The action of a group $G$ on a topological space $X$ can be viewed as a functor $F: G \to \mathcal{Top}$ with $F(\*)=X$. (Here I'm viewing a group as a category with one object, $ \* $, and the morphisms are isomorphisms labeled by the group elements.)


We can extend this idea and define the action of a groupoid $\mathcal{G}$ on a space to be a functor $F:\mathcal{G} \to\mathcal{Top}$. 


Are there any naturally occurring examples of a groupoid action on a space? (Other than the ones where the groupoid is actually a group.)



Groupoid actions on spaces
Perhaps the most natural example is given by universal covers?


Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X\_x$ be the universal covering of 
$X$ taken at $x$ (the fiber at $y \in X$ is the homotopy classes of paths $[0,1]\to X$ which start
at $x$ and end at $y$, where we are taking homotopy classes relative to $\lbrace0,1\rbrace$).


Let $\pi$ be the fundamental groupoid of $X$. Then there is a functor $\pi\to \text{Top}$ given on objects by $x\mapsto \tilde X\_x$. On morphisms of $\pi$ from $x$ to $y$, the functor is given by the map $\tilde X\_x \to \tilde X\_y$ that is induced by concatenating paths. 


