Proposition 3.3.5.3. Let $f: X \rightarrow Y$ be an anodyne morphism of simplicial sets. Then the induced map $\operatorname{Sd}(f): \operatorname{Sd}(X) \rightarrow \operatorname{Sd}(Y)$ is also anodyne.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 3.3.5.3. Let $S$ be the collection of all morphisms of simplicial sets $f: X \rightarrow Y$ for which the induced map $\operatorname{Sd}(f): \operatorname{Sd}(X) \rightarrow \operatorname{Sd}(Y)$ is anodyne. Since the subdivision functor $\operatorname{Sd}$ preserves colimits, the collection $S$ is weakly saturated (in the sense of Definition 1.5.4.12). To prove Proposition 3.3.5.3, it will suffice to show that $S$ contains every horn inclusion. Fix a positive integer $n$ and another integer $0 \leq i \leq n$. We will complete the proof by showing that the inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ induces an anodyne map $\operatorname{Sd}( \Lambda ^{n}_{i} ) \rightarrow \operatorname{Sd}( \Delta ^ n )$.
Let $J = [n] \setminus \{ i\} $, let $P(J)$ denote the collection of all subsets of $J$, partially ordered by inclusion. Set $P_{-}(J) = P(J) \setminus \{ J \} $, $P_{+}(J) = P(J) \setminus \{ \emptyset \} $, and $P_{\pm }(J) = P(J) \setminus \{ \emptyset , J \} $. In what follows, we identify $\operatorname{Sd}( \Delta ^ n )$ with the nerve of the partially ordered set $\operatorname{Chain}[n]$ of nonempty subsets of $[n]$, and $\operatorname{Sd}( \Lambda ^{n}_{i} )$ with the nerve of the partially ordered subset of $\operatorname{Chain}[n]$ obtained by removing the elements $[n]$ and $J$ (Proposition 3.3.3.17). The construction $J_0 \mapsto J_0 \cup \{ i\} $ determines an inclusion of partially ordered sets $P(J) \rightarrow \operatorname{Chain}[n]$, hence a monomorphism of simplicial sets
\[ g: \operatorname{\raise {0.1ex}{\square }}^{J} = \operatorname{N}_{\bullet }( P(J) ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Chain}[n]) = \operatorname{Sd}( \Delta ^ n ). \]
Let $Z \subseteq \operatorname{Sd}( \Delta ^{n} )$ be the union of $\operatorname{Sd}( \Lambda ^{n}_{i} )$ with the image of $g$. An elementary calculation shows that the inverse image $g^{-1}( \operatorname{Sd}( \Lambda ^{n}_{i} ) )$ can be identified with the nerve of the subset $P_{-}(J) \subseteq P(J)$, so that we have a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( P_{-}(J) ) \ar [r] \ar [d] & \operatorname{Sd}( \Lambda ^{n}_{i} ) \ar [d] \\ \operatorname{N}_{\bullet }( P(J) ) \ar [r]^-{g} & Z. } \]
The left vertical map is anodyne by virtue of Lemma 3.3.5.9, so the right vertical map is anodyne as well. Let $h: [1] \times P_{+}(J) \rightarrow \operatorname{Chain}[n]$ be the map of partially ordered sets given $h(0, J_0) = J_0$ and $h(1, J_0) = J_0 \cup \{ i \} $. Then $h$ determines a map of simplicial sets $\Delta ^1 \times \operatorname{N}_{\bullet }( P_{+}(J) ) \rightarrow \operatorname{Sd}( \Delta ^ n )$. An elementary calculation shows that this map of simplicial sets fits into a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ (\{ 1\} \times \operatorname{N}_{\bullet }( P_{+}(J) )) \coprod _{ \{ 1\} \times \operatorname{N}_{\bullet }( P_{\pm }(J) ) } ( \Delta ^1 \times \operatorname{N}_{\bullet }( P_{\pm }(J) )) \ar [r] \ar [d] & Z \ar [d] \\ \Delta ^1 \times \operatorname{N}_{\bullet }( P_{+}(J) ) \ar [r]^-{h} & \operatorname{Sd}( \Delta ^ n ). } \]
The left vertical map in this diagram is anodyne by virtue of Proposition 3.1.2.9, so the inclusion $Z \hookrightarrow \operatorname{Sd}( \Delta ^ n )$ is also anodyne. It follows that the composite map $\operatorname{Sd}( \Lambda ^{n}_{i} ) \hookrightarrow Z \hookrightarrow \operatorname{Sd}( \Delta ^ n )$ is anodyne, as desired. $\square$