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:

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$\pi(xg)=s(g),$
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$xu(\pi(x))=x,$ and
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$(xg)h=x(gh),$ 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