Action groupoid

In mathematics, an action groupoid (or transformation groupoid) is a groupoid that encodes a group action.

Definition

Given any right group action

$X\times G\to X,$

its action groupoid is the small category defined as follows:

• the objects are elements of $X$,
• the morphisms from $x$ to $y$ are the elements of $X\times G$;
• the composition between $x\stackrel{g}{\to }y$ and $y\stackrel{h}{\to }z$ is $x\stackrel{hg}{\to }z$.

Since a groupoid is often depicted using two arrows, the action groupoid can be written as

$X\times G\stackrel{s}{\underset{t}{\rightrightarrows }}X$

where $s,t$ denote the source and the target of a morphism in $𝒢$; thus, $s(x,g)=x$ is the projection and $t(x,g)=xg$ is the given group action. Moreover

• the unit of $x\in M$ is $(x,e)$;
• the inverse of $(x,g)$ is $(xg,g^{-1})$.

The analogous definition can be given for left group actions.

Properties

Several concepts related to a group action $X\times G\to X$ can be presented via its action groupoid $𝒢:=X\times G\rightrightarrows X$:

• the isotropy group $G_x:=\{g\in G|xg=x\}\subseteq G$ at $x\in X$ coincides with the isotropy group ${𝒢}_x:=𝒢(x,x)$ of $𝒢\rightrightarrows X$ at $x\in X$;
• the orbit ${𝒪}_x=\{y\in M|\mathrm{\exists }g\in G\text{ with }y=xg\}\subseteq M$ of $x\in X$ coincides with the orbit $s(t^{-1}(x))$ of $𝒢\rightrightarrows X$ at $x\in X$;
• the orbit space $M/G$ of the group action coincides with the orbit space of $𝒢\rightrightarrows X$.

As a consequence, a group action is transitive if and only if its action groupoid is transitive.

Topological setting

If $G$ is a topological group and the $G$-action is a continuous group action, then its action groupoid $𝒢:=X\times G\rightrightarrows X$ is a topological groupoid. In such case

• the group action is proper if and only if $𝒢\rightrightarrows X$ is proper;
• $𝒢\rightrightarrows X$ is source $k$-connected if and only if $G$ is $k$-connected;

Smooth setting

If $G$ is a Lie group and the $G$-action is a Lie group action, then its action groupoid $𝒢:=X\times G\rightrightarrows X$ is a Lie groupoid. In such case

• $𝒢\rightrightarrows X$ is étale if and only if $G$ is discrete;
• $𝒢\rightrightarrows X$ is effective if the $G$-action is free and $G$ is discrete;
• if the group action is transitive, then $𝒢\rightrightarrows X$ is isomorphic to the gauge groupoid associated to the principal $G_x$-bundle $G\to {𝒪}_x=X$ (for any point $x\in X$).

The Lie algebroid of the action groupoid $𝒢:=X\times G\rightrightarrows X$ is the action algebroid associated to the infinitesimal action of the Lie algebra $𝔤=\mathrm{L}\mathrm{i}\mathrm{e}(G)$ on $X$.

In an ∞-category

Let $C$ be an ∞-category and $G$ a groupoid object in it. Then a group action or an action groupoid on an object $X$ in $C$ is the simplicial diagram

$\cdots \underset{\rightrightarrows }{\rightrightarrows }X\times G\times G\underrightarrow{\rightrightarrows }X\times G\rightrightarrows X$

that satisfies the axioms similar to an action groupoid in the usual case.

References

Works cited

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Further reading

• https://ncatlab.org/nlab/show/action+groupoid
• https://mathoverflow.net/questions/130950/groupoids-vs-action-groupoids
• https://www.math.sci.hokudai.ac.jp/~wakate/mcyr/2023/pdf/uchimura_tomoki.pdf in Japanese