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

${\displaystyle X\times G\to X,}$

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

• objects are elements of ${\displaystyle X}$,
• morphisms from ${\displaystyle x}$ to ${\displaystyle y}$ are the actions of elements ${\displaystyle g}$ in ${\displaystyle G}$ such that ${\displaystyle y=xg}$,
• compositions for ${\displaystyle x{\overset {g}{\to }}y}$ and ${\displaystyle y{\overset {h}{\to }}z}$ is ${\displaystyle x{\overset {hg}{\to }}z}$.^[1]

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

${\displaystyle X\times G\,{\overset {s}{\underset {t}{\rightrightarrows }}}\,X}$

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

In an ∞-category

Let ${\displaystyle C}$ be an ∞-category and ${\displaystyle G}$ a groupoid object in it. Then a group action or an action groupoid on an object X in C is the simplicial diagram^[2]

${\displaystyle \cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,X\times G\times G\,{\underset {\rightarrow }{\rightrightarrows }}\,X\times G\,\rightrightarrows \,X}$

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