nLab
action groupoid

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

Given an action $\rho :S\times G\to S$ 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 $(g,s):s\to gs$.

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

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

$$\begin{array}{ccc}& & S\times G\\ & {}^{s:=p_1}\swarrow & & {\searrow }^{t=\rho }\\ S& & & & S\end{array}$$\array{ && S \times G \\ & {}^{s := p_1}\swarrow && \searrow^{t = \rho} \\ S &&&& S }

with composition

$$(S\times G){\times }_{t,s}(S\times G)\simeq S\times G\times G\to S\times G$$(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

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

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 $\rho$,

$$S//G\simeq {\mathrm{colim}}_{BG}\rho .$$S//G \simeq colim_{\mathbf{B}G} \rho \,.

The universal cocone consists of cells of the form

$$\begin{array}{ccccc}S& & \stackrel{\rho (g)}{\to }& & S\\ & \searrow & \stackrel{\simeq }{\Leftarrow }& \swarrow \\ & & S//G\end{array},$$\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 $s\mapsto (s\stackrel{g}{\to }\rho (s,g))$.

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

$$\begin{array}{ccc}S//G& \to & {\mathrm{Sets}}_{*}\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Sets}\end{array}.$$\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 $S//G$.

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