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

Given any right group action

${\displaystyle X\times G\to X,}$

its action groupoid is the small category defined as follows:

• the objects are elements of ${\displaystyle X}$,
• the morphisms from ${\displaystyle x}$ to ${\displaystyle y}$ are the elements of ${\displaystyle X\times G}$;
• the composition between ${\displaystyle x{\overset {g}{\to }}y}$ and ${\displaystyle y{\overset {h}{\to }}z}$ is ${\displaystyle x{\overset {hg}{\to }}z}$.^[1]

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

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

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

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

The analogous definition can be given for left group actions.

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

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

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

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

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

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

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

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

Let ${\displaystyle C}$ be an ∞-category and ${\displaystyle G}$ a groupoid object in it. Then a group action or an action groupoid on an object ${\displaystyle X}$ in ${\displaystyle C}$ is the simplicial diagram^[2]

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

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

• 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