Example 9.3.2.5. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ (Construction 5.5.1.1). Then:
The Kan complex $X$ is a discrete object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 9.3.2.1) if and only if every connected component of $X$ is contractible: that is, the projection map $X \rightarrow \pi _0(X)$ is a homotopy equivalence.
The Kan complex $X$ is a subterminal object of the $\infty $-category $\operatorname{\mathcal{S}}$ (in the sense of Definition 9.3.2.2) if and only if $X$ is either empty or contractible.
See Example 9.3.1.4.