TITLE: Geometric interpretation of the fundamental groupoid
QUESTION [6 upvotes]: Motivation
The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nice enough spaces the (integral) homology groups count $n$-dimensional holes.

A groupoid is a category where all morphisms are invertible. Given a space $X$, the fundamental groupoid of $X$, $\Pi_1(X)$, is the category whose objects are the points of $X$ and the morphisms are homotopy classes of maps rel end points. It's clear that $\Pi_1(X)$ is a groupoid and the group object at $x \in X$ is simply the fundamental group $\pi_1(X,x)$. My question is:

Is there a geometric interpretation $\Pi_1(X)$ analogous to the geometric interpretation of homotopy groups and homology groups explained above?

REPLY [2 votes]: Thanks Andrew for the nice comments! 
In relation to the comment on G-spaces by Donu, I should point out that Chapter 11 of "Topology and groupoids" is on  "Orbit spaces, orbit groupoids". But I doublt many topologists are aware of the latter concept! 
My new jointly authored book `Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids' is published in August 2011 by the European Math. Soc and distributed in the Amercas by the AMS from October 2011. It is a kind of sequel to the above book, exploring some major uses of groupoids in higher homotopy theory.  A kind of new foundation for algebraic topology at the border between homotopy and homology, and applications of some higher order Seifert-van Kampen Theorems. In particular, the 2-dimensional version allows some quite explicit calculations of homotopy 2-types.  
See my web site for more infomration.