Path components of an anima

Definition 2.4.17 (Path components of an anima). Recall from Example 1.8.14(1) that the inclusion \(\Set \hookrightarrow \An \) admits a left adjoint \(\pi _0\colon \An \to \Set \), sending an anima \(X\) to its set of path components \(\pi _0(X)\). The unit of the adjunction gives a map of animae \(X \to \pi _0(X)\), and for an element \([x] \in \pi _0(X)\) we define the corresponding path component of \(X\) as the fiber of this map over \([x]\). We say that \(X\) is connected if \(\pi _0(X)\) is a singleton.