Are fundamental groups and skeletal categories related?

Solution 1:

There is a simple relation between the two.

If $X$ is path-connected, then the fundamental group of $X$, expressed as a 1-element category, is a skeleton of its fundamental groupoid.

The more relevant analogy here is the preorder associated with a category, which is an example of $(-1)$-truncation. Given a category $C$, we can give the objects a pre-order by writing $a \leq b$ iff there is some morphism $a \to b$. We can then quotient the objects by equivalence to get a partial order.

Analogously, the fundamental groupoid is a higher-order truncation of the fundamental 2-groupoid. In the fundamental 2-groupoid, we also have paths between paths. But when we pass to the fundamental groupoid, we suppress the paths between between paths (or, equivalently, pretend that there is at most one path between paths). We can then quotient the paths by equivalence to get a set of paths between any two points up to homotopy equivalence.