# Contents

## 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 equal in $S/G$, in the action groupoid they are just isomorphic. We may think of the action groupoid as a 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 $S⋊G$. 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 $X$ (see “Topology and Groupoids”, Chapter 11).

## Definition

### In category theory

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

• an object is an element of $S$

• a morphism from $s\in S$ to $s\prime \in S$ is a group element $g\in G$ with $gs=s\prime$. So, a general morphism is a pair $\left(g,s\right):s\to gs$.

• The composite of $\left(g,s\right):s\to gs=s\prime$ and $\left(g\prime ,s\prime \right):s\to g\prime s\prime$ is $\left(g\prime g,s\right):s\to g\prime gs$.

Equivalently, we may define the action groupoid $S//G$ to be the groupoid

$\begin{array}{ccc}& & S×G\\ & {}^{s:={p}_{1}}↙& & {↘}^{t=\rho }\\ S& & & & S\end{array}$\array{ && S \times G \\ & {}^{s := p_1}\swarrow && \searrow^{t = \rho} \\ S &&&& S }

with composition

$\left(S×G\right){×}_{t,s}\left(S×G\right)\simeq S×G×G\to S×G$(S \times G) \times_{t,s} (S \times G) \simeq S \times G \times G \to S \times G

given by the product in $G$.

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

$S//G:=\left\{s\stackrel{g}{\to }\rho \left(s,g\right)\mid s\in S,g\in G\right\}.$S//G:=\{s\stackrel{g}{\to} \rho(s,g) | s\in S, g\in G\}.

### In (∞,1)-category theory

###### Definition

Let $C$ be an ($\infty$,1)-category, let $G\in \mathrm{Grpd}\left(C\right)$ be a groupoid object in $C$, let $X\in C$ be an object. Then the simplicial object

$...\stackrel{\to }{\stackrel{\to }{\to }}X{×}_{{G}_{0}}G{×}_{p}G\stackrel{\to }{\to }X{×}_{{G}_{0}}G\to X$... \stackrel{\to}{\stackrel{\to}{\to}} X\times_{G_0}G\times_p G \stackrel{\to}{\to} X\times_{G_0}G \to X

such that the degree-wise projections give a simplicial map

$\begin{array}{ccccc}...X{×}_{{G}_{0}}G{×}_{p}G& \stackrel{\to }{\to }& X{×}_{{G}_{0}}G& \to & X\\ ↓& & ↓& & {↓}^{a}\\ G{×}_{p}G& \stackrel{\to }{\to }& G& \stackrel{p}{\to }& {G}_{0}\end{array}$\array{ ... X\times_{G_0}G\times_p G & \stackrel{\to}{\to} & X\times_{G_0}G & \to & X \\ \downarrow&&\downarrow&&\downarrow^a \\ G\times_p G & \stackrel{\to}{\to} & G & \xrightarrow{p} & G_0 }

is called an action of $G$ on $X$. The colimit $\mathrm{colim}\phantom{\rule{thickmathspace}{0ex}}X{×}_{{G}_{0}}^{{×}_{•}}$ is called action $\infty$-groupoid of $G$ on $X$.

## 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

$\rho :BG\to \mathrm{Sets}$\rho : \mathbf{B}G \to Sets

from the group $G$ regarded as a one-object groupoid, denoted $BG$.

This functor sends the single object of $BG$ to the set $S$.

### As a pseudo colimit

$S//G$ is the 2-colimit of $\rho$,

$S//G\simeq {\mathrm{colim}}_{BG}\rho \phantom{\rule{thinmathspace}{0ex}}.$S//G \simeq colim_{\mathbf{B}G} \rho \,.

The universal cocone consists of cells of the form

$\begin{array}{ccccc}S& & \stackrel{\rho \left(g\right)}{\to }& & S\\ & ↘& \stackrel{\simeq }{⇐}& ↙\\ & & S//G\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ S &&\stackrel{\rho(g)}{\to}&& S \\ & \searrow &\stackrel{\simeq}{\Leftarrow}& \swarrow \\ && S//G } \,,

where the 2-morphism is uniquely specified and in components given by $s↦\left(s\stackrel{g}{\to }\rho \left(s,g\right)\right)$.

### As associated universal bundle

Let ${\mathrm{Set}}_{*}$ be the category of pointed sets and ${\mathrm{Sets}}_{*}\to \mathrm{Sets}$ be the canonical forgetful functor. We can think of this as the “universal $\mathrm{Set}$-bundle”.

Then $S//G$ is the pullback

$\begin{array}{ccc}S//G& \to & {\mathrm{Sets}}_{*}\\ ↓& & ↓\\ BG& \stackrel{\rho }{\to }& \mathrm{Sets}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ S//G &\to& Sets_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Sets } \,.

One place where we discussed this is the comment It was David Roberts who apparently first noticed….

Notice also that an action of $G$ on the set $S$ gives rise to a morphism $p:S⋊G\to G$ 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.

### As a stack

In the case where the action is 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 $S//G$.

## Action $\infty$-groupoid

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

Let $G$ be an ∞-group in that $BG$ is an ∞-groupoid with a single object. An action of $G$ on an (∞,1)-category is an (∞,1)-functor

$\rho :BG\to \left(\infty ,1\right)\mathrm{Cat}$\rho : \mathbf{B}G \to (\infty,1)Cat

to (∞,1)Cat. This takes the single object of $BG$ to some $\left(\infty ,1\right)$-category $V$.

Again we want to define the action groupoid $V//G$ as the (∞,1)-categorical colimit over the action:

$C//G:=\underset{\to }{\mathrm{lim}}\rho \phantom{\rule{thinmathspace}{0ex}}.$C//G := \lim_\to \rho \,.

By the result described here this is, as before, equivalent to the pullback of the “universal $\left(\infty ,1\right)\mathrm{Cat}$-bundle” $Z\to \left(\infty ,1\right)\mathrm{Cat}$, namely to the coCartesian fibration

$\begin{array}{ccc}V//G& \to & Z\\ ↓& & ↓\\ BG& \stackrel{\rho }{\to }& \left(\infty ,1\right)\mathrm{Cat}\end{array}$\array{ V//G &\to& Z \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat }

classified by $\rho$ under the (∞,1)-Grothendieck construction. As before, we can continue a fiber sequence to the left by adjoining the $\left(\infty ,1\right)$-categorical pullback along the point inclusion $*\to BG$

$\begin{array}{ccccc}V& \to & V//G& \to & Z\\ ↓& & ↓& & ↓\\ *& \to & BG& \stackrel{\rho }{\to }& \left(\infty ,1\right)\mathrm{Cat}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ V&\to& V//G &\to& Z \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat } \,.

The resulting total $\left(\infty ,1\right)$-pullback rectangle is the fiber of $Z\to \left(\infty ,1\right)\mathrm{Cat}$ over the $\left(\infty ,1\right)$-category $C$, which is $V$ itself, as indicated.

## References

• P.J. Higgins, 1971, “Categories and Groupoids”, van Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195.

• R. Brown, “Topology and groupoids”, Booksurge, 2006.

• John Armstrong’s article, Groupoids (and more group actions)

• John Baez, TWF 249

Revised on April 25, 2012 16:45:41 by Stephan Alexander Spahn (79.227.176.198)