([[cohesion|cohesive]] [[infinity-stack|∞-stacky]] [[sheaf cohomology]]) of [[globally equivariant homotopy theory]], in Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory} below.

In order to bring out the conceptual appearance of [[orbifolds]] further \hyperlink{OnOrbifolds}{below}, we take the liberty of referring to what otherwise is known as the ``[[global orbit category]]'' instead as the \emph{categories of singularities} (Def. \ref{CategoryOfSingularities} below).

$\,$

\begin{defn}
\label{CategoryOfSingularities}\hypertarget{CategoryOfSingularities}{}
\textbf{(category of singularities)}

Write

\begin{displaymath}
Singularities
  \;\coloneqq\;
  Groupoids^{cn}_{1,fin}
  \hookrightarrow
  Groupoids_\infty
\end{displaymath}
for the [[(2,1)-category]] of [[connected object|connected]] [[homotopy type with finite homotopy groups|finite groupoids]]. A [[skeleton]] has [[objects]] labeled by [[finite groups]] $G$, and we will denote these objects

\begin{displaymath}
\mathbb{B}G
  \;\in\;
  Singularities
\end{displaymath}
to distinguish them from their image as [[delooping]] groupoids $\mathbf{B} G \in$ [[∞Grpd]]. (As we consider [[(∞,1)-presheaves]] on $Singularities$ with values in [[∞Groupoids]], in Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory} below, these two objects become crucially different, albeit closely related.)

\end{defn}
\begin{remark}
\label{SingularitiesInTheLiterature}\hypertarget{SingularitiesInTheLiterature}{}
The category $Singularities$ in Def. \ref{CategoryOfSingularities}, when generalized from [[finite groups]] to [[compact Lie groups]], is called

\begin{itemize}%
\item ``$Orb$ version 1'' in \href{orbispace#HenriquesGepner07}{Henriques-Gepner 07}


\item ``$Glo$'' in \hyperlink{Rezk14}{Rezk 14, 2.2}


\item ``$Orb$'' in \href{orbispace#Koerschgen16}{Körschgen 16}, \href{orbispace#Schwede17}{Schwede 17}, \href{global+equivariant+homotopy+theory#Schwede18}{Schwede 18}.



\end{itemize}
\end{remark}
\begin{prop}
\label{CohesionOfGlobalEquivariantHomotopyTheory}\hypertarget{CohesionOfGlobalEquivariantHomotopyTheory}{}
\textbf{([[global equivariant homotopy theory]] [[cohesive (∞,1)-topos|cohesive]] over [[base (∞,1)-topos]])}

Let $\mathbf{H}$ be any [[(∞,1)-topos]] and consider the [[(∞,1)-category of (∞,1)-presheaves]] on the category of singularities (Def. \ref{CategoryOfSingularities}) over the [[base (∞,1)-topos]] $\mathbf{H}$, hence the [[(∞,1)-functor (∞,1)-category]]

\begin{displaymath}
\mathbf{H}_{sing}
  \;\coloneqq\;
  Sh_\infty\big( Singularities, \mathbf{H}\big)
  \;=\;
  Funct\big( \Singularities^{op}, \mathbf{H}\big)
  \,.
\end{displaymath}
This is a [[cohesive (∞,1)-topos]] over the [[base (∞,1)-topos]] $\mathbf{H}$ in that the [[global section]]-[[geometric morphisms]] enhances to an [[adjoint quadruple]] of [[adjoint (∞,1)-functors]]

\begin{equation}
\big(
    \Pi_{sing}
    \dashv
    Disc_{sing}
    \dashv
    \Gamma_{sing}
    \dashv
    coDisc_{sing}
  \big)
  \;\colon\;
  \mathbf{H}_{sing}
    \leftrightarrow
  \mathbf{H}
\label{SingularitiesAdjointQuadruple}\end{equation}
such that

\begin{enumerate}%
\item $Disc_{sing}, coDisc_{sing} \;\colon\; \mathbf{H} \to \mathbf{H}_{sing}$ are [[fully faithful (∞,1)-functors]];


\item $\Pi_{sing}$ preserves [[finite products]].



\end{enumerate}
hence inducing an [[adjoint triple]] of [[adjoint modalities]]

\begin{displaymath}
&#643;_{sing}
  \dashv 
  \flat_{sing}
  \dashv
  \sharp_{sing}
  \;\colon\;
  \mathbf{H}_{sing} \to \mathbf{H}_{sing}
\end{displaymath}
(``[[shape modality|shape]]'', ``[[flat modality|flat]]'', ``[[sharp modality|sharp]]'' for singularities).

Moreover, for $G$ a [[finite group]] regarded under the inclusion

\begin{displaymath}
G \in Grp \overset{Disc}\hookrightarrow Grp(\mathbf{H}) \overset{Disc_{sing}}{\hookrightarrow} Grp\left(\mathbf{H}_{sing}\right)
\end{displaymath}
and writing

\begin{displaymath}
\mathbf{B}G \in \mathbf{H}_{sing}
\end{displaymath}
for its [[delooping]] under $Grp\left( \mathbf{H}_{sing} \right) \underoverset{\simeq}{\mathbf{B}}{\longrightarrow} \mathbf{H}^{\ast/}_{cn}$,

in constrast to the [[(∞,1)-Yoneda embedding]]

\begin{displaymath}
\mathbb{B}G 
  \overset{y}{\longrightarrow}
  Sh_\infty\left( Singularities, \infty \mathrm{Grpd}\right)
  \overset{Disc}{\longrightarrow}
  Sh_\infty\left( Singularities, \mathbf{H}\right)
\end{displaymath}
we have

\begin{equation}
\begin{aligned}
     &#643;_{sing} \mathbb{B}G 
     & \simeq\; \ast/G = \ast
     \\
     \flat_{sing} \mathbb{B}G 
     &\simeq\; \ast \sslash G = \mathbf{B}G
  \end{aligned}
\label{ShapeOfOrbifoldSingularity}\end{equation}
\end{prop}
\begin{proof}
This is immediate by general properties of left/right [[(∞,1)-Kan extension]], using the evident fact that $Singularities$ has [[finite products]] (the [[terminal object in an (∞,1)-category|terminal object]] is $\mathbb{B}1$ and the binary [[Cartesian product]] is give by forming [[direct product groups]]: $\left(\mathbb{B}G_1\right) \times \left( \mathbb{B}G_2\right) \simeq \mathbb{B}\left( G_1 \times G_2\right)$ ). The directly analogous 1-categorical argument is at \emph{[[infinity-cohesive site]]}.

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
For $\mathbf{H} =$ [[∞Groupoids]] the [[cohesion]] of Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory} is that of plain [[globally equivariant homotopy theory]] (\hyperlink{Rezk14}{Rezk 14, 4.1}), i.e. without any geometric determination ([[geometrically discrete ∞-groupoids]]).

\end{example}
Conversely we have:

\begin{prop}
\label{SingularitiesAsCoDisc}\hypertarget{SingularitiesAsCoDisc}{}
\textbf{([[orbifold]] [[singularities]] are the [[codiscrete object|codiscrete]] aspect of [[homotopy quotients]]})

Let $\mathbf{H}$ itself be a [[cohesive (∞,1)-topos]] over [[∞Groupoids]]

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma \dashv coDisc)
  \;\colon\;
  \mathbf{H} \leftrightarrow \infty Groupoids
  \,.
\end{displaymath}
Then in the situation of Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}

\begin{displaymath}
coDisc_{sing} \mathbf{B}G
  \;\simeq\;
  \mathbb{B}G
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Let $\mathcal{S}$ be a cohesive [[(∞,1)-site]] of definition for $\mathbf{H}$, so that

\begin{displaymath}
\mathbf{H}_{sing}
  \;\simeq\;
  Sh_\infty
  \left(
    \mathcal{S} \times Singularities
    \;,\;
    \infty Groupoids
  \right)
\end{displaymath}
and $\Pi(S) \simeq \ast$ for $S \in \mathcal{S} \overset{y}{\hookrightarrow} \mathbf{H}$.

Then as [[(∞,1)-presheaves]] regarded this way we have

\begin{displaymath}
\begin{aligned}
    coDisc_{sing} \mathbf{B}G
    \;\colon\;
    S \times \mathbb{B}K 
    & \mapsto
    \mathbf{H}_{sing}\left( S \times \mathbb{B}K, coDisc_{sing} Disc \mathbf{B}G \right)
    \\
    & \simeq
    \mathbf{H}\big( 
       S 
         \times 
       \underset{
         \simeq \mathbf{B}K
       }{
       \underbrace{
         \Gamma_{sing}\left( \mathbb{B}K\right)
       }}
       , 
       Disc \mathbf{B}G 
    \big)    
    \\
    & \simeq
    \infty Groupoids\big( 
       \underset{
         \simeq \mathbf{B}K
       }{
       \underbrace{
         \Pi(S \times \mathbf{B}K)
       }}
       \,,
       \mathbf{B}G 
    \big)        
    \\
    & \simeq
    \infty Groupoids\left(
       \mathbf{B} K, \mathbf{B}G
    \right)
    \\
    & \simeq
    \mathbf{H}_{sing}\big(
       \mathbb{B}K, 
       \mathbb{B}G
    \big)
    \\
    & \simeq
    \mathbf{H}_{sing}\big(
       S \times \mathbb{B}K, 
       \mathbb{B}G
    \big)
  \end{aligned}
\end{displaymath}
Here we used the various [[(∞,1)-adjunctions]] and the [[(∞,1)-Yoneda lemma]], and the claim in turn follows by the [[(∞,1)-Yoneda lemma]].

\end{proof}
$\,$

\hypertarget{EquivariantCohesiveToposes}{}\subsubsection*{{$G$-Equivariant cohesive toposes}}\label{EquivariantCohesiveToposes}

In order to speak of $G$-[[equivariant homotopy theory]] (Def. \ref{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos} below) inside [[globally equivariant homotopy theory]] (Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory} above) we need a certain concept of faithfulness (Def. \ref{SingularitiesFaithful} below).

For that purpose, recall that in an [[(∞,1)-topos]] the [[pair]] of [[classes]] of [[n-connected object in an (∞,1)-topos|n-connected morphisms]] and [[n-truncated object in an (∞,1)-category|n-truncated morphisms]] for an [[orthogonal factorization system in an (∞,1)-category|orthogonal factorization system]] for all $n \in \{-2,-1\} \sqcup \mathbb{N} \sqcup \{\infty\}$.

In particular this says that a [[1-morphism]] in an [[(∞,1)-topos]] is [[n-truncated object in an (infinity,1)-category|0-truncated]] precisely if it has the [[right lifting property]] against every morphism that is [[n-connected object of an (infinity,1)-topos|0-connected]].

$\,$

Just for the record:

\begin{lemma}
\label{GammaPreservesnTruncatedMorphisms}\hypertarget{GammaPreservesnTruncatedMorphisms}{}
\textbf{($\Gamma_{sing}$ preserves [[n-truncated object in an (∞,1)-category|n-truncated morphisms]])}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] with $\mathbf{H}_{sing}$ its [[globally equivariant homotopy theory]] from Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}.

Then in particular the functor \eqref{SingularitiesAdjointQuadruple}

\begin{displaymath}
\Gamma_{sing}
  \;\colon\;
  \mathbf{H}_{sing}
  \longrightarrow
  \mathbf{H}
\end{displaymath}
preserves [[n-truncated object in an (∞,1)-category|n-truncated morphisms]] for all $n$.

\end{lemma}
\begin{proof}
By the [[(n-connected, n-truncated) factorization system]] and the [[adjunctions]] in \eqref{SingularitiesAdjointQuadruple} the statement is equivalent to

\begin{displaymath}
Disc_{sing}
  \;\colon\;
  \mathbf{H}
  \longrightarrow
  \mathbf{H}_{sing}
\end{displaymath}
preserving [[n-connected object in an (∞,1)-topos|n-connected morphisms]]. These are [[effective epimorphisms in an (∞,1)-category]] satisfying an extra condition. Both the definition of effective epimorphisms as well as that extra conditions are entirely formulated in terms of [[(∞,1)-limits]] and [[(∞,1)-colimits]] (\href{n-connected+object+of+an+infinity-topos#CharacterizationByDiagonal}{this Prop.}). Since $Disc_{sing}$ is both a left and a right [[adjoint (∞,1)-functor]] by \eqref{SingularitiesAdjointQuadruple} it preserves all these.

\end{proof}
\begin{example}
\label{EsoAndFullToFaithfulFactoriazation}\hypertarget{EsoAndFullToFaithfulFactoriazation}{}
\textbf{(on [[groupoids]] [[(n-connected, n-truncated) factorization system|(0-connected, 0-truncated)]] is [[(eso and full, faithful) factorization system|(eso and full, faithful)]])}

In the [[(∞,1)-topos]] [[∞Groupoids]] the [[n-truncated object in an (infinity,1)-category|1-truncated objects]] are equivalently [[groupoids]] in the sense of [[small categories]] with all [[morphisms]] [[invertible morphism|invertible]]

\begin{displaymath}
Groupoids \simeq \infty Groupoids_{\leq 1} \hookrightarrow \infty Groupoids
\end{displaymath}
Under this identification the [[1-morphisms]] between 1-truncated objects correspond equivalently [[functors]], and we have that these [[1-morphisms]] are

\begin{enumerate}%
\item [[n-truncated object in an (infinity,1)-category|0-truncated]] precisely if they correspond to [[faithful functors]];


\item [[n-connected object of an (infinity,1)-topos|0-connected]] precisely if they correspond to [[essentially surjective functor|essentially surjective]] and [[full functors]].



\end{enumerate}
In particular a morphism of [[delooping]] groupoids

\begin{equation}
\itexarray{
     \mathbf{B} G'
     \\
     \downarrow^{\mathrlap{\mathbf{B} p}}
     \\
     \mathbf{B} G
  }
\label{BGPrimeToBG}\end{equation}
is a 0-connected morphism in $\infty Groupoids$ precisely if the corresponding [[group homomorphism]] $p \colon G' \to G$ is [[surjective function|surjective]].

\end{example}
Therefore one might say ``[[faithful morphism]]'' for every [[n-truncated object in an (infinity,1)-category|0-truncated]] morphism in an [[(∞,1)-topos]]. But the terminology ``faithful'' is used with other meanings, too, and we need to refer to these variants

\begin{defn}
\label{SingularitiesFaithful}\hypertarget{SingularitiesFaithful}{}
\textbf{($Singularities$-faithful morphisms)}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] with $\mathbf{H}_{sing} \coloneqq Sh_\infty\left( Singularities, \mathbf{H}\right)$ its [[globally equivariant homotopy theory]] according to Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}. We say that a morphism $\mathcal{X} \overset{f}{\to} \mathcal{Y}$ in $\mathbf{H}_{sing}$ is \emph{$Singularities$-faithful} if it has the [[right lifting property]] against morphisms of the form

\begin{equation}
\left(
    \itexarray{
      \mathbb{B}G'
      \\
      \downarrow^{\mathrlap{\mathbb{B}p}}
      \\
      \mathbb{B}G
    }
    \phantom{{}^{\mathbb{B}^p}}
  \right)
  \;\in\;
  Singularities \overset{Yoneda}{\hookrightarrow} 
  Sh_\infty\left(
    Singularities,
    \infty Groupoids
  \right)
  \overset{Disc}{\hookrightarrow}
  Sh_\infty\left(
    Singularities,
    \mathbf{H}
  \right)
  \;=\;
  \mathbf{H}_{sing}
\label{Singularities0Connected}\end{equation}
where $p \;\colon\; G' \to G$ is a \emph{[[surjective function|surjective]]} [[group homomorphism]].

\end{defn}
For the case $\mathbf{H} =$ [[∞Groupoids]] this is the definition of \emph{faithful maps} in \hyperlink{Rezk14}{Rezk 14. Prop. 3.4.1}.

\begin{quote}%
It seems that the morphisms \eqref{Singularities0Connected} are not in general 0-connected in $Sh_\infty(Singularities, \infty Groupoids)$. Them being 0-connected should come down to the statement that for $p \colon G' \to G$ a surjective group homomorphism and $H \subset G$ any subgroup, there always is a lift of $H$ to $G'$ and that any two such lifts are conjugate to each other, in $G'$. But already the first condition fails in general, since not every epimorphism of groups is a [[split epimorphism]].

Nevertheless and in any case we have the following, which is all we will need:


\end{quote}
\begin{prop}
\label{coDiscSingOf0TruncatedMorphismsIsSingularitiesFaithful}\hypertarget{coDiscSingOf0TruncatedMorphismsIsSingularitiesFaithful}{}
\textbf{($coDisc_{sing}$ of a 0-truncated morphism is $Singularities$-faithful)}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] with $\mathbf{H}_{sing}$ its [[globally equivariant homotopy theory]] according to Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}.

If a morphism $X \overset{f}{\to} Y$ in $\mathbf{H}$ is [[n-truncated object in an (infinity,1)-category|0-truncated]], then its image under $coDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing}$ is $Singularities$-faithful (Def. \ref{SingularitiesFaithful}).

\end{prop}
\begin{proof}
This follows by the adjunctions \eqref{SingularitiesAdjointQuadruple}, the relations \eqref{ShapeOfOrbifoldSingularity} and the fact \eqref{BGPrimeToBG}:

\begin{displaymath}
\phantom{{}^{\mathbb{B}p}}
  \itexarray{
    \mathbb{B}G' &\longrightarrow& coDisc_{sing}(X)
    \\ 
    {}^{\mathllap{\mathbb{B}p}}\big\downarrow 
      &\nearrow&   
    \big \downarrow^{\mathrlap{ coDisc_{sing}(f) }}
    \\
    \mathbb{B}G
    &\longrightarrow&
    coDisc_{sing}(Y) 
  }
  \phantom{AAAA}  
  \;\Leftrightarrow\;\;
  \phantom{A}
  \phantom{{}^{\mathbb{B}p}}
  \itexarray{
    \Gamma_{sing}\left(\mathbb{B}G'\right) 
      &\longrightarrow& 
    X
    \\ 
    {}^{\mathllap{\Gamma_{sing}\left(\mathbb{B}p\right)}}\big\downarrow 
      &\nearrow&   
    \big \downarrow^{\mathrlap{ f }}
    \\
    \Gamma_{sing}\left(\mathbb{B}G\right)
    &\longrightarrow&
    Y
  }
  \phantom{AA}
  \;\;\simeq\;\;
  \phantom{A}
  \itexarray{
    \mathbf{B} G'
      &\longrightarrow& 
    X
    \\ 
    {}^{\mathllap{\mathbf{B}p}}\big\downarrow 
      &\nearrow&   
    \big \downarrow^{\mathrlap{ f }}
    \\
    \mathbf{B} G
    &\longrightarrow&
    Y
  }
\end{displaymath}
\end{proof}
\begin{defn}
\label{GlobalOrbitCategory}\hypertarget{GlobalOrbitCategory}{}
\textbf{([[global orbit category]])}

Write

\begin{displaymath}
GlobalOrbits
  \;\coloneqq\;
  Singularities^{faith}
\end{displaymath}
for the wide non-[[full sub-(infinity,1)-category]] of $Singularities$ (Def. \ref{CategoryOfSingularities}) with the same [[objects]] $\mathbb{B}G$ but the [[1-morphisms]] required to be 0-truncated as morphisms of $\infty$-groupods, hence to be faithful functors of groupoids (Example \ref{EsoAndFullToFaithfulFactoriazation}), hence to come from \emph{[[injective function|injective]]} [[group homomorphisms]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The category $GlobalOrbits$ in Def. \ref{GlobalOrbitCategory}, when generalized from [[finite groups]] to [[compact Lie groups]], is called

\begin{itemize}%
\item ``$Orb$ version 2'' in \href{orbispace#HenriquesGepner07}{Henriques-Gepner 07}


\item ``$Orb$'' in \hyperlink{Rezk14}{Rezk 14, 4.5}



\end{itemize}
and is \emph{not} the category called ``$Orb$'' in \href{orbispace#Koerschgen16}{Körschgen 16}, \href{orbispace#Schwede17}{Schwede 17}, \href{global+equivariant+homotopy+theory#Schwede18}{Schwede 18} (see Remark \ref{SingularitiesInTheLiterature}).

\end{remark}
\begin{prop}
\label{GorbitCategory}\hypertarget{GorbitCategory}{}
\textbf{([[slice (∞,1)-category|slice]] of [[global orbit category|global orbits]] is $G$-[[orbit category|orbits]])}

For $G$ a [[finite group]], the [[slice (∞,1)-category|slice]] of the [[global orbit category]] from Def. \ref{GlobalOrbitCategory} over the object $\mathbb{B}G$ is [[equivalence of (∞,1)-categories|equivalent]] to the $G$-[[orbit category]]

\begin{displaymath}
GlobalOrbits_{/\mathbb{B}G}
  \;\simeq\;
  G Orbits
\end{displaymath}
\end{prop}
\begin{defn}
\label{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos}\hypertarget{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos}{}
\textbf{($G$-[[equivariant homotopy theory]] of a [[cohesive (∞,1)-topos]])}

For $\mathbf{H}$ a [[cohesive (∞,1)-topos]] and $G$ a [[finite group]] we say that the \emph{$G$-[[equivariant homotopy theory]] of $\mathbf{H}$} is the [[(∞,1)-presheaf (∞,1)-topos]] on the $G$-[[orbit category]] (Def. \ref{GorbitCategory}) over $\mathbf{H}$:

\begin{displaymath}
\begin{aligned}
    \mathbf{H}_G  
    &\coloneqq\;
    Sh_\infty\big( G Orbits , \mathbf{H} \big)
    \\
    & \simeq
    Sh_\infty\big( GlobalOrbits_{/\mathbb{B}G} , \mathbf{H} \big)
    \\
    & \simeq
    Sh_\infty\big( GlobalOrbits , \mathbf{H} \big)_{/\mathbb{B}G}
  \end{aligned}
\end{displaymath}
On the right we are displaying immediate [[equivalence of (∞,1)-categories|equivalences]], the first by Prop. \ref{GorbitCategory}, the second by the general slicing behaviour of $\infty$-toposes (\href{slice+homotopy-topos#SlicingCommutesWithPassingToPresheaves}{this Prop.}).

\end{defn}
The relation between the [[global homotopy theory]] $\mathbf{H}_{sing}$ (Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}) and the $G$-[[equivariant homotopy theory]] $\mathbf{H}_G$ (Def. \ref{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos}) is the topic of \hyperlink{Rezk14}{Rezk 14}:

\begin{defn}
\label{NormalSubgroupClassifier}\hypertarget{NormalSubgroupClassifier}{}
\textbf{([[normal subgroup]] classifier, \hyperlink{Rezk14}{Rezk 14, 4.1})}

For $\mathbf{H}$ a [[cohesive (∞,1)-topos]], let

\begin{displaymath}
\mathcal{N}
  \;\in\;
  Sh\big(Singularities, Sets \big)
  \hookrightarrow
  Sh_\infty\big(Singularities, \infty Groupoids \big)
  \overset{}{\hookrightarrow}
  Sh_\infty\big( 
    Singularities, \mathbf{H}
  \big)
\end{displaymath}
be the presheaf which to any [[finite group]] $\mathbb{B}G$ assigns the [[set]] (i.e. [[0-groupoid]]) of [[normal subgroups]] of $G$.

\end{defn}
\begin{prop}
\label{SingularitiesFaithfulSliceOverNormalSubgroupClassifier}\hypertarget{SingularitiesFaithfulSliceOverNormalSubgroupClassifier}{}
\textbf{(\hyperlink{Rezk14}{Rezk 14})}

For $\mathbf{H}$ an [[(∞,1)-topos]], with $\mathbf{H}_{sing} \coloneqq Sh_\infty\big( Singularities, \mathbf{H}\big)$ its [[global equivariant homotopy theory]] according to Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}, there is an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\itexarray{
    Sh_\infty\big( GlobalOrbits, \mathbf{H}\big)
    &\overset{\simeq}{\longrightarrow}&
    \left(
      Sh_\infty\big( Singularities, \mathbf{H}\big)_{/\mathcal{N}}
    \right)_{\text{singfaith}}
    \;=\;
    \left(
    \left( 
      \mathbf{H}_{sing}
    \right)_{/\mathcal{N}}
    \right)_{singfaith}
  }
\end{displaymath}
between the [[(∞,1)-presheaf (∞,1)-category]] on the [[global orbit category]] according to Def. \ref{GlobalOrbitCategory} and the [[full sub-(∞,1)-category]] of the [[slice (∞,1)-topos]] of $\mathbf{H}_{sing}$ over the normal subgroup classifier $\mathcal{N}$ (Def. \ref{NormalSubgroupClassifier}) on those morphisms to $\mathcal{N}$ which are $Singularities$-faithful according to Def. \ref{SingularitiesFaithful}.

Moreover, if $G$ is a [[finite group]] then slicing over $\mathbb{B}G$ this yields an equivalence

\begin{displaymath}
\itexarray{
    \mathbf{H}_G
    \;\simeq\;
    \left(
      \left( 
        \mathbf{H}_{sing}
      \right)_{/\mathbb{B}G}
    \right)_{singfaith}
  }
\end{displaymath}
between the $G$-[[equivariant homotopy theory]] $\mathbf{H}_G$ (Def. \ref{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos}) and the [[full sub-(∞,1)-category]] on the $Singularities$-faithful objects of the [[slice (∞,1)-category|slice]] of the [[global homotopy theory]] $\mathbf{H}_{sing}$ (Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}) over $\mathbb{B}G$.

\end{prop}
\hypertarget{OnOrbifolds}{}\subsubsection*{{Orbifolds}}\label{OnOrbifolds}

With [[cohesion|cohesive]] [[globally equivariant homotopy theory]] in place, there is now an elegant definition of [[orbifolds]] (Def. \ref{Orbifold} below) which, being fully \emph{[[synthetic mathematics|synthetic]]} makes manifest good defining properties of the [[category]] of [[orbifolds]] (Remark \ref{OrbifoldsInGloballyEquivariantStillEquivalentToGeometricGroupoids} below).

Not directly evident is that under passing to [[shape modality|shapes]] (underlying [[globally equivariant homotopy theory|globally equivariant]] [[geometrically discrete ∞-groupoids]]) this definition is compatible with the standard embedding of [[G-spaces]] into [[globally equivariant homotopy theory]]. That this indeed is the case is confirmed in Example \ref{GlobalHomotopyQuotientsOfSmoothManifolds} below.

$\,$

\begin{defn}
\label{Orbifold}\hypertarget{Orbifold}{}
\textbf{([[orbifold]]})

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] with $\mathbf{H}_{sing}$ its corresponding [[globally equivariant homotopy theory]] according to Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}.

For $G \in Grp(\mathbf{H})$ a [[finite group]] we say that an \emph{orbifold with singularities ([[isotropy groups]]) in $G$} is an [[object]] of the [[slice (∞,1)-topos]]

\begin{displaymath}
\mathcal{X}
  \;\in\;
  \left(\mathbf{H}_{sing}\right)_{/\mathbb{B}G}
\end{displaymath}
which is

\begin{enumerate}%
\item [[n-truncated object in an (infinity,1)-category|0-truncated]] (as an object of the [[slice (infinity,1)-topos|slice]]);


\item $\sharp_{sing}$-[[modal object|modal]] (hence ``$Singularity$-[[codiscrete object|codiscrete]]'').



\end{enumerate}
Given such an orbifold, we say that its \emph{underlying [[geometric stack|geometric groupoid]]} is its $Singularities$-[[flat modality|flat]] aspect:

\begin{equation}
\flat_{sing}\left( \mathcal{X}\right)
  \;\in\;
  \mathbf{H}_{/\mathbf{B}G}
  \overset{Disc_{sing}}{\hookrightarrow}
  \left(
    \mathbf{H}_{sing}
  \right)_{/\mathbb{B}G}
\label{OrbifoldUnderlyingGroupoid}\end{equation}
where on the right we used \eqref{ShapeOfOrbifoldSingularity}.

If $\mathbf{H}$ is moreover [[differential cohesion|differentially cohesive]] and $V \in Grp(\mathbf{H})$ is a [[∞-group|group object]], then an orbifold $\mathcal{X}$ is called a \emph{$V$-orbifold} if its underlying [[geometric stack|geometric groupoid]] $\flat_{sing}\left( \mathcal{X}\right)$ \eqref{OrbifoldUnderlyingGroupoid} is a [[V-manifold]].

\end{defn}
\begin{example}
\label{GlobalHomotopyQuotientOrbifold}\hypertarget{GlobalHomotopyQuotientOrbifold}{}
\textbf{(global [[homotopy quotient]]-[[orbifolds]])}

Let $X \in \mathbf{H}$ be [[n-truncated object in an (infinity,1)-category|0-truncated]] and equipped with a $G$-[[∞-action|action]], with [[homotopy quotient]] $(X \sslash \mathbf{B}G \to \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}$. Then

\begin{displaymath}
\mathcal{X}
  \;\coloneqq\;
  coDisc_{sing}
  \left(
    \itexarray{
      X \sslash G
      \\
      \downarrow
      \\
      \mathbf{B}G
    }
  \right)
  \;\simeq\;
  \left(
    \itexarray{
      coDisc_{sing}\left( X\sslash G\right)
      \\
      \downarrow
      \\
      \mathbb{B}G
    }
  \right)
\end{displaymath}
is an [[orbifold]] with isotropy groups in $G$, according to Def. \ref{Orbifold}. Here on the right we identified the slice using Prop. \ref{SingularitiesAsCoDisc}.

\end{example}
As a further special case:

\begin{example}
\label{GlobalHomotopyQuotientsOfSmoothManifolds}\hypertarget{GlobalHomotopyQuotientsOfSmoothManifolds}{}
\textbf{(global [[homotopy quotient]]-[[orbifolds]] of [[smooth manifolds]])}

Let $\mathbf{H} \coloneqq$ [[Smooth∞Groupoids]]. For $G$ a [[finite group]], let $X$ be a [[smooth manifold]] equipped with a smooth $G$-[[action]]. Under the canonical embedding into $\mathbf{H}$ the corresponding [[action groupoid]] is a 0-truncated object

\begin{displaymath}
\left(
  \itexarray{
    X \sslash G
    \\
    \downarrow
    \\
    \mathbf{B}G
  }
  \right)
  \;\in\;
  Smooth\infty Groupoids_{/\mathbf{B}G}
\end{displaymath}
as in Example \ref{GlobalHomotopyQuotientOrbifold}, and hence its $Singularities$-[[codiscrete object|codiscrete]] image is a [[orbifold]] in the sense of Def. \ref{Orbifold}:

\begin{displaymath}
\mathcal{X}
  \;\coloneqq\;
  \left(
  \itexarray{
    coDisc_{sing}\big(X \sslash G \big)
    \\
    \downarrow
    \\
    \mathbb{B}G
  }
  \right)
  \;\in\;
  \left( 
    Smooth\infty Groupoids_{sing}
  \right)_{/\mathbb{B}G}
  \,.
\end{displaymath}
We claim that the [[shape modality|shape]] of this orbifold in plain [[global equivariant homotopy theory]] coincides with the global equivariant homotopy type associated with the [[G-space]] underlying $X$

\begin{displaymath}
&#643; coDisc_{sing}\left( 
   X \sslash G
  \right)
  \;\simeq\;
  \Delta_G\big( X \big)
  \;\in\;
  Sh_\infty\big( 
    Singularities, \infty Groupoids
  \big)
  \,,
\end{displaymath}
where on the right $\Delta_G$ is as in \hyperlink{Rezk14}{Rezk 14, 3.2}

\end{example}
\begin{proof}
The key point is that the assumption of 0-truncation of $\mathcal{X}$ and the restriction to [[finite groups|finite]] (hence [[discrete groups|discrete]]) groups ensures that $coDisc_{sing}$ forms the correct [[fixed point]] [[sheaves]], whose separate [[shape modality|shape]]/[[geometric realization]] then coincides with the relevant fixed point spaces of $X$.

In detail, as $\infty$-groupoid-valued presheaves on the product site [[CartSp]] $\times Singularities$ we have

\begin{displaymath}
\begin{aligned}
    coDisc_{sing}\big( X\sslash G\big)
    \;\colon\;
    \mathbb{R}^n \times \mathbb{B}K
    & \mapsto\;
    \infty Groupoids\big(
      \mathbf{B}K,
      \underset{
        \in 1Groupoid
      }{
        \underbrace{  
          \underset{\in Set}{
            \underbrace{C^\infty(\mathbb{R}^n, X)}
          } 
        }
      }
      \sslash G
    \big)  
    \\
    & \simeq \;
    \underset{
      \simeq Grpd(Set)
    }{
    \underbrace{
      1 Groupoids
     }
     }
     \big(
      \mathbf{B}K, 
      C^\infty(\mathbb{R}^n, X) \sslash G
    \big)
    \\
    & \simeq \;
    \underset{\phi \in Groups(K,G)}{\bigsqcup} 
      C^\infty(\mathbb{R}^n, X^{\phi(K)}) \sslash G
  \end{aligned}
\end{displaymath}
where in the first step we used the [[adjunction]] $(\Gamma_{sing} \dashv coDisc_{sing})$ as in Prop. \ref{SingularitiesAsCoDisc}. Hence as smooth $\infty$-groupoid valued presheaves on just $Singularities$ this is

\begin{displaymath}
\Big(
  coDisc_{sing}\big( 
    X \sslash G
  \big)
  \;\colon\;
  \mathbb{B}K
  \;\mapsto\;
    \underset{\phi \in Groups(K,G)}{\bigsqcup} 
    X^{\phi(K)} \sslash G  
  \Big)
  \;\in\;
  Sh_\infty\big( Singularities, Smooth\infty Groupoids \big)
  \,.
\end{displaymath}
Now [[shape modality|shape]]  is [[left adjoint]], hence preserves the [[coproducts]] and the [[homotopy quotient]] by $G$ and finally also $G$ itself ($G$ being [[discrete group|discrete]], and  preserving the point (the [[terminal object]]), by [[cohesion]]), so that in conclusion

\begin{displaymath}
\Big(
  &#643;
  coDisc_{sing}\big( 
    X \sslash G
  \big)
  \;\colon\;
  \mathbb{B}K
  \;\mapsto\;
    \underset{\phi \in Groups(K,G)}{\bigsqcup} 
     &#643; \left(X^{\phi(K)}\right) \sslash G  
  \Big)
  \;\in\;
  Sh_\infty\big( Singularities, \infty Groupoids \big)
  \,.
\end{displaymath}
But this is exactly the formula for $\Delta_G (X)$, as in \hyperlink{Rezk14}{Rezk 14, 3.2}.

\end{proof}
\begin{prop}
\label{OrbifoldsInEquivariantHomotopyTheory}\hypertarget{OrbifoldsInEquivariantHomotopyTheory}{}
\textbf{([[orbifolds]] are in [[cohesion|cohesive]] [[equivariant homotopy theory]])}

An [[orbifold]] $\mathcal{X}$ with isotropy groups in $G$, according to Def. \ref{Orbifold} is $Singularities$-faithful over $\mathbb{B}G$ (Def. \ref{SingularitiesFaithful}) and hence inside the inclusion (from Prop. \ref{SingularitiesFaithfulSliceOverNormalSubgroupClassifier}) of the $G$-[[equivariant homotopy theory]] of $\mathbf{H}$ (Def. \ref{GEquivariantHomotopyTheoryOfCohesiveInfinityTopos}) into the [[globally equivariant homotopy theory]] of $\mathbf{H}$:

\begin{displaymath}
\mathcal{X}
  \;\in\;
  \mathbf{H}_G
  \;\hookrightarrow\;
  \left(
    \mathbf{H}_{sing}
  \right)_{/\mathbb{B}G}
\end{displaymath}
\end{prop}
\begin{proof}
By Prop. \ref{SingularitiesFaithfulSliceOverNormalSubgroupClassifier} we need to check that $\mathcal{X} \to \mathbb{B}G$ is $Singularities$-faithful.

Now, by the first defining assumption on $\mathcal{X}$ (Def. \ref{Orbifold}) and by Lemma \ref{GammaPreservesnTruncatedMorphisms}, we have that $\Gamma_{sing}(\mathcal{X} \to \mathbb{B}G)$ is [[n-truncated object in an (infinity,1)-category|0-truncated]]. By [[cohesion]] \eqref{SingularitiesAdjointQuadruple} we have $coDisc_{sing} \circ \Gamma_{sing} \;\simeq\; Id$ and hence $\mathcal{X} \to \mathbb{B}G$ is the image under $coDisc_{sing}$ of a 0-truncated morphism. With this the statement follows by Prop. \ref{coDiscSingOf0TruncatedMorphismsIsSingularitiesFaithful}.

\end{proof}
\begin{remark}
\label{OrbifoldsInGloballyEquivariantStillEquivalentToGeometricGroupoids}\hypertarget{OrbifoldsInGloballyEquivariantStillEquivalentToGeometricGroupoids}{}
\textbf{([[orbifolds]] inside [[globally equivariant homotopy theory]] are still [[equivalence of (infinity,1)-categories|equivalent]] to [[cohesive (infinity,1)-topos|cohesive groupoids]])}

Since $coDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing}$ is a [[full sub-(∞,1)-category]]-inclusion, the [[(2,1)-category]] of $V$-[[orbifolds]] inside $\mathbf{H}_{sing}$ according to Def. \ref{Orbifold} is equivalent to its pre-image in $\mathbf{H}$, hence will coincide, for suitable choices of $\mathbf{H}$ and $V \in Grp(\mathbf{H})$, with traditional definition of [[(2,1)-categories]] of orbifolds regarded as certain [[geometric stacks|geometric groupoids]]. But by embedding this into the larger [[global homotopy theory]] $\mathbf{H}_{sing}$ of $\mathbf{H}$ more general [[coefficient]]-objects for orbifold cohomology become available, and this brings in the previously missing [[Bredon cohomology|Bredon]]-[[equivariant cohomology]]-aspect of orbifold cohomology.

\end{remark}
$\,$

\hypertarget{orbifold_cohomology}{}\subsubsection*{{Orbifold cohomology}}\label{orbifold_cohomology}

With [[orbifolds]] properly realized in [[cohesion|cohesive]] [[globally equivariant homotopy theory]] by the \hyperlink{OnOrbifolds}{above}, the proper definition of \emph{orbifold cohomology} is now immediate, by the general logic of [[cohomology]] in [[(∞,1)-toposes]], this is Def. \ref{OrbifoldCohomology} below.

What needs checking is that this definition, in the special case that the [[coefficient]] object is [[geometrically discrete infinity-groupoid|geometrically discrete]] reproduces [[Bredon cohomology|Bredon]] [[equivariant cohomology]] of the underlying bare [[homotopy type]] of the orbifold. But this follows readily with the general considerations above, this is Example \ref{OrbifoldCohomologyWithGeometricallyDiscreteCoefficients} below.

$\,$

\begin{defn}
\label{OrbifoldCohomology}\hypertarget{OrbifoldCohomology}{}
\textbf{(orbifold cohomology)}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] and write $\mathbf{H}_{sing}$ for its [[globally equivariant homotopy theory]] as in Prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}.

Let $G$ be a [[finite group]] and consider an [[orbifold]] with [[isotropy groups]]/[[singularities]] in $G$, according to Def. \ref{Orbifold}:

\begin{displaymath}
\mathcal{X} \;\in\; \left( \mathbf{H}_{sing} \right)
  \,.
\end{displaymath}
Then for

\begin{displaymath}
\mathcal{A} \;\in\; \left( \mathbf{H}_{sing} \right)
\end{displaymath}
any other object (\emph{not} necessarily itself an orbifold, and typically far from being so) the \emph{orbifold cohomology} of $\mathcal{X}$ with [[coefficients]] in $\mathcal{A}$ is the [[cohomology]] as given by the ambient [[(∞,1)-topos]], hence

\begin{displaymath}
H\big(
    \mathcal{X},
    \mathcal{A}
  \big)
  \;\coloneqq\;
  \left(
    \mathbf{H}_{sing}
  \right)_{/\mathbb{B}G}
  \big( \mathcal{X}, \mathcal{A}\big)
  \,.
\end{displaymath}
\end{defn}
We check the two desiderata for a good definition of orbifold cohomology discussed \hyperlink{Idea}{above}:

\begin{enumerate}%
\item It is clear that on objects in the inclusion $Disc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing}$ orbifold cohomology reduces to geometric cohomology on $\mathbf{H}$.


\item In the other extreme, for orbifold cohomology with geometrically discrete coefficients, we check that we re-obtain [[Bredon cohomology|Bredon]] [[equivariant cohomology]] of the underlying [[G-spaces]]:



\end{enumerate}
\begin{example}
\label{OrbifoldCohomologyWithGeometricallyDiscreteCoefficients}\hypertarget{OrbifoldCohomologyWithGeometricallyDiscreteCoefficients}{}
\textbf{([[orbifold cohomology]] with [[geometrically discrete infinity-groupoid|geometrically discrete]] [[coefficients]] is $G$-[[equivariant cohomology]])}

Let $\mathbf{H} =$ [[Smooth∞Groupoids]] and consider a [[smooth manifold]] $X$ equipped with smooth [[action]] by a [[finite group]] $G$, regarded as an [[orbifold]] as in Example \ref{GlobalHomotopyQuotientsOfSmoothManifolds}:

\begin{displaymath}
\mathcal{X}
  \;\coloneqq\;
  \left(
  \itexarray{
    coDisc_{sing}\big( X \sslash G \big)
    \\
    \downarrow
    \\
    \mathbb{B}G
  }
  \right)
  \;\in\;
  \Big( 
    Smooth \infty Groupoids_{sing}
  \Big)_{/\mathbb{B}G}
  \,.
\end{displaymath}
Let moreover

\begin{displaymath}
\mathcal{A}
  \;\in\;
  \Big( 
    \infty Groupoids
  \Big)_{G}
  \overset{Disc}{\hookrightarrow}
  \Big( 
    Smooth \infty Groupoids
  \Big)_{G}
  \hookrightarrow
  \Big( 
    Smooth \infty Groupoids_{sing}
  \Big)_{/\mathbb{B}G}
\end{displaymath}
be any [[geometrically discrete infinity-groupoid|geometrically discrete]] [[coefficient]] object in $G$-[[equivariant homotopy theory]], included into the [[slice (infinity,1)-topos|slice]] of the [[global equivariant homotopy theory]] via Prop. \ref{GorbitCategory}.

Then the [[orbifold cohomology]] of $\mathcal{X}$ with [[coefficients]] in $\mathcal{A}$, according to Def. \ref{OrbifoldCohomology}, coincides with the [[Bredon cohomology|Bredon]] $G$-[[equivariant cohomology]] of the [[G-space]] underlying $X$:

\begin{displaymath}
H\big(
   \mathcal{X}, Disc(\mathcal{A})
  \big)
  \;\simeq\;
  H_G\big( 
    X, \mathcal{A}
  \big)
  \,.
\end{displaymath}
\end{example}
\begin{proof}
We compute

\begin{displaymath}
\begin{aligned}
    \left(Smooth \infty Groupoids_{sing}\right)_{/\mathbb{B}G}
    \left(
      \mathcal{X}, Disc(\mathcal{A})
    \right)
    & \simeq
    Smooth \infty Groupoids_G
    \left(
      \mathcal{X}, Disc(\mathcal{A})
    \right)    
    \\
    & \simeq
    \infty Groupoids_G
    \left(
      &#643;_{sing}\mathcal{X}, \mathcal{A}
    \right)    
  \end{aligned}
\end{displaymath}
where the first step is by Prop. \ref{OrbifoldsInEquivariantHomotopyTheory}, while the second is by the [[cohesion]] [[adjunction]] $&#643; \dashv Disc$ for [[Smooth∞Groupoids]]. By Example \ref{GlobalHomotopyQuotientsOfSmoothManifolds} we have that $&#643; \mathcal{X} \in \infty Groupoids_G$ is indeed the [[G-space]] $X$ regarded in $G$-[[equivariant homotopy theory]].

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

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


\item [[Gromov-Witten theory]]



\end{itemize}
History

According to \hyperlink{Abramovich05}{Abramovich 05, p. 42}:

\begin{quote}%
On December 7, 1995 [[Maxim Kontsevich]] delivered a history-making lecture at Orsay, titled \emph{String Cohomology}. This is what is now know, after \hyperlink{ChenRuan00}{Chen-Ruan 00}, as \emph{orbifold cohomology}, Kontsevich's lecture notes described the orbifold and [[quantum cohomology]] of a global quotient [[orbifold]]. Twisted sectors, the age grading, and a version of orbifold stable maps for global quotients are all there.


\end{quote}
The same lecture also introduced \emph{[[motivic integration]]}.

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

Discussion of orbifold cohomology in the context of [[Bredon cohomology|Bredon]]/[[global equivariant homotopy theory|global]] [[equivariant cohomology]] includes

\begin{itemize}%
\item [[Dorette Pronk]], [[Laura Scull]], \emph{Translation Groupoids and Orbifold Bredon Cohomology}, Canad. J. Math. 62(2010), 614-645 (\href{https://arxiv.org/abs/0705.3249}{arXiv:0705.3249}, \href{https://doi.org/10.4153/CJM-2010-024-1}{doi:10.4153/CJM-2010-024-1})


\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]]}, New Mathematical Monographs, 34 Cambridge University Press, 2018 (\href{https://arxiv.org/abs/1802.09382}{arXiv:1802.09382})



\end{itemize}
related to results of

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


\item [[Charles Rezk]], \emph{[[Global Homotopy Theory and Cohesion]]}, 2014



\end{itemize}
Discussion of cohomology of [[inertia orbifolds]] is due to

\begin{itemize}%
\item Weimin Chen, [[Yongbin Ruan]], \emph{A New Cohomology Theory for Orbifold}, Commun. Math. Phys. 248 (2004) 1-31 (\href{http://arxiv.org/abs/math/0004129}{arXiv:math/0004129})

\end{itemize}
See also

\begin{itemize}%
\item [[Dan Abramovich]], \emph{Lectures on Gromov-Witten invariants of orbifolds} (\href{http://arxiv.org/abs/math/0512372}{arXiv:math/0512372})

\end{itemize}
[[!redirects orbifold cohomologies]]

[[!redirects orbifold cohomology group]] [[!redirects orbifold cohomology groups]]

[[!redirects orbispace cohomology]] [[!redirects orbispace cohomologies]]

[[!redirects orbispace cohomology group]] [[!redirects orbispace cohomology groups]]



\end{document}