In mathematics, an action groupoid (or transformation groupoid) is a groupoid that encodes a group action.
Definition Given any right group action
its action groupoid is the small category defined as follows: the objects are elements of , the morphisms from to are the elements of ; the composition between and is .https://www.matem.unam.mx/~omar/groupoids/day1.html
Since a groupoid is often depicted using two arrows, the action groupoid can be written as
where denote the source and the target of a morphism in ; thus, is the projection and is the given group action. Moreover
the unit of is ; the inverse of is .
The analogous definition can be given for left group actions.
Properties Several concepts related to a group action can be presented via its action groupoid :
the isotropy group at coincides with the isotropy group of at ; the orbit of coincides with the orbit of at ; the orbit space of the group action coincides with the orbit space of .
As a consequence, a group action is transitive if and only if its action groupoid is transitive.
Topological setting If is a topological group and the -action is a continuous group action, then its action groupoid is a topological groupoid. In such case
the group action is proper if and only if is proper; is source -connected if and only if is -connected;
Smooth setting If is a Lie group and the -action is a Lie group action, then its action groupoid is a Lie groupoid. In such case
is étale if and only if is discrete; is effective if the -action is free and is discrete; if the group action is transitive, then is isomorphic to the gauge groupoid associated to the principal -bundle (for any point ).
The Lie algebroid of the action groupoid is the action algebroid associated to the infinitesimal action of the Lie algebra on .
In an ∞-category Let be an ∞-category and a groupoid object in it. Then a group action or an action groupoid on an object in is the simplicial diagram
that satisfies the axioms similar to an action groupoid in the usual case.
References
Works cited
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