# Are fundamental groups and skeletal categories related

algebraic-topologycategory-theory

For any two topological spaces $$X, Y$$ that are homotopy-equivalent with $$f: X\to Y$$, we have that their fundamental groups $$\pi_1(X, x_0) \cong \pi_1(Y, f(x_0))$$ are isomorphic. Similarly, we have that for any equivalent categories $$\mathbb{C} \simeq \mathbb{D}$$ that their skeletal categories are isomorphic $$\text{sk}\ \mathbb{C} \cong \text{sk}\ \mathbb{D}$$. The construction of both skeletal categories and the fundamental group relates objects to their equivalence classes, so is there some "deeper" relation between these two?

Now, I know that one of these statements relates homotopies and group isomorphisms, while the other relates category equivalences and category isomorphisms, so the analogy may break down at some point. It's just that since Category Theory grew off of Algebraic Topology, and that most of the notation and concepts are very similar in both subjects that I wondered whether there is a connection between these 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.