Kerodon

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Proposition 1.1.4.13. Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $X_{\bullet }$ is discrete (Definition 1.1.4.9). That is, $X_{\bullet }$ is isomorphic to a constant simplicial set $\underline{S}_{\bullet }$.

$(2)$

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 map $d_0: X_{n} \rightarrow X_{n-1}$ is a bijection.

$(4)$

The simplicial set $X_{\bullet }$ has dimension $\leq 0$, in the sense of Definition 1.1.3.9. That is, $X_{\bullet }$ does not contain any nondegenerate $n$-simplices for $n > 0$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 1.1.4.3, and the implication $(2) \Rightarrow (3)$ is immediate. To prove that $(3) \Rightarrow (4)$, we observe that if the face map $d_0: X_{n} \rightarrow X_{n-1}$ is bijective, then the degeneracy operator $s_0: X_{n-1} \rightarrow X_{n}$ is also bijective (since it is a right inverse of $d_0$). In particular, $s_0$ is surjective, so every $n$-simplex of $X_{\bullet }$ is degenerate.

We complete the proof by showing that $(4) \Rightarrow (1)$. If $X_{\bullet }$ is a simplicial set of dimension $\leq 0$ and $S = X_0$ is the set of vertices of $X_{\bullet }$, then Proposition 1.1.3.13 supplies an isomorphism of simplicial sets $\coprod _{v \in S} \Delta ^{0} \simeq X_{\bullet }$, whose domain can be identified with the constant simplicial set $\underline{S}_{\bullet }$ (by virtue of Remark 1.1.4.12 and Example 1.1.4.4). $\square$