Lemma 8.5.8.12. Let $n \geq 0$ be an integer and let $K$ be a simplicial set which satisfies the following condition for $0 \leq m \leq n$:
- $(\ast _ m)$
For every integer $0 < i < m$, every morphism of simplicial sets $\sigma : \Lambda ^{m}_{i} \rightarrow K$ can be extended to an $m$-simplex of $K$.
Then there exists an inner anodyne morphism $\iota : K \hookrightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}$ is an $\infty $-category and $\iota $ is bijective on simplices of dimension $< n$.
Proof.
We construct $\operatorname{\mathcal{E}}$ as the colimit of a sequence of inner anodyne maps
\[ K = K(0) \hookrightarrow K(1) \hookrightarrow K(2) \hookrightarrow K(3) \hookrightarrow \cdots \]
Assume that $K(t)$ has been constructed for some $t \geq 0$, and let $S$ be the collection of all maps $\sigma : \Lambda ^{m}_{i} \rightarrow K(t)$ where $0 < i < m$ and $m > n$. For every $\sigma \in S$, let us write $C_{\sigma }$ for the simplicial set $\Lambda ^{m}_{i}$ which is the source of $\sigma $, and $D_{\sigma }$ for the simplex $\Delta ^{m}$. We then construct a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{\sigma \in S} C_{\sigma } \ar [r] \ar [d] & K(t) \ar [d] \\ \coprod _{\sigma \in S} D_{\sigma } \ar [r] & K(t+1). } \]
By construction, the morphism $K(t) \rightarrow K(t+1)$ is inner anodyne and bijective on simplices of dimension $< n$. To complete the proof, it will suffice to show that the colimit $\operatorname{\mathcal{E}}= \varinjlim _{t} K(t)$ is an $\infty $-category. Fix a pair of integers $0 < i < m$ and a morphism $\sigma : \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{E}}$; we wish to show that $\sigma $ can be extended to an $m$-simplex of $\operatorname{\mathcal{E}}$. Since $\Lambda ^{m}_{i}$ is a finite simplicial set, $\sigma $ factors (uniquely) through $K(t)$ for some integer $t \gg 0$. By construction, if $m > n$ then $\sigma $ can be extended to an $m$-simplex of $K(t+1)$. We may therefore assume that $m \leq n$. In this case, we can take $k = 0$, in which case the existence of the desired extension follows from assumption $(\ast _ m)$.
$\square$