\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*{infinity-action} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra} [[!include higher algebra - 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{NotionsInHigherRepresentationTheory}{Notions in higher representation theory}\dotfill \pageref*{NotionsInHigherRepresentationTheory} \linebreak \noindent\hyperlink{Invariants}{Invariants}\dotfill \pageref*{Invariants} \linebreak \noindent\hyperlink{Quotients}{Coinvariants / Quotients}\dotfill \pageref*{Quotients} \linebreak \noindent\hyperlink{CartesianClosedMonoidalStructure}{Conjugation actions}\dotfill \pageref*{CartesianClosedMonoidalStructure} \linebreak \noindent\hyperlink{internal_object_of_homomorphisms}{Internal object of homomorphisms}\dotfill \pageref*{internal_object_of_homomorphisms} \linebreak \noindent\hyperlink{stabilizer_groups}{Stabilizer groups}\dotfill \pageref*{stabilizer_groups} \linebreak \noindent\hyperlink{Linearization}{Linearization}\dotfill \pageref*{Linearization} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{ExamplesPermutationRepresentations}{Discrete group actions on sets}\dotfill \pageref*{ExamplesPermutationRepresentations} \linebreak \noindent\hyperlink{group_actions_in_an_topos}{$\infty$-group actions in an $\infty$-topos}\dotfill \pageref*{group_actions_in_an_topos} \linebreak \noindent\hyperlink{TrivialAction}{Trivial action}\dotfill \pageref*{TrivialAction} \linebreak \noindent\hyperlink{fundamental_action}{Fundamental action}\dotfill \pageref*{fundamental_action} \linebreak \noindent\hyperlink{adjoint_action}{Adjoint action}\dotfill \pageref*{adjoint_action} \linebreak \noindent\hyperlink{AutomorphismAction}{Automorphism action}\dotfill \pageref*{AutomorphismAction} \linebreak \noindent\hyperlink{ConjugationActions}{Conjugation actions}\dotfill \pageref*{ConjugationActions} \linebreak \noindent\hyperlink{GeneralCovariance}{General covariance}\dotfill \pageref*{GeneralCovariance} \linebreak \noindent\hyperlink{SemidirectProductGroups}{Semidirect product groups}\dotfill \pageref*{SemidirectProductGroups} \linebreak \noindent\hyperlink{GModules}{$G$-Modules}\dotfill \pageref*{GModules} \linebreak \noindent\hyperlink{ExamplesActionsInASlice}{Actions in a slice}\dotfill \pageref*{ExamplesActionsInASlice} \linebreak \noindent\hyperlink{codiscretization_of_actions}{Co-Discretization of Actions}\dotfill \pageref*{codiscretization_of_actions} \linebreak \noindent\hyperlink{infinitesimally_actions_of_algebroids}{Infinitesimally: actions of $L_\infty$-algebroids}\dotfill \pageref*{infinitesimally_actions_of_algebroids} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{model_category_presentation}{Model category presentation}\dotfill \pageref*{model_category_presentation} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ReferencesForDiscreteGeometry}{For discrete geometry}\dotfill \pageref*{ReferencesForDiscreteGeometry} \linebreak \noindent\hyperlink{ForActionsOfTopologicalGroups}{For actions of topological groups}\dotfill \pageref*{ForActionsOfTopologicalGroups} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The notion of \emph{$\infty$-action} is the notion of \emph{[[action]]} ([[module]]/[[representation]]) in [[homotopy theory]]/[[(∞,1)-category theory]], from [[algebra]] to [[higher algebra]]. Notably a [[monoid object in an (∞,1)-category]] $A$ may \emph{act} on another object $N$ by a [[morphism]] $A \otimes N \to N$ which satisfies an action property up to [[coherence law|coherent]] higher [[homotopy]]. If the $\infty$-action is suitably linear in some sense, this is also referred to as \emph{[[∞-representation]]}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} We discuss the actions of [[∞-groups]] in an [[(∞,1)-topos]], following \hyperlink{NSS}{NSS}. (For [[groupoid ∞-actions]] see there.) Let $\mathbf{H}$ be an [[(∞,1)-topos]]. Let $G \in Grp(\mathbf{H})$ be an [[group object in an (∞,1)-category]] in $\mathbf{H}$, hence a homotopy-[[simplicial object]] on $\mathbf{H}$ of the form \begin{displaymath} \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} G \stackrel{\longrightarrow}{\longrightarrow} * \right) \end{displaymath} satisfying the groupoidal [[Segal conditions]]. hence an \emph{[[∞-group]]}. \begin{defn} \label{}\hypertarget{}{} An \textbf{action} (or \emph{$\infty$-action}, for emphasis) of $G$ on an object $V \in \mathbf{H}$ is a [[groupoid object in an (∞,1)-category]] which is equivalent to one of the form \begin{displaymath} \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} V \times G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} V \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} V \right) \end{displaymath} such that the projection maps \begin{displaymath} \itexarray{ \cdots &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}& V \times G \times G &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}& V \times G &\stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}}& V \\ && \downarrow && \downarrow && \downarrow \\ \cdots &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}& G \times G &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}& G &\stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}}& * } \end{displaymath} constitute a morphism of groupoid objects $V\sslash G \to *\sslash G$. The [[(∞,1)-category]] of such actions is the slice of groupoid objects over $*\sslash G$ on these objects. \end{defn} There is an equivalent formulation which does not invoke the notion of [[groupoid object in an (∞,1)-category]] explicitly. This is based on the fundamental fact, discussed at \emph{[[∞-group]]}, that [[delooping]] constitutes an [[equivalence of (∞,1)-categories]] \begin{displaymath} \mathbf{B} : Grp(\mathbf{H}) \to \mathbf{H}^{*/}_{\geq 1} \,. \end{displaymath} form [[group objects in an (∞,1)-category]] to the [[(∞,1)-category]] of [[connected object in an (∞,1)-topos|connected]] [[pointed objects]] in $\mathbf{H}$. \begin{prop} \label{}\hypertarget{}{} Every $\infty$-action $\rho : V \times G \to V$ has a classifying morphism $\mathbf{c}_\rho : V \sslash G \to \mathbf{B}G$ in that there is a [[fiber sequence]] \begin{displaymath} \itexarray{ V \\ \downarrow \\ V \sslash G &\stackrel{\overline{\rho}}{\to}& \mathbf{B}G } \end{displaymath} such that $\rho$ is the $G$-action on $V$ regarded as the corresponding $G$-[[principal ∞-bundle]] modulated by $\overline{\rho}$. \end{prop} This allows to characterize $\infty$-actions in the following convenient way. See (\hyperlink{NSS}{NSS}) for a detailed discussion. \begin{defn} \label{GActionByFiberSequence}\hypertarget{GActionByFiberSequence}{} For $V \in \mathbf{H}$ an object, a $G$-$\infty$-action $\rho$ on $V$ is a [[fiber sequence]] in $\mathbf{H}$ of the form \begin{displaymath} \itexarray{ V &\to& V \sslash G \\ && \downarrow^{\mathrlap{\overline{\rho}}} \\ && \mathbf{B}G } \,. \end{displaymath} The [[(∞,1)-category]] of $G$-actions in $\mathbf{H}$ is the [[slice (∞,1)-topos]] of $\mathbf{H}$ over $\mathbf{B}G$: \begin{displaymath} Act_{\mathbf{H}}(G) \coloneqq \mathbf{H}_{/\mathbf{B}G} \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} A $\rho \in Act_{\mathbf{H}}(G)$ corresponds to a morphism denoted $\overline{\rho} : V\sslash G \to \mathbf{B}G$ in $\mathbf{H}$ hence to an object $\overline{\rho} \in \mathbf{H}_{/\mathbf{B}G}$. A morphism $\phi : \rho_1 \to \rho_2$ in $Act_{\mathbf{H}}(G)$ corresponds to a diagram \begin{displaymath} \itexarray{ V_1 \sslash G &&\stackrel{}{\to}&& V_2 \sslash G \\ & {}_{\mathllap{\overline{\rho_1}}}\searrow && \swarrow_{\mathrlap{\overline{\rho_2}}} \\ && \mathbf{B}G } \end{displaymath} in $\mathbf{H}$. \end{remark} \begin{remark} \label{}\hypertarget{}{} The bundle $\overline{\rho}$ in def. \ref{GActionByFiberSequence} is the universal $\rho$-[[associated infinity-bundle|associated]] $V$-[[fiber ∞-bundle]]. \end{remark} \begin{remark} \label{DefinitionInTypeTheory}\hypertarget{DefinitionInTypeTheory}{} In the form of def. \ref{GActionByFiberSequence} $\infty$-actions have a simple formulation in the [[internal language]] of [[homotopy type theory]]: a $G$-action on $V$ is simply a [[dependent type]] over $\mathbf{B}G$ with fiber $V$: \begin{displaymath} * : \mathbf{B}G \vdash V(*) : Type \,. \end{displaymath} \end{remark} \hypertarget{NotionsInHigherRepresentationTheory}{}\subsection*{{Notions in higher representation theory}}\label{NotionsInHigherRepresentationTheory} We discuss some basic [[representation theory|representation theoretic]] notions of $\infty$-actions. In summary, for $\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type$ an action of $G$ on $V$, we have \begin{itemize}% \item the [[dependent sum]] \begin{displaymath} \vdash \sum_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \end{displaymath} is the [[quotient]] $V\sslash G$ of $V$ by $G$; \item the [[dependent product]] \begin{displaymath} \vdash \prod_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \end{displaymath} is the collection of [[invariants]] ([[homotopy fixed points]]) of the action. \end{itemize} And for $V_1, V_2$ two actions we have \begin{itemize}% \item the [[dependent product]] over the [[dependent type|dependent]] [[function type]] \begin{displaymath} \vdash \prod_{\mathbf{c} : \mathbf{B}G} (V_1(\mathbf{c}) \to V_2(\mathbf{c})) : Type \end{displaymath} is the collection of $G$-[[homomorphisms]] ($G$-[[equivariance|equivariant]] maps); \item the [[dependent sum]] over the [[dependent type|dependent]] [[function type]] \begin{displaymath} \vdash \sum_{\mathbf{c} : \mathbf{B}G} (V_1(\mathbf{c}) \to V_2(\mathbf{c})) : Type \end{displaymath} is the [[quotient]] of \emph{all} functions $V_1 \to V_2$ by the [[conjugation action]] of $G$. \end{itemize} \hypertarget{Invariants}{}\subsubsection*{{Invariants}}\label{Invariants} \begin{defn} \label{TypeOfInvariants}\hypertarget{TypeOfInvariants}{} The [[invariants]] ([[homotopy fixed points]]) of a $G$-$\infty$-action $\rho$ are the [[sections]] of the morphism $V \sslash G \to \mathbf{B}G$, \begin{displaymath} Invariants(V) = \prod_{\mathbf{B}G \to *} (V \sslash G \to \mathbf{B}G) \,, \end{displaymath} where $\prod_{\mathbf{B}G \to *} : \mathbf{H}_{/\mathbf{B}G} \to \mathbf{H}$ is the [[direct image]] of the [[base change geometric morphism]]. In [[homotopy type theory]] [[syntax]] for \begin{displaymath} \mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type \end{displaymath} an action as in remark \ref{DefinitionInTypeTheory}, its type of [[invariants]] is the [[dependent product]] \begin{displaymath} \vdash \prod_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} This is the [[internal limit]] in $\mathbf{H}$ of the [[internal diagram]] \begin{displaymath} \rho \colon \mathbf{B}G \to Type \,. \end{displaymath} See at \emph{\href{internal+%28co-%29limit#BorelConstructionHomotopyQuotients}{internal limit -- Examples -- Homotopy Invariants}}. \end{remark} \hypertarget{Quotients}{}\subsubsection*{{Coinvariants / Quotients}}\label{Quotients} From def. \ref{GActionByFiberSequence} we read off: \begin{defn} \label{}\hypertarget{}{} The [[quotient]] of a $G$-action \begin{displaymath} \mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type \end{displaymath} is the [[dependent sum]] \begin{displaymath} \vdash \sum_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} This is the [[internal colimit]] in $\mathbf{H}$ of the [[internal diagram]] \begin{displaymath} \rho \colon \mathbf{B}G \to Type \,. \end{displaymath} See at \emph{\href{internal+%28co-%29limit#BorelConstructionHomotopyQuotients}{internal limit -- Examples -- Homotopy Coinvariants}}. \end{remark} \hypertarget{CartesianClosedMonoidalStructure}{}\subsubsection*{{Conjugation actions}}\label{CartesianClosedMonoidalStructure} \begin{remark} \label{}\hypertarget{}{} By def. \ref{GActionByFiberSequence}, and basic facts disussed at \emph{[[slice (∞,1)-topos]]}, the [[(∞,1)-category]] $Act_{\mathbf{H}}(G)$ is an [[(∞,1)-topos]] and in particular is a [[cartesian closed (∞,1)-category]]. \end{remark} We describe here aspects of the [[cartesian product]] and [[internal hom]] of $\infty$-actions given this way. The following statements are essentially immediate consequences of basic [[homotopy type theory]]. \begin{prop} \label{}\hypertarget{}{} For $(V_1, \rho_1), (V_2, \rho_2) \in Act(G)$ their [[cartesian product]] is a $G$-action on the product of $V_1$ with $V_2$ in $\mathbf{H}$. \end{prop} \begin{proof} Let \begin{displaymath} \itexarray{ V_i &\to& V_i \sslash G \\ && \downarrow^{\bar \rho_i} \\ && \mathbf{B}G } \end{displaymath} be the [[principal ∞-bundles]] exhibiting the two actions. Along the lines of the discussion at [[locally cartesian closed category]] we find that $(V_1, \rho_1) \times (V_2, \rho_2) \in Act(G)$ is given in $\mathbf{H}$ by the [[(∞,1)-pullback]] \begin{displaymath} \sum_{\mathbf{B}G} \bar \rho_1 \times \bar \rho_2 \simeq V_1\sslash G \times_{\mathbf{B}G} V_2 \sslash G \end{displaymath} in $\mathbf{H}$, with the product action being exhibited by the [[principal ∞-bundle]] \begin{displaymath} \itexarray{ V_1 \times V_2 &\to& V_1\sslash G \times_{\mathbf{B}G} V_2 \sslash G \\ && \downarrow^{\mathrlap{\overline{ \rho_1 \times \rho_2 }}} \\ && \mathbf{B}G } \,. \end{displaymath} Here the [[homotopy fiber]] on the left is identified as $V_1 \times V_2$ by using that [[(∞,1)-limits]] commute over each other. \end{proof} \begin{prop} \label{InternalHomAction}\hypertarget{InternalHomAction}{} For $\rho_1, \rho_2 \in Act(G)$ their [[internal hom]] $[\rho_1, \rho_2] \in Act_{\mathbf{H}}(G)$ is a $G$-action on the [[internal hom]] $[V_1, V_2] \in \mathbf{H}$. \end{prop} \begin{proof} Taking fibers \begin{displaymath} pt_{\mathbf{B}G}^* : \mathbf{H}_{/\mathbf{B}G} \to \mathbf{H} \end{displaymath} is the [[inverse image]] of an [[etale geometric morphism]], hence is a [[cartesian closed functor]] (see the \emph{\href{cartesian+closed+functor#Examples}{Examples}} there for details). Therefore it preserves [[exponential objects]]: \begin{displaymath} \begin{aligned} pt_{\mathbf{B}G}^* [\bar \rho_1, \bar \rho_2] & \simeq [pt_{\mathbf{B}G}^* \bar \rho_1, pt_{\mathbf{B}G}^* \bar \rho_2] \\ & \simeq [V_1, V_2] \end{aligned} \,. \end{displaymath} \end{proof} \begin{remark} \label{}\hypertarget{}{} The above internal-hom action \begin{displaymath} \itexarray{ [V_1,V_2] &\to& V_1 \sslash G \times_{\mathbf{B}G} V_2 \sslash G \\ && \downarrow^{\mathrlap{\overline{[\rho_1,\rho_2]}}} \\ && \mathbf{B}G } \end{displaymath} encodes the [[conjugation action]] of $G$ on $[V_1, V_2]$ by pre- and post-composition of [[functions]] $V_1 \to V_2$ with the $G$-action on $V_1$ and on $V_2$, respectively. See also at \emph{\hyperlink{ConjugationActions}{Conjugation actions}} below. \end{remark} \hypertarget{internal_object_of_homomorphisms}{}\subsubsection*{{Internal object of homomorphisms}}\label{internal_object_of_homomorphisms} \begin{remark} \label{}\hypertarget{}{} The [[invariant]], def. \ref{TypeOfInvariants} of the conjugation action, prop. \ref{InternalHomAction} are the action [[homomorphisms]]. (See also at \hyperlink{ConjugationActions}{Examples - Conjugation actions}.) \end{remark} Therefore \begin{defn} \label{}\hypertarget{}{} For $\bar \rho_i : V_i \sslash G \to \mathbf{B}G$ two $G$-actions, the \textbf{object of homomorphisms} is \begin{displaymath} \prod_{\mathbf{B}G \to *}[\bar \rho_1, \bar \rho_2] \in \mathbf{H} \,. \end{displaymath} In the [[syntax]] of [[homotopy type theory]] \begin{displaymath} \vdash \prod_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type \,. \end{displaymath} \end{defn} \hypertarget{stabilizer_groups}{}\subsubsection*{{Stabilizer groups}}\label{stabilizer_groups} See at \emph{[[stabilizer group]]}. \hypertarget{Linearization}{}\subsubsection*{{Linearization}}\label{Linearization} We discuss \emph{linearization} of $\infty$-actions using the axioms of [[differential cohesion]]. Let $0 \colon \ast \to V$ be a [[pointed object]]. Let $G$ be an $\infty$-group acting on $V$ \begin{displaymath} \itexarray{ V &\longrightarrow& V/G \\ && \downarrow \\ && \mathbf{B}G } \end{displaymath} such that this action preserves the point of $V$, i.e. such that the point is an [[invariant]] of the action. This means equivalently that there is a lift as given by the diagonal morphism in \begin{displaymath} \itexarray{ \ast &\stackrel{0}{\longrightarrow}& V &\longrightarrow & V/G \\ & \searrow& & \nearrow& \downarrow \\ && \ast &\longrightarrow& \mathbf{B}G } \end{displaymath} which in turn means that the action factors through an action of the [[stabilizer group]] $Stab_G(0)$ \begin{displaymath} \itexarray{ \ast &\longrightarrow& \mathbf{B}Stab_G(0) \\ \downarrow &\nearrow& \downarrow & \searrow \\ \mathbf{B}G &\longrightarrow& V/G &\longrightarrow& \mathbf{B}G } \end{displaymath} (using that the left morphism is a [[1-epimorphism]] and the right morphism a [[1-monomorphism]]). It follows by the [[pasting law]] the top squares in the following diagram is a [[homotopy pullback]] \begin{displaymath} \itexarray{ \ast &\longrightarrow& \ast/G \\ {}^{\mathllap{0}}\downarrow && \downarrow \\ V &\longrightarrow& V/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G } \end{displaymath} exhibiting that the $G$-action on $V$ restricts to the trivial action on the point $0$ of $V$. Now let $\int_{inf}$ denote the [[infinitesimal shape modality]]. Since it preserves the top homotopy pullback, it follows that applying the [[orthogonal factorization system]] ($\int_{inf}$-equivalences, [[formally etale morphisms]]) to the top vertical morphisms produces a pasting diagram of homotopy pullbacks of the form \begin{displaymath} \itexarray{ \ast &\longrightarrow& \ast/G \\ \downarrow && \downarrow \\ \mathbb{D}^V_0 &\longrightarrow& \mathbb{D}^V_0/G \\ \downarrow && \downarrow \\ \downarrow && \downarrow \\ V &\longrightarrow& V/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G } \end{displaymath} where $\mathbb{D}^V_0$ is the [[infinitesimal disk]] around $0$ in $V$. Here the cartesian subdiagram \begin{displaymath} \itexarray{ \mathbb{D}^V_0 &\longrightarrow& \mathbb{D}^V_0/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G } \end{displaymath} hence exhibits a $G$-action on $\mathbb{D}^V_0$. Any $G$-action on an infinitesimal disk is a linear action, given by a homomorphism $G \to GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V_0)$ to the [[automorphism infinity-group]] of the [[infinitesimal disk]], the [[general linear group]] of the [[tangent space]] of $V$ at 0. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{ExamplesPermutationRepresentations}{}\subsubsection*{{Discrete group actions on sets}}\label{ExamplesPermutationRepresentations} As the simplest special case, we discuss how the traditional concept of [[discrete groups]] acting on a [[sets]] (``[[permutation representations]]'') is recoverd from the above general abstract concepts. Write [[Grpd]] for the [[(2,1)-category]] of [[groupoids]], the [[full sub-(infinity,1)-category]] of [[∞Grpd]] on the [[1-truncated objects]]. We write \begin{displaymath} X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} X_0) \end{displaymath} for a [[groupoid object]] given by an explicit choice of set of objects and of morphisms and then write $X \in Grpd$ for the object that this presents in the $(2,1)$-category. Given any such $X$, we recover a presentation by choosing any [[essentially surjective functor]] $S \to X$ (an [[atlas]]) out of a set $S$ (regarded as a groupoid) and setting \begin{displaymath} X_\bullet = (S \underset{X}{\times} S \stackrel{\longrightarrow}{\longrightarrow} S) \end{displaymath} hence taking $S$ as the set of objects and the [[homotopy fiber product]] of $S$ with itself over $X$ as the set of morphism. For $G$ a [[discrete group]], then $\mathbf{B}G$ denotes the [[groupoid]] presented by $(\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow}\ast)$ with [[composition]] operation given by the product in the group. Of the two possible ways of making this identification, we agree to use \begin{displaymath} \itexarray{ && \ast \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast } \,. \end{displaymath} \begin{defn} \label{Action1Groupoid}\hypertarget{Action1Groupoid}{} Given a [[discrete group]] $G$ and an [[action]] $\rho$ of $G$ on a [[set]] $S$ \begin{displaymath} \rho \colon S \times G \longrightarrow S \end{displaymath} then the corresponding \emph{[[action groupoid]]} is \begin{displaymath} (S//G)_\bullet \coloneqq \left( S\times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S \right) \end{displaymath} with [[composition]] given by the product in $G$. Hence the [[objects]] of $S$ are the elements of $S$, and the morphisms $s \stackrel{}{\longrightarrow } t$ are labeled by elements $g\in G$ and are such that $t = \rho(s)(g)$. \end{defn} Schematically: \begin{displaymath} (S//G)_\bullet = \left\{ \itexarray{ && \rho(s)(g) \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ s && \underset{g_1 g_2}{\longrightarrow} && \rho(s)(g_1 g_2) } \right\} \,. \end{displaymath} \begin{example} \label{}\hypertarget{}{} For the unique and trivial $G$-action on the singleton set $\ast$, we have \begin{displaymath} \ast//G \simeq \mathbf{B}G \,. \end{displaymath} \end{example} This makes it clear that: \begin{prop} \label{MapFromActionGroupoidOnSetBackToBG}\hypertarget{MapFromActionGroupoidOnSetBackToBG}{} In the situation of def. \ref{Action1Groupoid}, there is a canonical morphism of groupoids \begin{displaymath} (p_\rho)_\bullet \;\colon\; (S//G)_\bullet \longrightarrow (\mathbf{B}G)_\bullet \end{displaymath} which, in the above presentation, forgets the labels of the objects and is the identity on the labels of the morphisms. This morphism is an [[isofibration]]. \end{prop} \begin{prop} \label{IntertwinersOfPermutationActionAsSliceHoms}\hypertarget{IntertwinersOfPermutationActionAsSliceHoms}{} For $G$ a [[discrete group]], given two $G$-[[actions]] $\rho_1$ and $\rho_2$ on sets $S_1$ and $S_2$, respectively, then there is a [[natural equivalence]] between the set of action [[homomorphisms]] (``[[intertwiners]]'') $\rho_1 \to \rho_2$, regarded as a groupoid with only identity morphisms, and the [[hom groupoid]] of the [[slice (infinity,1)-category|slice]] $Grpd_{/\mathbf{B}G}$ between their [[action groupoids]] regarded in the slice via the maps from prop. \ref{MapFromActionGroupoidOnSetBackToBG} \begin{displaymath} G Act(\rho_1,\rho_2) \simeq Grpd_{/\mathbf{B}G}(p_{\rho_1}, p_{\rho_2}) \,. \end{displaymath} \end{prop} \begin{proof} One quick way to see this is to use, via the discussion at \emph{[[slice (infinity,1)-category]]}, that the [[hom-groupoid]] in the slice is given by the [[homotopy pullback]] of unsliced hom-groupoids \begin{displaymath} \itexarray{ Grpd_{/\mathbf{B}G}(p_{\rho_1}, p_{\rho_2}) &\longrightarrow& Grpd(S_1//G, S_2//G) \\ \downarrow &(pb)& \downarrow^{\mathrlap{Grpd(S_1//G,p_{\rho_2})}} \\ \ast &\stackrel{}{\longrightarrow}& Grpd(S_1//G, \mathbf{B}G) } \,. \end{displaymath} Now since $(p_{\rho_2})_\bullet$ is an [[isofibration]], so is $Grpd((S_1//G)_\bullet, (p_{\rho_2})_\bullet)$, and hence this is computed as an ordinary pullback (in the above presentation). That in turn gives the [[hom-set]] in the 1-categorical slice. This consists of functors \begin{displaymath} \phi_\bullet \colon (S_1//G)_\bullet \longrightarrow (S_1//G)_\bullet \end{displaymath} which strictly preserves the $G$-labels on the morphisms. These are manifestly the intertwiners. \begin{displaymath} \phi_\bullet \;\colon\; \left( \itexarray{ s \\ \downarrow^{\mathrlap{g}} \\ \rho(s)(g) } \right) \mapsto \left( \itexarray{ \phi(s) \\ \downarrow^{\mathrlap{g}} \\ \phi(\rho(s)(g)) & = \rho(\phi(s))(g) } \right) \,. \end{displaymath} \end{proof} \begin{prop} \label{}\hypertarget{}{} The [[homotopy fiber]] of the morphism in prop. \ref{MapFromActionGroupoidOnSetBackToBG} is [[equivalence of groupoids|equivalent]] to the set $S$, regarded as a groupoid with only identity morphisms, hence we have a [[homotopy fiber sequence]] of the form \begin{displaymath} \itexarray{ S &\longrightarrow& S//G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,. \end{displaymath} \end{prop} \begin{proof} In the presentation $(S//G)_\bullet$ of def. \ref{Action1Groupoid}, $p_\rho$ is an [[isofibration]], prop. \ref{MapFromActionGroupoidOnSetBackToBG}. Hence the [[homotopy fibers]] of $p_\rho$ are equivalent to the ordinary fibers of $(p_\rho)_\bullet$ computed in the 1-category of 1-groupoids. Since $(p_\rho)_\bullet$ is the identity on the labels of the morphisms in this presentation, this ordinary fiber is precisely the sub-groupoid of $(S//G)_\bullet$ consisting of only the identity morphismss, hence is the set $S$ regarded as a groupoid. \end{proof} Conversely, the following construction extract a group action from a homotopy fiber sequence of groupoids of this form. \begin{defn} \label{ActionMapFromFiberSequenceSetToGroupoidToBG}\hypertarget{ActionMapFromFiberSequenceSetToGroupoidToBG}{} Given a [[homotopy fiber sequence]] of [[groupoids]] of the form \begin{displaymath} \itexarray{ S &\stackrel{i}{\longrightarrow}& E \\ && \downarrow^{\mathrlap{p}} \\ && \mathbf{B}G } \end{displaymath} such that $S$ is [[equivalence of groupoids|equivalent]] to a [[set]] $S$, define a $G$-[[action]] on this set as follows. Consider the [[homotopy fiber product]] \begin{displaymath} S \underset{E}{\times} S \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}} S \end{displaymath} of $i$ with itself. By the [[pasting law]] applied to the total homotopy pullback diagram \begin{displaymath} \itexarray{ S \underset{E}{\times} S &\longrightarrow& S \\ \downarrow && \downarrow^{\mathrlap{i}} \\ S &\stackrel{i}{\longrightarrow}& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \ast &\longrightarrow& \mathbf{B}G } \;\;\;\; \simeq \;\;\;\; \itexarray{ S\times G &\stackrel{p_1}{\longrightarrow}& S \\ \downarrow && \downarrow \\ G &\stackrel{}{\longrightarrow}& \ast \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G } \end{displaymath} there is a canonical [[equivalence of groupoids]] \begin{displaymath} S \underset{E}{\times} S \simeq S \times G \end{displaymath} such that one of the two canonical maps from the fiber product to $S$ is projection on the first factor. The \emph{other} map under this equivalence we denote by $\rho$: \begin{displaymath} S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} The functor $i \colon S \to E$ is clerly [[essentially surjective functor|essentially surjective]] (every connected component of $E$ has a homotopy fiber under its map to $\mathbf{B}G$). This implies that $E$ is presented by \begin{displaymath} E_\bullet \coloneqq (S \underset{E}{\times}S \stackrel{\overset{p_1}{\longrightarrow}}{\underset{p_2}{\longrightarrow}} S) \end{displaymath} and hence, via the construction in def. \ref{ActionMapFromFiberSequenceSetToGroupoidToBG}, by \begin{displaymath} E_\bullet \simeq (S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S) \,. \end{displaymath} \end{remark} But this already exhibits $E$ as an [[action groupoid]], in particular it mans that $\rho$ is really an [[action]]: \begin{prop} \label{ActionGroupoidFromFiberSequence}\hypertarget{ActionGroupoidFromFiberSequence}{} The morphism $\rho$ constructed in def. \ref{ActionMapFromFiberSequenceSetToGroupoidToBG} is a $G$-[[action]] in that it satisfies the action propery, which says that the [[diagram]] (of [[sets]]) \begin{displaymath} \itexarray{ S\times G \times G &\stackrel{(id,(-)\cdot(-))}{\longrightarrow}& S \times G \\ \downarrow^{\mathrlap{(\rho,id)}} && \downarrow^{\mathrlap{\rho}} \\ S \times G &\stackrel{\rho}{\longrightarrow}& S } \end{displaymath} [[commuting diagram|commutes]]. \end{prop} \begin{prop} \label{}\hypertarget{}{} For $G$ a [[discrete group]], there is an [[equivalence of categories]] \begin{displaymath} G Act(Set) \stackrel{\simeq}{\longrightarrow} (Grpd_{/\mathbf{BG}})_{\leq 0} \end{displaymath} between the category of [[permutation representations]] of $G$ and the full subcategory of the [[slice (infinity,1)-category|slice (2,1)-category]] of [[Grpd]] over $\mathbf{B}G$ on the [[0-truncated objects]]. This equivalence takes an action to its [[action groupoid]]. \end{prop} \begin{proof} By remark \ref{ActionGroupoidFromFiberSequence} the construction of action groupoids is [[essentially surjective functor|essentially surjective]]. By prop. \ref{IntertwinersOfPermutationActionAsSliceHoms} it is [[fully faithful functor|fully faithful]]. \end{proof} \hypertarget{group_actions_in_an_topos}{}\subsubsection*{{$\infty$-group actions in an $\infty$-topos}}\label{group_actions_in_an_topos} Let $\mathbf{H}$ be an [[(∞,1)-topos]] and let $G \in Grp(\mathbf{H})$ be an [[∞-group]] in $\mathbf{H}$. The following lists some fundamental classes of examples of $\infty$-actions of $G$, and of other canonical $\infty$-groups. By the discussion \hyperlink{PropertiesOfGroupActionsInTopos}{above} these actions may be given by the classifying morphisms. \hypertarget{TrivialAction}{}\paragraph*{{Trivial action}}\label{TrivialAction} Consider the [[étale geometric morphism]] \begin{displaymath} Act_{\mathbf{H}}(G) \coloneqq \mathbf{H}_{/\mathbf{B}G} \stackrel{\overset{p^* \coloneqq (-) \times \mathbf{B}G}{\leftarrow}}{\underset{}{\to}} \mathbf{H} \,. \end{displaymath} \begin{defn} \label{TrivialAction}\hypertarget{TrivialAction}{} For $V \in \mathbf{H}$ any object, the \textbf{trivial action} of $G$ on $V$ is $p^* V \in Act_{\mathbf{H}}(G)$, exhibited by the split fiber sequence \begin{displaymath} \itexarray{ V &\to& V \times \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,. \end{displaymath} \end{defn} \hypertarget{fundamental_action}{}\paragraph*{{Fundamental action}}\label{fundamental_action} The \emph{right $\infty$-action} of $G$ on itself is given by the fiber sequence \begin{displaymath} \itexarray{ G \\ \downarrow \\ * &\to& \mathbf{B}G } \end{displaymath} which exhibits $\mathbf{B}G$ as the [[delooping]] of $G$. \begin{displaymath} G \sslash G \simeq * \,. \end{displaymath} \hypertarget{adjoint_action}{}\paragraph*{{Adjoint action}}\label{adjoint_action} The fiber sequence \begin{displaymath} \itexarray{ G \\ \downarrow \\ \mathcal{L} \mathbf{B}G &\stackrel{ev_*}{\to}& \mathbf{B}G } \end{displaymath} given by the [[free loop space object]] $\mathcal{L}\mathbf{B}G$ exhibits the higher [[adjoint action]] of $G$ on itself: \begin{displaymath} G \sslash_{Ad}G \simeq \mathcal{L}\mathbf{B}G \,. \end{displaymath} \hypertarget{AutomorphismAction}{}\paragraph*{{Automorphism action}}\label{AutomorphismAction} \begin{defn} \label{AutomorphAction}\hypertarget{AutomorphAction}{} For $V \in \mathbf{H}$ any object, there is a canonical action of the internal [[automorphism infinity-group]] $\mathbf{Aut}(V)$: \begin{displaymath} \itexarray{ V \\ \downarrow \\ V \sslash \mathbf{Aut}(V) &\to& \mathbf{B} \mathbf{Aut}(V) } \end{displaymath} \end{defn} \hypertarget{ConjugationActions}{}\paragraph*{{Conjugation actions}}\label{ConjugationActions} We discuss the simple case of the [[cartesian closed category]] of $G$-sets (G-[[permutation representations]]) for $G$ an ordinary [[discrete group]] as a simple illustration of the internal hom of $\infty$-actions, prop. \ref{InternalHomAction}. This example spells out everything completely in components: \begin{example} \label{}\hypertarget{}{} Let $\mathbf{H} =$ [[∞Grpd]], let $G \in Grp(\infty Grpd)$ be an ordinary [[discrete group]] and let $V, \Sigma, X$ be [[sets]] equipped with $G$-[[action]] ([[permutation representations]]). In this case $[\Sigma,X]$ is simply the set of [[functions]] $f : \Sigma \to X$ of sets. Its $G$-action as the internal hom of $G$-actions given, for every $g \in G$ and $\sigma \in \Sigma$, by \begin{displaymath} g(f)(\sigma) = g(f(g^{-1}(\sigma))) \,, \end{displaymath} (where we write generically $g(-)$ for the given action on the set specified implicitly by the type of the argument). Hence a morphism of $G$-actions \begin{displaymath} \phi : V \to [\Sigma,X] \end{displaymath} is a function $\phi$ of the underlying sets such that for all $V \in V$, $g \in G$ and all $\sigma \in \Sigma$ we have \begin{equation} \phi(g(v))(\sigma) = g(\phi(v)(g^{-1}(\sigma)) \,. \label{Equation1}\end{equation} On the other hand, a morphism of actions \begin{displaymath} \psi : V \times \Sigma \to X \end{displaymath} is a function of the underlying sets, such that for all these terms we have \begin{displaymath} \psi(g(v), g(\sigma)) = g(\psi(v,\sigma)) \end{displaymath} which is equivalent to \begin{equation} \psi(g(v), \sigma) = g(\psi(v,g^{-1}(\sigma))) \,. \label{Equation2}\end{equation} Comparison of \eqref{Equation1} and \eqref{Equation2} shows that the identification \begin{displaymath} \psi(v,\sigma) \coloneqq \phi(v)(\sigma) \end{displaymath} establishes a [[natural equivalence]] (a [[natural bijection]] of sets in this case) \begin{displaymath} Act_{\mathbf{H}}(G)(V, [\Sigma,X]) \simeq Act_{\mathbf{H}}(G)(V \times \Sigma, X) \,, \end{displaymath} showing how $[\Sigma,X]$ is indeed the [[internal hom]] of $G$-actions. \end{example} \begin{remark} \label{}\hypertarget{}{} Generally, for $G$ a [[discrete ∞-group]] we have an [[equivalence of (∞,1)-categories]] \begin{displaymath} \infty Grpd_{/\mathbf{B}G} \simeq \infty Func(\mathbf{B}G, \infty Grpd) \end{displaymath} (by the [[(∞,1)-Grothendieck construction]]), and hence \begin{displaymath} Act_{\infty Grpd}(G) \simeq \infty Func(\mathbf{B}G, \infty Grpd) \end{displaymath} is the [[(∞,1)-category]] of [[∞-permutation representations]]. \end{remark} \hypertarget{GeneralCovariance}{}\paragraph*{{General covariance}}\label{GeneralCovariance} Let $X \in \mathbf{H}$ be a [[moduli infinity-stack]] for field in a [[gauge theory]] or [[sigma-model]]. Let $\Sigma \in \mathbf{H}$ be the corresponding [[spacetime]] or [[worldvolume]], respectively. We have the automorphism action, def. \ref{AutomorphAction} \begin{displaymath} \itexarray{ \Sigma &\to& \Sigma \sslash \mathbf{Aut}(\Sigma) \\ && \downarrow \\ && \mathbf{B} \mathbf{Aut}(\Sigma) } \,. \end{displaymath} The slice $\mathbf{H}_{/\mathbf{Aut}(\Sigma)} = Act_{\mathbf{H}}(\mathbf{Aut}(\Sigma))$ is the context of types which are \emph{[[general covariance|generally covariant]]} over $\Sigma$. On $X$ consider the trivial $\mathbf{Aut}(\Sigma)$-action, def. \ref{TrivialAction}. Then the internal-hom action of prop. \ref{InternalHomAction} \begin{displaymath} [\Sigma, X]\sslash \mathbf{Aut}(\Sigma) \simeq [\Sigma \sslash \mathbf{Aut}(\Sigma), X \times \mathbf{B}\mathbf{Aut}(\Sigma)]_{\mathbf{B}\mathbf{Aut}(\Sigma)} \end{displaymath} is the configuration space of fields on $\Sigma$ modulo automorphisms (diffeomorphisms, in [[smooth infinity-groupoid|smooth cohesion]]) of $\Sigma$. This is the configuration space of ``[[general covariance|generally covariant]]'' field theory on $\Sigma$. \hypertarget{SemidirectProductGroups}{}\paragraph*{{Semidirect product groups}}\label{SemidirectProductGroups} Let $G, A \in Grp(\mathbf{H})$ be 0-truncated group objects and let $\rho$ be an action of $G$ on $A$ by group homomorphisms. This is equivalently an action of $G$ on $\mathbf{B}A$, hence a fiber sequence \begin{displaymath} \itexarray{ \mathbf{B}A &\to& \mathbf{B} (G \ltimes A) \\ && \downarrow \\ && \mathbf{B}G } \,. \end{displaymath} The corresponding [[action groupoid]] $(\mathbf{B}A)\sslash G \simeq \mathbf{B}( G \ltimes A)$ is the delooping of the corresponding [[semidirect product group]]. \hypertarget{GModules}{}\paragraph*{{$G$-Modules}}\label{GModules} \begin{defn} \label{InfinityModuleOverAnInfinityGroup}\hypertarget{InfinityModuleOverAnInfinityGroup}{} For $G \in Grp(\mathbf{H})$ the $\infty$-category of \textbf{$G$-[[modules]]} is \begin{displaymath} Stab( \mathbf{H}_{/\mathbf{B}G}) \simeq Stab(G Act) \,, \end{displaymath} the [[stabilization]] of the $\infty$-category of $G$-actions. \end{defn} \begin{example} \label{}\hypertarget{}{} For $G$ and $A$ 0-truncated groups, $A$ an [[abelian group]] with $G$-[[module]] structure, the semidirect product group $G \ltimes A$ from \hyperlink{SemidirectProductGroups}{above} exhibits $A$ as a $G$-module in the sense of def. \ref{InfinityModuleOverAnInfinityGroup}. \end{example} \hypertarget{ExamplesActionsInASlice}{}\paragraph*{{Actions in a slice}}\label{ExamplesActionsInASlice} Consider an object $B \in \mathbf{H}$ and an object \begin{displaymath} L \in \mathbf{H}_{/B} \end{displaymath} in the slice. By the discussion of [[conjugation actions]] \href{CartesianClosedMonoidalStructure}{above}, the [[automorphism ∞-group]] of $L$ as an object in $\mathbf{H}$ is the [[dependent product]] over the [[automorphism ∞-group]] $\mathbf{Aut}_{\mathbf{H}}(L)\in \mathbf{H}_{/B}$ in the slice. \begin{displaymath} \mathbf{Aut}_{\mathbf{H}}(L) \coloneqq \underset{B}{\prod} \mathbf{Aut}(L) \in \mathrm{Grp}(\mathbf{H}) \,. \end{displaymath} By [[adjunction]] there is a canonical morphism from the re-pullback of this to the slice automorphism group \begin{displaymath} \epsilon \colon B^\ast \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) \longrightarrow \mathbf{B} \mathbf{Aut}(L) \,. \end{displaymath} Hence the canonical $\mathbf{Aut}(L)$-action on $L$ in the slice pulls back to give an action of $B^\ast \mathbf{Aut}_{\mathbf{H}}(L)$ on $L$: \begin{displaymath} \itexarray{ L &\longrightarrow& L//(B^\ast\mathbf{Aut}_{\mathbf{H}}(L)) &\longrightarrow& L//\mathbf{Aut}(L) \\ \downarrow && \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}B^\ast \mathbf{Aut}_{\mathbf{H}}(L) &\stackrel{\epsilon}{\longrightarrow}& \mathbf{B} \mathbf{Aut}(L) } \end{displaymath} \begin{prop} \label{AutomorphismActionInSliceInducingActionOnSum}\hypertarget{AutomorphismActionInSliceInducingActionOnSum}{} Underlying the $B^\ast\mathbf{Aut}_{\mathbf{H}}(L)$-action on $L$ is an $\mathbf{Aut}_{\mathbf{H}}(L)$-action on \begin{displaymath} X \coloneqq \underset{B}{\sum} L \end{displaymath} and \begin{displaymath} \underset{B}{\sum} \left(L//B^\ast\mathbf{Aut}_{\mathbf{H}}(L)\right) \;\simeq\; X//\mathbf{Aut}_{\mathbf{H}}(L) \end{displaymath} \end{prop} \begin{proof} Applying $\underset{B}{\sum}$ to the Cartesian diagram that defines the $\infty$-action on $L$ \begin{displaymath} \itexarray{ L &\longrightarrow& L//\mathbf{Aut}_{\mathbf{H}}(L) \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}B^\ast \mathbf{Aut}_{\mathbf{H}}(L) } \end{displaymath} yields \begin{displaymath} \itexarray{ X &\longrightarrow& \underset{X}{\sum} \left( L//\mathbf{Aut}_{\mathbf{H}}(L) \right) \\ \downarrow && \downarrow \\ B &\longrightarrow& \underset{B}{\sum} B^\ast \mathbf{B} \mathbf{Aut}_{\mathbf{H}}(L) } \end{displaymath} which is still Cartesian, by \href{dependent+sum#AbsoluteDependentSumPreservesFiberProducts}{this proposition}. Use that the bottom left object here is equivalently $B \simeq \underset{B}{\sum} B^\ast (\ast)$ and form the [[pasting diagram|pasting]] with the [[naturality square]] of the $(\underset{B}{\sum}\dashv B^\ast)$-[[counit of an adjunction|counit]]. \begin{displaymath} \itexarray{ X &\longrightarrow& \underset{B}{\sum} \left(L//\mathbf{Aut}_{\mathbf{H}}(L)\right) \\ \downarrow && \downarrow \\ \underset{B}{\sum}B^\ast \ast &\longrightarrow& \underset{B}{\sum}B^\ast \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) } \,. \end{displaymath} By \href{dependent+sum#NaturalitySquareOfCounitIsPullback}{this proposition} also this naturality square is Cartesian. Hence by the [[pasting law]] the total rectangle is Cartesian. This exhibits the $\mathbf{Aut}_{\mathbf{H}}(L)$-action on $X = \underset{B}{\sum} L$. \end{proof} \begin{remark} \label{}\hypertarget{}{} Stated more intuitively, prop. \ref{AutomorphismActionInSliceInducingActionOnSum} says that sliced automorphisms of the form \begin{displaymath} \mathbf{Aut}_{\mathbf{H}}(L) = \left\{ \itexarray{ X & & \stackrel{\simeq}{\longrightarrow} & & X \\ & {}_{\mathllap{L}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{L}} \\ && B } \right\} \end{displaymath} act on $X$ by the evident restriction to the horizontal equivalences, \begin{displaymath} \left\{ \itexarray{ X & & \stackrel{\simeq}{\longrightarrow} & & X } \right\} \end{displaymath} and that forming the homotopy quotient of this action on $L$ makes $L$ [[descent]] to the homotopy quotient of $X$ by this action to yield \begin{displaymath} \itexarray{ X // \mathbf{Aut}_{\mathbf{H}}(L) \\ \downarrow^{\mathrlap{L//\mathbf{Aut}_{\mathbf{H}}(L)}} \\ B } \,. \end{displaymath} (For instance if here $B$ is a [[moduli stack]] for some [[prequantum n-bundles]], then this says that the [[quantomorphism n-group]] acting on this gives higher and pre-quantized ``[[symplectic reduction]]'' of these bundles to the quotient space.) \end{remark} \hypertarget{codiscretization_of_actions}{}\paragraph*{{Co-Discretization of Actions}}\label{codiscretization_of_actions} Let $\mathbf{H}$ be a [[local (∞,1)-topos]] (for instance a [[cohesive (∞,1)-topos]]) and write $\sharp$ for its [[sharp modality]]. Write $\sharp_n$ for the [[n-image]] of itd [[unit of a monad|unit]]. \begin{prop} \label{CoDiscretizationOfActions}\hypertarget{CoDiscretizationOfActions}{} Given an [[∞-group]] $G$ in $\mathbf{H}$ and a $G$-action, def. \ref{GActionByFiberSequence}, on some $X$, then $\sharp_n G$ is itself canonically an $\infty$-group equipped with a canonically induced action on $\sharp_n X$ such that the projection $X \to \sharp_n X$ carries the structure of a homomorphism of $G$-actions. \end{prop} We indicate two proofs, the first non-elementary (making use of the [[Giraud-Rezk-Lurie theorem]]), the second [[elementary (infinity,1)-topos|elementary]]. (Following \href{http://nforum.ncatlab.org/discussion/4576/nimage/?Focus=51743#Comment_51743}{this} discussion.) \begin{proof} Observe that $\sharp_n$ preserves [[products]], since $\sharp$ does (being a [[right adjoint]]) and by \href{n-image#nImagePreservesProducts}{this proposition}. Now use that the [[homotopy quotient]] $V/G$ is the realization of the [[simplicial object in an (infinity,1)-category|simplicial object]] $(V/G)_\bullet = G^{\times_{\bullet}} \times V$. So applying $\sharp_n$ to this yields a simplicial object $((\sharp_n V)/(\sharp_n G))_\bullet = (\sharp_n G)^{\times_{\bullet}} \times (\sharp_n V)$ which exhibits the desired action. \end{proof} \begin{proof} Generally, let $A:B\to Type$ be any [[dependent type]] family (speaking [[homotopy type theory]]). We claim that there is an induced family $A^{\sharp_n} : \sharp_{n+1} B \to Type$ such that $A^{\sharp_n}(\eta_{n+1}(b)) = \sharp_n (A(b))$ for any $b:B$, where $\eta_{n+1} : B \to \sharp_{n+1} B$ is the inclusion. Applying this when $A \to B$ is $V/G \to \mathbf{B}G$ and when $b$ is (necessarily) [[generalized the|the]] basepoint of $\mathbf{B}G$ gives the desired action on the desired type. First of all, we have the composite $B \xrightarrow{A} Type \xrightarrow{\sharp} Type_{\sharp}$, where $Type_{\sharp} = \sum_{X:Type} is\sharp(X)$. Since $Type_{\sharp}$ is itself $\sharp$ (since $\sharp$ is lex), this factors through $\sharp B$, giving a type family $A^\sharp : \sharp B \to Type_{\sharp}$ such that $A^{\sharp}(\eta(b)) = \sharp (A(b))$ for any $b:B$, where $\eta:B\to \sharp B$ is the unit of $\sharp$. Now fix $y:\sharp B$ and $x:A^\sharp(y)$. For any $b:B$ and $p:\eta(b)=y$, we can define the type ${\big\Vert \sum_{(a:A(b))} p_\ast (\eta(a)) = x\big\Vert}_n$. This is an $n$-type, and since the [[type of types|type of]] [[truncated types]] $n\text{-}Type$ is an $(n+1)$-type, as a function of $(b,p) : \sum_{b:B} \eta(b)=y$, this construction factors through $\big\Vert \sum_{b:B} \eta(b)=y\big\Vert_{n+1}$. Thus, for $y:\sharp B$ and $x:A^\sharp(y)$ and $\xi : {\big\Vert \sum_{(b:B)} \eta(b)=y\big\Vert}_{n+1}$ we have a type $P(y,x,\xi)$, such that \begin{displaymath} P\big(y,x,{|(b,p)|}_{n+1}\big) = {\left\Vert \sum_{(a:A(b))} p_\ast (\eta(a)) = x\right\Vert}_n. \end{displaymath} Now by definition, $\sharp_{n+1} B \coloneqq \sum_{(y:\sharp B)} {\big\Vert \sum_{(b:B)} \eta(b)=y\big\Vert}_{n+1}$. Thus, we can define $A^{\sharp_n} : \sharp_{n+1} B \to Type$ by $A^{\sharp_n}(y,\xi) = \sum_{x:A^\sharp(y)} P(y,x,\xi)$. And since $\eta_{n+1}(b) = (\eta(b),{|(b,1)|}_{n+1})$, we have $A^{\sharp_n}(\eta_{n+1}(b)) = \sum_{x:\sharp(A(b))} {\big\Vert \sum_{(a:A(b))} \eta(a)) = x\big\Vert}_n$, which is $\sharp_{n}(A(b))$ by definition. \end{proof} \hypertarget{infinitesimally_actions_of_algebroids}{}\subsubsection*{{Infinitesimally: actions of $L_\infty$-algebroids}}\label{infinitesimally_actions_of_algebroids} See \emph{[[Lie infinity-algebroid representation]]}. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{model_category_presentation}{}\subsubsection*{{Model category presentation}}\label{model_category_presentation} In the context of [[geometrically discrete ∞-groupoids]] a [[model category]] structure presenting the [[(∞,1)-category]] of $\infty$-actions is the \emph{[[Borel model structure]]} (\hyperlink{DDK80}{DDK 80}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[action]], \textbf{$\infty$-action} \begin{itemize}% \item [[faithful ∞-action]] \item [[stabilizer ∞-group]] \end{itemize} \item [[module]], [[∞-module]] \item [[representation]], [[∞-representation]] \item [[associated bundle]], [[associated ∞-bundle]] \item [[induced representation]] \item [[equivariant homotopy theory]] \item [[equivariant cohomology]] \begin{itemize}% \item [[group cohomology]] \end{itemize} \end{itemize} [[!include homotopy type representation theory -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Actions of [[A-∞ algebras]] in some [[symmetric monoidal (∞,1)-category]] are discussed in section 4.2 of \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Higher Algebra]]} \end{itemize} Aspects of actions of [[∞-groups]] in an [[∞-topos]] in the contect of [[associated ∞-bundles]] are discussed in section I 4.1 of \begin{itemize}% \item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]}, Journal of Homotopy and Related Structures, June 2014 (\href{http://arxiv.org/abs/1207.0248}{arXiv:1207.0248}) \end{itemize} Discussion in [[homotopy type theory]] is in \begin{itemize}% \item [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], \emph{Higher Groups in Homotopy Type Theory} (\href{https://arxiv.org/abs/1802.04315}{arXiv:1802.04315}) \end{itemize} \hypertarget{ReferencesForDiscreteGeometry}{}\subsubsection*{{For discrete geometry}}\label{ReferencesForDiscreteGeometry} For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-[[infinity-actions]] is (via the [[Borel model structure]]) is due to \begin{itemize}% \item E. Dror, [[William Dwyer]], [[Daniel Kan]], \emph{Equivariant maps which are self homotopy equivalences}, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670--672 (\href{http://www.jstor.org/stable/2043448}{jstor:2043448}) \end{itemize} This is mentioned for instance as exercise 4.2 in \begin{itemize}% \item [[William Dwyer]], \emph{Homotopy theory of classifying spaces}, Lecture notes Copenhagen (June, 2008) \href{http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf}{pdf} \end{itemize} An alternative proof in terms of [[relative categories]] is in \begin{itemize}% \item Amit Sharma, \emph{On the homotopy theory of $G$-spaces} (\href{http://arxiv.org/abs/1512.03698}{arXiv:1512.03698}) \end{itemize} Closely related discussion of homotopy fiber sequences and homotopy action but in terms of [[Segal spaces]] is in section 5 of \begin{itemize}% \item [[Matan Prezma]], \emph{Homotopy normal maps} (\href{http://arxiv.org/pdf/1011.4708v7.pdf}{arXiv}) \end{itemize} There, conditions are given for a morphism $A_\bullet \to B_\bullet$ to a [[reduced Segal space]] to have a fixed homotopy fiber, and hence encode an action of the loop group of $B$ on that fiber. \hypertarget{ForActionsOfTopologicalGroups}{}\subsubsection*{{For actions of topological groups}}\label{ForActionsOfTopologicalGroups} That $G$-actions for $G$ a [[topological group]] in the sense of [[G-spaces]] in [[equivariant homotopy theory]] (and hence with $G$ \emph{not} regarded as the geometrically discrete [[∞-group]] of its underying [[homotopy type]] ) are equivalently objects in the [[slice (∞,1)-topos]] over $\mathbf{B}G$ is [[Elmendorf's theorem]] together with the fact, highlighted in this context in \begin{itemize}% \item [[Charles Rezk]], \emph{[[Global Homotopy Theory and Cohesion]]}, 2014 \end{itemize} that \begin{displaymath} G Space \simeq PSh_\infty(Orb_G) \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb)_{/\mathbf{B}G} \end{displaymath} is therefore the slice of the $\infty$-topos over the [[global orbit category]] by $\mathbf{B}G$. [[!include equivariant homotopy theory -- table]] See at \emph{[[equivariant homotopy theory]]} for more references along these lines. [[!redirects ∞-action]] [[!redirects infinity-actions]] [[!redirects ∞-actions]] \end{document}