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Lemma (Sparse Horns). Let $n \geq 0$ be an integer and let $S$ be a subset of $[n] = \{ 0 < 1 < \cdots < n \} $. Let $K \subseteq \Delta ^ n$ be the simplicial subset spanned by those nondegenerate simplices which do not contain every element of $S$. Suppose that there exist $0 \leq i < j < k \leq n$ such that $i,k \in S$, $j \notin S$. Then the inclusion $K \hookrightarrow \Delta ^ n$ is inner anodyne.

Proof of Lemma Let $P$ denote the collection of all subsets $S' \subseteq [n]$ which contain $S \cup \{ j\} $. Choose a linear ordering

\[ \{ S(1) \leq \cdots \leq S(c) \} \]

of $P$ with the property that if $S(a) \subseteq S(b)$, then $a \leq b$. Let For $0 \leq b \leq c$, let $K(b) \subseteq \Delta ^ n$ denote the union of $K$ with the faces $\{ \operatorname{N}_{\bullet }( S(a) ) \subseteq \Delta ^ n \} _{ 1 \leq a \leq b}$. We then have inclusion maps

\[ K = K(0) \subseteq K(1) \subseteq K(2) \subseteq \cdots \subseteq K(c-1) \subseteq K(c) = \Delta ^ n. \]

It will therefore suffice to show that, for every positive integer $b \leq c$, the inclusion map $K(b-1) \hookrightarrow K(b)$ is inner anodyne.

Let us identify $\operatorname{N}_{\bullet }( S(b) )$ with the image of a nondegenerate simplex $\sigma : \Delta ^ m \hookrightarrow \Delta ^ n$. Let $L \subseteq \Delta ^ m$ be the inverse image $\sigma ^{-1}( S(b-1) )$, so that we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ L \ar [r] \ar [d] & \Delta ^ m \ar [d]^{\sigma } \\ S(b-1) \ar [r] & S(b). } \]

It will therefore suffice to show that the inclusion map $L \subseteq \Delta ^ m$ is inner anodyne. Because $S(b)$ contains the integers $i < j < k$, we can write $j = \sigma (\overline{j})$ for some $0 < \overline{j} < n$. We conclude by observing that $L$ can be identified with the inner horn $\Lambda ^{m}_{\overline{j} } \subseteq \Delta ^ m$. $\square$