$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.2.3.4. Let $(S, \leq )$ and $(T, \leq )$ be linearly ordered sets, and let $f: S \rightarrow T$ be a nondecreasing function. The following conditions are equivalent:
- $(1)$
The function $f: S \rightarrow T$ is cofinal in the sense of Definition 4.7.1.26. That is, for every element $t \in T$, there exists an element $s \in S$ satisfying $t \leq f(s)$.
- $(2)$
The induced morphism of simplicial sets $\operatorname{N}_{\bullet }(S) \rightarrow \operatorname{N}_{\bullet }(T)$ is right cofinal, in the sense of Definition 7.2.1.1.
Proof.
For each $t \in T$, set $S_{\geq t} = \{ s \in S: t \leq f(s) \} $, which we regard as a linearly ordered subset of $S$. Using Theorem 7.2.3.1, we can rewrite conditions $(1)$ and $(2)$ as follows:
- $(1')$
For each element $t \in T$, the linearly ordered set $S_{\geq t}$ is nonempty.
- $(2')$
For each element $t \in T$, the linearly ordered set $S_{\geq t}$ has weakly contractible nerve.
The implication $(2') \Rightarrow (1')$ is immediate, and the reverse implication follows from Corollary 3.2.8.5.
$\square$