Warning 1.2.1.9. Let $S$ be a simplicial set. As we will soon see, the set $\pi _0(S_{})$ admits many different descriptions:
We can identify $\pi _0( S_{} )$ with the set of connected components of $S_{}$ (Definition 1.2.1.8).
We can identify $\pi _0( S_{} )$ with a colimit of the diagram $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ given by the simplicial set $S_{}$ (Remark 1.2.1.20).
We can identify $\pi _0( S_{} )$ with the quotient of the set of vertices of $S$ by an equivalence relation $\sim $ generated by the set of edges of $S$ (Remark 1.2.1.23).
We can identify $\pi _0( S_{} )$ with the set of connected components of the directed graph $\mathrm{Gr}( S_{} )$ introduced in ยง1.1.6 (Variant 1.2.1.24).
If $S_{}$ is a Kan complex, we can identify $\pi _0( S )$ as the set of isomorphism classes of objects in the fundamental groupoid $\pi _{\leq 1}(S_{})$ (Remark 1.4.6.13).
Because of this abundance of perspectives, it often will be convenient to view $I = \pi _0( S_{} )$ as an abstract index set which is equipped with a bijection
rather than as the set of connected components itself.