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?
Best Answer
I'm not sure how to answer this, because it already seems pretty geometric to me. So let me answer a slightly different question: what is the fundamental groupoid good for? Since one knows that the fundamental group and groupoid are equivalent as categories for path connected spaces, it's tempting to view the groupoid as giving nothing new. But in fact, there are situations when it seems more natural. For example, if a group $G$ acts continuously on a space $X$, then unless one knows something more, we only get an outer action of $G$ on $\pi_1(X,x)$ i.e. it's only well defined up to inner automorphisms. However, $G$ will act on the fundamental groupoid $\Pi_1(X)$ on the nose, and in fact, the previous statement becomes easier to see from this point of view.