Path-connectedness in the language of category theory?

Question

A non-empty topological space $X$ is connected iff the functor $\operatorname{Hom}_{\mathsf{Top}}(X,-)$ preserves all coproducts. Is there a similar category-theoretic characterization of a (non-empty) path-connected topological space?

Answer 1

Denote by $\bullet$ any one-point space, then you have two maps (of sets) $$\mathrm{ev}_i: \mathrm{Hom}_{\mathbf{Top}}([0,1],X) \longrightarrow \mathrm{Hom}_{\mathbf{Top}}(\bullet,X) = X \ \ \ \ f \longmapsto (\bullet \longmapsto f(i))$$ for any topological space, where $i=0$ or $1$. Let me consider the coequalizer of these two maps, we denote by $\pi_0(X)$ this coequalizer. In the category of sets, coequalizer of two maps is the quotient of the codomain by the smallest equivalence relation that "patching" two given maps. That being said, the coequalizer of the maps above is $$\pi_0(X) = X/(\mathrm{ev}_0(f) \sim \mathrm{ev}_1(f)) \ (f:[0,1] \longrightarrow X).$$ which is simply $X$ mod out by the relation that two points are equivalent iff there exists a path between them. Therefore it is trivial iff for every two points, there exists a path connecting them, and this is the definition of a path-connected space.

Hence, $X$ is path-connected iff $\pi_0(X)=\bullet$ (or $1$) and moreover, from the very definition, $\pi_0$ is a functor from $\mathbf{Top}$ to $\mathbf{Sets}$.