Proposition 1.1.5.14 (00G3)

Proposition 1.1.5.14. Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$: The simplicial set $X_{\bullet }$ is discrete (Definition 1.1.5.10). That is, $X_{\bullet }$ is isomorphic to a constant simplicial set $\underline{S}$.
$(2)$: For every morphism $\alpha : [m] \rightarrow [n]$ in the category $\operatorname{{\bf \Delta }}$, the induced map $X_{n} \rightarrow X_{m}$ is a bijection.
$(3)$: For every positive integer $n$, the $0$th face operator $d^{n}_0: X_{n} \rightarrow X_{n-1}$ is a bijection.
$(4)$: The simplicial set $X_{\bullet }$ has dimension $\leq 0$, in the sense of Definition 1.1.3.1. That is, $X_{\bullet }$ does not contain any nondegenerate $n$-simplices for $n > 0$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 1.1.5.3, and the implication $(2) \Rightarrow (3)$ is immediate. To prove that $(3) \Rightarrow (4)$, we observe that if the face operator $d^{n}_0: X_{n} \rightarrow X_{n-1}$ is bijective, then the degeneracy operator $s^{n-1}_0: X_{n-1} \rightarrow X_{n}$ is also bijective (since it is a right inverse of $d^{n}_0$). In particular, $s^{n-1}_0$ is surjective, so every $n$-simplex of $X_{\bullet }$ is degenerate. The implication $(4) \Rightarrow (1)$ follows from Example 1.1.5.6. $\square$