In mathematics, an '''action groupoid''' or a '''transformation groupoid''' is a groupoid that expresses a group action. Namely, given a (right) group action :<math>X \times G \to X,</math> we get the groupoid <math>\mathcal{G}</math> (= a category whose morphisms are all invertible) where *objects are elements of <math>X</math>, *morphisms from <math>x</math> to <math>y</math> are the actions of elements <math>g</math> in <math>G</math> such that <math>y = xg</math>, *compositions for <math>x \overset{g}\to y</math> and <math>y \overset{h}\to z</math> is <math>x \overset{hg}\to z</math>.<ref>https://www.matem.unam.mx/~omar/groupoids/day1.html</ref>
A groupoid is often depicted using two arrows. Here the above can be written as: :<math>X \times G \,\overset{s}\underset{t}\rightrightarrows\, X</math> where <math>s, t</math> denote the source and the target of a morphism in <math>\mathcal{G}</math>; thus, <math>s(x, g) = x</math> is the projection and <math>t(x, g) = xg</math> is the given group action (here the set of morphisms in <math>\mathcal{G}</math> is identified with <math>X \times G</math>).
== In an ∞-category == Let <math>C</math> be an ∞-category and <math>G</math> a groupoid object in it. Then a group action or an action groupoid on an object ''X'' in ''C'' is the simplicial diagram<ref>{{harvnb|Khan|2023|loc=Remark 4.2.4.}}</ref> :<math>\cdots \, \underset{\rightrightarrows}\rightrightarrows \, X \times G \times G \, \underset{\rightarrow}\rightrightarrows \, X \times G \, \rightrightarrows\, X</math> <!-- sorry I didn’t know how to better format this. -->that satisfies the axioms similar to an action groupoid in the usual case.
== References == {{reflist}} === Works cited === {{refbegin}} *{{citation | first = Adeel A. | last = Khan | year = 2023 | title = Lectures on Algebraic Stacks | url = https://www.preschema.com/papers/stacksncts.pdf }} {{refend}}
== 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
{{algebra-stub}} Category:Algebraic structures