# PlanetPhysics/Groupoid Action

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Let $\displaystyle \grp$
be a groupoid and ${\displaystyle X}$ a topological space. A groupoid action , or $\displaystyle \grp$
-action, on ${\displaystyle X}$ is given by two maps: the anchor map  ${\displaystyle \pi :X\longrightarrow G_{0}}$ and a map ${\displaystyle \mu :X\times _{G_{0}}G_{1}\longrightarrow X,}$ with the latter being defined on pairs ${\displaystyle (x,g)}$ such that ${\displaystyle \pi (x)=t(g)}$, written as ${\displaystyle \mu (x,g)=xg}$. The two maps are subject to the following conditions:

• ${\displaystyle \pi (xg)=s(g),}$
• ${\displaystyle xu(\pi (x))=x,}$ and
• ${\displaystyle (xg)h=x(gh),}$ whenever the operations are defined.

{\mathbf Note:} The groupoid action generalizes the concept of group action in a non-trivial way