Kerodon

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Proposition 4.2.4.10. For every pair of integers $0 \leq j < n$, the horn $\Lambda ^{n}_{j}$ admits a finite filtration

\[ \{ 0\} = X(0) \subset X(1) \subset \cdots \subset X(k) = \Lambda ^{n}_{j}, \]

where each of the inclusion maps $X(a-1) \hookrightarrow X(a)$ can be realized as a pushout of a horn inclusion $\Lambda ^{m}_{i} \hookrightarrow \Delta ^ m$ for $0 \leq i < m < n$. In particular, the inclusion $\{ 0\} \hookrightarrow \Lambda ^{n}_{j}$ is left anodyne.

Proof. Let us say that a monomorphism of simplicial sets $A \hookrightarrow B$ is good if it can be written as a composition of finitely many morphisms, each of which is a pushout of a horn inclusion $\Lambda ^{m}_{i} \hookrightarrow \Delta ^ m$ for $0 \leq i < m < n$. We wish to show that the inclusion map $\{ 0\} \hookrightarrow \Lambda ^{n}_{j}$ is good. Our proof proceeds by induction on $n$. It follows from our inductive hypothesis that the inclusion map $\{ 0\} \hookrightarrow \Lambda ^{j}_{0}$ is good; in particular, the inclusion map $\{ 0\} \hookrightarrow \Delta ^{j}$ is good. For every integer $d \geq 0$, let $Y(d) \subseteq \Delta ^ n$ be the simplicial subset whose nondegenerate simplices have vertex set $J \subseteq [n]$ satisfying one of the following conditions:

  • The set $J$ is contained in $[j] = \{ 0 < 1 < \cdots < j \} $.

  • The set $J \setminus \{ j\} $ has cardinality $< d$.

We have inclusion maps

\[ \Delta ^{j} = Y(0) \subseteq Y(1) \subseteq Y(2) \subseteq \cdots \subseteq Y(n-1) = \Lambda ^{n}_{j}. \]

It will therefore suffice to show that for $0 < d < n$, the inclusion map $Y(d-1) \hookrightarrow Y(d)$ is good. Let $S$ be the collection of all nondegenerate $d$-simplices of $Y(d)$. By construction, for each $\sigma \in S$, there is a unique integer $0 \leq i_{\sigma } < d$ satisfying $\sigma ( i_{\sigma } ) = j$. Unwinding the definitions, we see that there is a pushout square

\[ \xymatrix { \coprod _{\sigma \in S} \Lambda ^{d}_{i_{\sigma }} \ar [r] & Y(d-1) \ar [d] \\ \coprod _{\sigma \in S} \Delta ^{d} \ar [r] & Y(d), } \]

so that the inclusion $Y(d-1) \hookrightarrow Y(d)$ can be written as a finite pushout of morphisms of the form $\Lambda ^{d}_{i} \hookrightarrow \Delta ^ d$ for $i < d$. $\square$