groupoid action

Definition 0.1.

Let $𝒢$ be a groupoid and $X$ a topological space. A groupoid action, or $𝒢$-action, on $X$ is given by two maps: the anchor map $\pi :X⟶G_0$ and a map $\mu :X{\times }_{G_0}{G}_1⟶X,$ 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