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 $x{\overset {g}{\to }}y$ and $y{\overset {h}{\to }}z$ is $x{\overset {hg}{\to }}z$.

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

$X\times G\,{\overset {s}{\underset {t}{\rightrightarrows }}}\,X$

where $s,t$ denote the source and the target of a morphism in ${\mathcal {G}}$; 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 ${\mathcal {G}}:=X\times G\rightrightarrows X$:

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

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 ${\mathcal {G}}:=X\times G\rightrightarrows X$ is a topological groupoid. In such case

• the group action is proper if and only if ${\mathcal {G}}\rightrightarrows X$ is proper;
• ${\mathcal {G}}\rightrightarrows X$ 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 ${\mathcal {G}}:=X\times G\rightrightarrows X$ is a Lie groupoid. In such case

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

The Lie algebroid of the action groupoid ${\mathcal {G}}:=X\times G\rightrightarrows X$ is the action algebroid associated to the infinitesimal action of the Lie algebra ${\mathfrak {g}}=\mathrm {Lie} (G)$ 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

$\cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,X\times G\times G\,{\underset {\rightarrow }{\rightrightarrows }}\,X\times G\,\rightrightarrows \,X$

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

References

Works cited

• Khan, Adeel A. (2023), Lectures on Algebraic Stacks (PDF)

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