## 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?