Proof.
The implication $(4) \Rightarrow (3)$ is trivial, the implication $(3) \Rightarrow (2)$ follows from Example 1.1.3.2, and the implication $(2) \Rightarrow (1)$ follows from Remarks 1.1.3.3 and 1.1.3.4. It will therefore suffice to show that $(1)$ implies $(4)$. Assume that $S$ has dimension $\leq k$, and let $T$ denote the colimit $\varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S, \leq k} } \Delta ^ n$; we wish to show that the tautological map $f: T \rightarrow S$ is an isomorphism of simplicial sets. Since $S$ has dimension $\leq k$, it follows immediately from the construction that the image of $f$ contains every nondegenerate simplex of $S$. Applying Remark 1.1.2.6, we deduce that $f$ is an epimorphism of simplicial sets. We will complete the proof by showing that $f$ is injective. Let $\tau $ and $\tau '$ be $\ell $-simplices of $T$ satisfying $f(\tau ) = f(\tau ')$; we wish to show that $\tau = \tau '$. Choose an object $( [n] , \sigma ) \in \operatorname{{\bf \Delta }}_{S, \leq k}$ and a lift of $\tau $ to an $\ell $-simplex $\widetilde{\tau }$ of $\Delta ^ n$, which we can identify with a nondecreasing function from $[\ell ]$ to $[n]$. Note that $\widetilde{\tau }$ factors uniquely as a composition $[\ell ] \xrightarrow { \alpha } [m] \xrightarrow { \beta } [n]$, where $\alpha $ is surjective and $\beta $ is injective. Replacing $n$ by $\ell $ and $\sigma $ by the associated $\ell $-simplex of $S$, we can reduce to the case where $\widetilde{\tau }: [\ell ] \twoheadrightarrow [n]$ is a surjection. Using Proposition 1.1.3.8, we can factor $\sigma $ as a composition
\[ \Delta ^{n} \xrightarrow {\gamma } \Delta ^{p} \xrightarrow { \rho } S, \]
where $\gamma $ is surjective and $\rho $ is a nondegerate $p$-simplex of $S_{\bullet }$. Replacing $([n], \sigma )$ by $( [p], \rho )$ and $\widetilde{\tau }$ by the composition $\gamma \circ \widetilde{\tau }$, we can further assume that $\sigma $ is a nondegenerate $n$-simplex of $S_{\bullet }$. Similarly, we may assume that $\tau '$ lifts to an $m$-simplex $\widetilde{\tau }'$ of $\Delta ^{n'}$, for some object $([n'], \sigma ' )$ of $\operatorname{{\bf \Delta }}_{S, \leq k}$ where $\sigma '$ is nondegenerate and $\widetilde{\tau }': [m] \twoheadrightarrow [n']$ is surjective. We then have an equality
\[ \sigma \circ \widetilde{\tau } = f( \tau ) = f( \tau ' ) = \sigma ' \circ \widetilde{\tau }'. \]
The uniqueness assertion of Proposition 1.1.3.8 then implies that $([n], \sigma ) = ( [n'], \sigma ' )$ and $\widetilde{\tau } = \widetilde{\tau }'$, so that $\tau $ and $\tau '$ are the same $m$-simplex of $T$.
$\square$