groupoid action

Definition 0.1.

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

• $\pi(xg)=s(g),$

• $xu(\pi(x))=x,$ and

• $(xg)h=x(gh),$ whenever the operations are defined.

Note: The groupoid action generalizes the concept of group action in a non-trivial way.

 Title groupoid action Canonical name GroupoidAction Date of creation 2013-03-22 19:19:23 Last modified on 2013-03-22 19:19:23 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 9 Author bci1 (20947) Entry type Definition Classification msc 22A22 Classification msc 18B40 Synonym action Related topic GroupAction Related topic Groupoid Related topic GroupoidRepresentation4 Defines anchor map