In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action
formula_1
we get the groupoid formula_2 (= a category whose morphisms are all invertible) where
A groupoid is often depicted using two arrows. Here the above can be written as:
formula_12
where formula_13 denote the source and the target of a morphism in formula_2; thus, formula_15 is the projection and formula_16 is the given group action (here the set of morphisms in formula_2 is identified with formula_18).
In an ∞-category.
Let formula_19 be an ∞-category and formula_7 a groupoid object in it. Then a group action or an action groupoid on an object "X" in "C" is the simplicial diagram 
formula_21
that satisfies the axioms similar to an action groupoid in the usual case.
Further reading.