Action groupoid

In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action

we get the groupoid (= a category whose morphisms are all invertible) where

• objects are elements of ,
• morphisms from to are the actions of elements in such that ,
• compositions for and is .

A groupoid is often depicted using two arrows. Here the above 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 (here the set of morphisms in is identified with ).

In an ∞-category

Let be an ∞-category and a groupoid object in it. Then a group action or an action groupoid on an object X in C 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