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

In category theory

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\}.

In (∞,1)-category theory

Definition

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

$$...\overrightarrow{\overrightarrow{\to }}X{\times }_{G_0}G{\times }_{p}G\overrightarrow{\to }X{\times }_{G_0}G\to X$$... \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

$$\begin{array}{ccccc}...X{\times }_{G_0}G{\times }_{p}G& \overrightarrow{\to }& X{\times }_{G_0}G& \to & X\\ \downarrow & & \downarrow & & {\downarrow }^a\\ G{\times }_{p}G& \overrightarrow{\to }& G& \stackrel{p}{\to }& G_0\end{array}$$\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 $\mathrm{colim}X{\times }_{G_0}^{{\times }_{•}}$ is called action $\mathrm{\infty }$-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 $\rho$,

$$S//G\simeq {\mathrm{colim}}_{BG}\rho .$$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 $S//G$.

Action $\mathrm{\infty }$-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

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

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

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

$$C//G:=\underrightarrow{\lim }\rho .$$C//G := \lim_\to \rho \,.

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

$$\begin{array}{ccc}V//G& \to & Z\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& (\mathrm{\infty },1)\mathrm{Cat}\end{array}$$\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 $(\mathrm{\infty },1)$-categorical pullback along the point inclusion $*\to BG$

$$\begin{array}{ccccc}V& \to & V//G& \to & Z\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & BG& \stackrel{\rho }{\to }& (\mathrm{\infty },1)\mathrm{Cat}\end{array}.$$\array{ V&\to& V//G &\to& Z \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat } \,.

The resulting total $(\mathrm{\infty },1)$-pullback rectangle is the fiber of $Z\to (\mathrm{\infty },1)\mathrm{Cat}$ over the $(\mathrm{\infty },1)$-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