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\begin{document}

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\section*{action groupoid}

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

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

[[!include homotopy - contents]]

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

[[!include representation theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_category_theory}{In category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{in_1category_theory}{In (∞,1)-category theory}\dotfill \pageref*{in_1category_theory} \linebreak
\noindent\hyperlink{interpretations}{Interpretations}\dotfill \pageref*{interpretations} \linebreak
\noindent\hyperlink{as_a_pseudo_colimit}{As a pseudo colimit}\dotfill \pageref*{as_a_pseudo_colimit} \linebreak
\noindent\hyperlink{as_associated_universal_bundle}{As associated universal bundle}\dotfill \pageref*{as_associated_universal_bundle} \linebreak
\noindent\hyperlink{as_a_stack}{As a stack}\dotfill \pageref*{as_a_stack} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_representation_theory}{Relation to representation theory}\dotfill \pageref*{relation_to_representation_theory} \linebreak
\noindent\hyperlink{ActionooGrpd}{Action $\infty$-groupoid}\dotfill \pageref*{ActionooGrpd} \linebreak
\noindent\hyperlink{some_comment}{Some comment}\dotfill \pageref*{some_comment} \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}

Given an [[action]] $\rho$ of a [[group]] $G$ on a [[set]] $S$, the action groupoid $S//G$ is a bit like the quotient set $S/G$ (the set of $G$-orbits). But, instead of taking elements of $S$ in the same $G$-orbit as being [[equality|equal]] in $S/G$, in the action groupoid they are just [[isomorphism|isomorphic]]. We may think of the action groupoid as a [[homological resolution|resolution]] of the usual quotient. When the action of $G$ on $S$ fails to be free, the action groupoid is generally better-behaved than the quotient set.

The action groupoid also goes by other names, including `[[weak quotient]]'. It is a special case of a `[[pseudo colimit]]', as explained below. It is also called a ``[[semidirect product]]'' and then written $S \rtimes G$. The advantage of this is that it accords with the generalisation to the action of a group $G$ on a groupoid $S$, which is relevant to orbit space considerations, since if $G$ acts on a space $X$ it also acts on the fundamental groupoid of $X$; this is fully developed in ``Topology and Groupoids'', Chapter 11.

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

\hypertarget{in_category_theory}{}\subsubsection*{{In category theory}}\label{in_category_theory}

Given an [[action]] $\rho : S \times G \to S$ of a group $G$ on the set $S$, the \emph{action groupoid} $S//G$ (or, more precisely, $S//_\rho G$) is the [[groupoid]] for which:

\begin{itemize}%
\item an object is an element of $S$


\item a morphism from $s \in S$ to $s' \in S$ is a group element $g \in G$ with $g s = s'$. So, a general morphism is a pair $(g,s) : s \to g s$.


\item The composite of $(g,s) : s \to g s = s'$ and $(g',s'): 
s' \to g's'$ is $(g' g, s) : s \to g' g s$.



\end{itemize}
Equivalently, we may define the \emph{action groupoid} $S//G$ to be the groupoid

\begin{displaymath}
\itexarray{
     && S \times G
     \\
     & {}^{s := p_1}\swarrow
     &&
     \searrow^{t = \rho}
     \\
     S
     &&&&
     S
  }
\end{displaymath}
with composition

\begin{displaymath}
(S \times G) \times_{t,s} (S \times G) \simeq S \times G \times G \to S \times G
\end{displaymath}
given by the product in $G$.

We can denote the morphisms in $S//G$ by

\begin{displaymath}
S//G:=\{s\stackrel{g}{\to} \rho(s,g) | s\in S, g\in G\}.
\end{displaymath}
\hypertarget{in_1category_theory}{}\subsubsection*{{In (∞,1)-category theory}}\label{in_1category_theory}

\begin{defn}
\label{ActionInfinityGroupoid}\hypertarget{ActionInfinityGroupoid}{}
Let $C$ be an ($\infty$,1)-category, let $G\in Grpd(C)$ be a groupoid object in $C$, let $X\in C$ be an object. Then the [[simplicial object]]

\begin{displaymath}
\itexarray{
      \cdots
    &
      \underoverset{\to}{\to}{\to}
    &
      X\times_{G_0}G\times_p G
    &
      \rightrightarrows
    &
      X\times_{G_0}G
    &
      \to
    &
      X
  }
\end{displaymath}
such that the degree-wise projections give a simplicial map

\begin{displaymath}
\itexarray{
        \cdots
      &
        \underoverset{\to}{\to}{\to}
      &
        X\times_{G_0}G\times_p G
      &
        \rightrightarrows
      &
        X\times_{G_0}G
      &
        \to
      &
        X
    \\
      &&
        \downarrow
      &&
        \downarrow
      &&
        \downarrow^a
    \\
        \cdots
      &
        \underoverset{\to}{\to}{\to}
      &
        G\times_p G
      &
        \rightrightarrows
      &
        G
      &
        \xrightarrow{p}
      &
        G_0
  }
\end{displaymath}
is called an \emph{action of} $G$ \emph{on} $X$. The colimit $colim\; X\times_{G_0}^{\times_\bullet}$ is called \emph{action $\infty$-groupoid of} $G$ \emph{on} $X$.

\end{defn}
\hypertarget{interpretations}{}\subsection*{{Interpretations}}\label{interpretations}

On top of the above explicit definitions, there are several useful ways to think of action groupoids.

Recall that the action $\rho$ is equivalently thought of as a functor

\begin{displaymath}
\rho : \mathbf{B}G \to Sets
\end{displaymath}
from the [[group]] $G$ regarded as a one-object groupoid, denoted $\mathbf{B}G$.

This functor sends the single object of $\mathbf{B}G$ to the set $S$.

\hypertarget{as_a_pseudo_colimit}{}\subsubsection*{{As a pseudo colimit}}\label{as_a_pseudo_colimit}

$S//G$ is the [[2-limit|2-colimit]] of $\rho$, i.e., the [[category of elements]] of $\rho$.

\begin{displaymath}
S//G \simeq colim_{\mathbf{B}G} \rho
  \,.
\end{displaymath}
The universal cocone consists of cells of the form

\begin{displaymath}
\itexarray{
     S &&\stackrel{\rho(g)}{\to}&& S
     \\
     & \searrow &\stackrel{\simeq}{\Leftarrow}& \swarrow
     \\
     && S//G
  }
  \,,
\end{displaymath}
where the 2-morphism is uniquely specified and in components given by $s \mapsto (s \stackrel{g}{\to} \rho(s,g))$.

\hypertarget{as_associated_universal_bundle}{}\subsubsection*{{As associated universal bundle}}\label{as_associated_universal_bundle}

Let $Set_*$ be the category of pointed sets and $Sets_* \to Sets$ be the canonical forgetful functor. We can think of this as the ``universal $Set$-bundle''.

Then $S//G$ is the pullback

\begin{displaymath}
\itexarray{
    S//G
    &\to&
    Sets_*
    \\
    \downarrow
    &&
    \downarrow
    \\
    \mathbf{B}G
    &\stackrel{\rho}{\to}&
    Sets
  }
  \,.
\end{displaymath}
One place where we discussed this is the comment \href{http://golem.ph.utexas.edu/category/2007/08/on_hess_and_lack_on_bundles_of_1.html#c019094}{It was David Roberts who apparently first noticed\ldots{}}.

Notice also that an action of $G$ on the set $S$ gives rise to a morphism $p: S \rtimes G \to G$ which has the property of unique path lifting, or in other words is a discrete opfibration. It is also called a covering morphism of groupoids, and models nicely covering maps of spaces.

Higgins used this idea to lift presentations of a group $G$ to presentations of the covering morphism of $G$ derived from the action of $G$ on cosets, and so to apply graph theory to obtain old and new subgroup theorems in group theory.

\hypertarget{as_a_stack}{}\subsubsection*{{As a stack}}\label{as_a_stack}

In the case where the action is [[internalization|internal]] to sets with structure, such as internal to [[Diff]] one wants to realize the action groupoid as a [[Lie groupoid]]. That Lie groupoid in turn may be taken to present a [[differentiable stack]] which then usually goes by the same name $S//G$.

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

\hypertarget{relation_to_representation_theory}{}\subsubsection*{{Relation to representation theory}}\label{relation_to_representation_theory}

The action groupoids $X//G$ of a group $G$ come equipped with a canonical map to $\mathbf{B}G \simeq \ast //G$. Regarded via this map as objects in the [[slice (infinity,1)-category|slice]] of groupoids over $\mathbf{B}G$, action groupoids are in fact [[equivalence|equivalent]] to the actions that they arise from.

For more on this see at \emph{[[infinity-action]]} and also at \emph{[[geometry of physics -- representations and associated bundles]]}.

\hypertarget{ActionooGrpd}{}\subsection*{{Action $\infty$-groupoid}}\label{ActionooGrpd}

All of this goes through almost verbatim for actions in the context of [[(∞,1)-category theory]].

Let $G$ be an [[∞-group]] in that $\mathbf{B}G$ is an [[∞-groupoid]] with a single object. An action of $G$ on an [[(∞,1)-category]] is an [[(∞,1)-functor]]

\begin{displaymath}
\rho : \mathbf{B}G \to (\infty,1)Cat
\end{displaymath}
to [[(∞,1)Cat]]. This takes the single object of $\mathbf{B}G$ to some $(\infty,1)$-category $V$.

Again we want to \textbf{define} the \emph{action groupoid} $V//G$ as the [[limit in a quasi-category|(∞,1)-categorical colimit]] over the action:

\begin{displaymath}
V//G := \lim_\to \rho
  \,.
\end{displaymath}
By the result \href{http://ncatlab.org/nlab/show/limit+in+a+quasi-category#WithValInooGrpd}{described here} this is, as before, equivalent to the pullback of the ``universal $(\infty,1)Cat$-bundle'' $Z \to (\infty,1)Cat$, namely to the [[coCartesian fibration]]

\begin{displaymath}
\itexarray{
    V//G &\to& Z
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat
  }
\end{displaymath}
classified by $\rho$ under the [[(∞,1)-Grothendieck construction]]. As before, we can continue a [[fiber sequence]] to the left by adjoining the $(\infty,1)$-categorical pullback along the point inclusion $* \to \mathbf{B}G$

\begin{displaymath}
\itexarray{
    V&\to& V//G &\to& Z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat
  }
  \,.
\end{displaymath}
The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $V$, which is $V$ itself, as indicated.

\hypertarget{some_comment}{}\subsection*{{Some comment}}\label{some_comment}

\begin{itemize}%
\item If the action of a Lie group $G$ on the manifold $X$ is free and proper, what you get is a manifold $X/G$.


\item If the action of a Lie group $G$ on the manifold $X$ is not necesssarily free and proper, what you get is a Lie groupoid, denoted (among other symbols) by $X//G$.



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

\begin{itemize}%
\item the [[groupoid cardinality]] of action groupoids is given by the [[class formula]]

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

\begin{itemize}%
\item P.J. Higgins, 1971, ``Categories and Groupoids'', van Nostrand, \{New York\}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1--195.


\item R. Brown, ``Topology and groupoids'', Booksurge, 2006, available from amazon; details \href{http://groupoids.org.uk/topgpds.html}{here}.


\item John Armstrong's article, \href{http://unapologetic.wordpress.com/2007/06/09/groupoids-and-more-group-actions/}{Groupoids (and more group actions)}


\item John Baez, \href{http://math.ucr.edu/home/baez/week249.html}{TWF 249}



\end{itemize}
[[!redirects action groupoids]] [[!redirects action group]] [[!redirects action groups]]



\end{document}