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

$X\times G\to X,$

we get the groupoid ${𝒢}$ (= a category whose morphisms are all invertible) where

• objects are elements of $X$,
• morphisms from $x$ to $y$ are the actions of elements $g$ in $G$ such that $y=xg$,
• compositions for $x\stackrel{g}{\to }y$ and $y\stackrel{h}{\to }z$ is $x\stackrel{hg}{\to }z$.

A groupoid is often depicted using two arrows. Here the above can be written as:

$X\times G\stackrel{s}{\underset{t}{\rightrightarrows }}X$

where $s,t$ denote the source and the target of a morphism in ${𝒢}$; thus, $s(x,g)=x$ is the projection and $t(x,g)=xg$ is the given group action (here the set of morphisms in ${𝒢}$ is identified with $X\times G$).

In an ∞-category

Let $C$ be an ∞-category and $G$ a groupoid object in it. Then a group action or an action groupoid on an object X in C is the simplicial diagram

$\cdots \underset{\rightrightarrows }{\rightrightarrows }X\times G\times G\underrightarrow{\rightrightarrows }X\times G\rightrightarrows X$

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

References

Works cited

• Khan, Adeel A. (2023), Lectures on Algebraic Stacks, https://www.preschema.com/papers/stacksncts.pdf 

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