Proposition 4.6.8.24. Let $K$ be a simplicial set. Then the morphism $\rho _{K}: \Phi (K) \rightarrow K$ of Construction 4.6.8.23 is a categorical equivalence of simplicial sets.
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Proof of Proposition 4.6.8.24. Let $K$ be a simplicial set. We wish to show that the map $\rho _{K}: \Phi (K) \rightarrow K$ of Construction 4.6.8.23 is a categorical equivalence of simplicial sets. Using Corollary 4.6.8.22, we can write $\rho _{K}$ as a filtered colimit of morphisms $\rho _{K_{\alpha }}: \Phi ( K_{\alpha } ) \rightarrow K_{\alpha }$, where $K_{\alpha }$ ranges over the collection of all finite simplicial subsets of $K$ (Remark 3.6.1.8). Since the collection of categorical equivalences is closed under the formation of filtered colimits (Corollary 4.5.7.2), it will suffice to show that each $\rho _{ K_{\alpha } }$ is a categorical equivalence. We may therefore replace $K$ by $K_{\alpha }$ and thereby reduce to the case where the simplicial set $K$ is finite.
Since $K$ is a finite simplicial set, it has dimension $\leq n$ for some integer $n \geq -1$. We proceed by induction on $n$. If $n=-1$, then both $K$ and $\Phi (K)$ are empty, and there is nothing to prove. We may therefore assume that $n \geq 0$ and that $\rho _{K'}$ is a categorical equivalence for every simplicial set $K'$ of dimension $< n$. We now proceed by induction on the number $m$ of nondegenerate $n$-simplices of $K$. If $m = 0$, then $K$ has dimension $\leq n-1$ and the desired result holds by virtue of our inductive hypothesis. We may therefore assume that $K$ has at least one nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow K$. Using Proposition 1.1.4.12, we see that there is a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ K' \ar [r] & K, } \]
where $S'$ is a simplicial set of dimension $\leq n$ with exactly $(m-1)$-nondegenerate $m$-simplices. We then have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Phi ( \operatorname{\partial \Delta }^ n ) \ar [rr] \ar [dd] \ar [dr]^{ \rho _{ \operatorname{\partial \Delta }^ n }} & & \Phi ( \Delta ^ n) \ar [dd] \ar [dr]^{ \rho _{ \Delta ^ n } } & \\ & \operatorname{\partial \Delta }^ n \ar [rr] \ar [dd] & & \Delta ^ n \ar [dd] \\ \Phi (K') \ar [rr] \ar [dr]^{ \rho _{K'} } & & \Phi (K) \ar [dr]^{ \rho _ K} & \\ & K' \ar [rr] & & K } \]
where the front and back faces are pushout squares (Corollary 4.6.8.22) in which the horizontal maps are monomorphisms (Remark 4.6.8.15), and are therefore categorical pushout squares (Example 4.5.4.12). Our inductive hypotheses guarantees that the maps $\rho _{K'}$ and $\rho _{\operatorname{\partial \Delta }^ n}$ are categorical equivalences. Consequently, to show that $\rho _{K}$ is a categorical equivalence, it will suffice to show that $\rho _{\Delta ^ n}$ is a categorical equivalence (Proposition 4.5.4.9). We may therefore replace $K$ by $\Delta ^ n$ and thereby reduce to the case where $K$ is a standard simplex.
If $n = 0$, then the map $\rho _{\Delta ^ n}: \Phi ( \Delta ^ n ) \rightarrow \Delta ^ n$ is an isomorphism (Example 4.6.8.12). We may therefore assume without loss of generality that $n > 0$, so that Lemma 4.6.8.30 supplies an isomorphism of simplicial sets $\Phi ( \Delta ^ n ) \simeq \Delta ^{0} \diamond \Phi ( \Delta ^{n-1} )$. Using this isomorphism, we can identify $\rho _{\Delta ^ n}$ with the composite map
\[ \Delta ^{0} \diamond \Phi ( \Delta ^{n-1} ) \xrightarrow { \operatorname{id}\diamond \rho _{ \Delta ^{n-1} } } \Delta ^{0} \diamond \Delta ^{n-1} \xrightarrow {c} \Delta ^{0} \star \Delta ^{n-1} \simeq \Delta ^ n, \]
where $c$ is the comparison map of Notation 4.5.8.3 (to check this, it suffices to observe that they agree on vertices). Our inductive hypothesis guarantees that $\rho _{ \Delta ^{n-1} }$ is a categorical equivalence of simplicial sets, so that the induced map $\Delta ^{0} \diamond \Phi ( \Delta ^{n-1} ) \xrightarrow { \operatorname{id}\diamond \rho _{ \Delta ^{n-1} }} \Delta ^0 \diamond \Delta ^{n-1}$ is also a categorical equivalence by virtue of Remark 4.5.8.7. We are therefore reduced to showing that $c$ is a categorical equivalence, which is a special case of Proposition 4.5.8.12. $\square$