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Proposition 5.3.3.8. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets and let $f_0: K_0 \rightarrow \operatorname{\mathcal{C}}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then every lifting problem

5.24
\begin{equation} \label{equation:slice-interior-fibration-more-precise} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d] \\ \Delta ^{m} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur] & \operatorname{\mathcal{C}}_{/f_0} } \end{gathered} \end{equation}

admits a solution provided that $m \geq 2$ and one of the following additional conditions is satisfied:

$(a)$

The simplicial set $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, $i=0$, and the composition

\[ \Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{m}_{i} \xrightarrow { \sigma _0} \operatorname{\mathcal{C}}_{/f} \]

is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

$(b)$

The integer $i$ satisfies $0 < i < m$ and the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Delta ^{m} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(c)$

The integer $i$ is equal to $m$ and, for every vertex $x \in K$, the composite map

\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \star \{ x\} \hookrightarrow \Lambda ^{m}_{i} \star K \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Proof. Unwinding the definitions, we can identify the diagram (5.24) with a morphism of simplicial sets

\[ \overline{f}: ( \Lambda ^{m}_{i} \star K ) \coprod _{ (\Lambda ^{m}_{i} \star K_0 ) } ( \Delta ^{m} \star K_0 ) \rightarrow \operatorname{\mathcal{C}}, \]

and we wish to show that $\overline{f}$ can be extended to a morphism $\Delta ^{m} \star K \rightarrow \operatorname{\mathcal{C}}$. Let $P$ be the collection of all pairs $(L,g)$, where $L$ is a simplicial subset of $K$ containing $K_0$ and $g: \Delta ^{m} \star L \rightarrow \operatorname{\mathcal{C}}$ is a morphism satisfying

\[ g|_{ \Delta ^{m} \star K_0} = \overline{f} |_{ \Delta ^{m} \star K_0} \quad \quad g|_{ \Lambda ^{m}_{i} \star L} = \overline{f} |_{ \Lambda ^{m}_{i} \star L}. \]

We regard $P$ as a partially ordered set, with $(L, g) \leq (L', g')$ if $L$ is contained in $L'$ and $g = g'|_{\Delta ^ m \star L}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma and therefore admits a maximal element $(L_{\mathrm{max}}, g_{\mathrm{max}} )$. We will complete the proof by showing that $L_{\mathrm{max}} = K$ (so that $g_{\mathrm{max}}$ is the desired extension of $\overline{f}$). Suppose otherwise. Then there is some nondegenerate simplex $\rho : \Delta ^{n} \rightarrow K$ which is not contained in $L_{\mathrm{max}}$. Choosing $\rho $ so that $n$ is as small as possible, we may assume without loss of generality that $\rho $ carries the boundary $\operatorname{\partial \Delta }^ n$ into $L_{\mathrm{max}}$. Let $L' \subseteq K$ be the simplicial subset given by the union of $L_{\mathrm{max}}$ together with the image of $\rho $, so that $\rho $ determines a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & L_{\mathrm{max}} \ar [d] \\ \Delta ^{n} \ar [r] & L'. } \]

We will show that $g_{\mathrm{max}}$ can be extended to a morphism of simplicial sets $g': \Delta ^{m} \star L' \rightarrow \operatorname{\mathcal{C}}$ satisfying $g'|_{ \Lambda ^{m}_{i} \star L'} = \overline{f}|_{ \Lambda ^{m}_{i} \star L'}$; thereby contradicting the maximality of $(L_{\mathrm{max}}, g_{\mathrm{max}} )$ and completing the proof of Proposition 5.3.3.8. Note that the composite maps

\[ \Lambda ^{m}_{i} \star \Delta ^{n} \xrightarrow { \operatorname{id}\star \rho } \Lambda ^{m}_{i} \star K \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

\[ \Delta ^{m} \star \operatorname{\partial \Delta }^ n \xrightarrow { \operatorname{id}\star \rho } \Delta ^{m} \star L_{\mathrm{max}} \xrightarrow { g_{\mathrm{max}}} \operatorname{\mathcal{C}} \]

can be amalgamated to a morphism of simplicial sets

\[ \tau _0: (\Lambda ^{m}_{i} \star \Delta ^{n}) \coprod _{ (\Lambda ^{m}_{i} \star \operatorname{\partial \Delta }^ n)} (\Delta ^ m \star \operatorname{\partial \Delta }^ n) \rightarrow \operatorname{\mathcal{C}}, \]

whose source can be identified with the horn $\Lambda ^{m+1+n}_{i} \subseteq \Delta ^{m+1+n}$ (Lemma 4.3.6.12). We wish to show that $\tau _0$ can be extended to a map

\[ \tau : \Delta ^{m} \star \Delta ^{n} \simeq \Delta ^{m+1+n} \rightarrow \operatorname{\mathcal{C}}. \]

If $0 < i \leq m$, the desired extension exists because the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Lambda ^{m+1+n}_{i} \xrightarrow {\tau _0} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (by virtue of assumption $(b)$ when $i < m$ or $(c)$ in the case $i = m$). If $i=0$, then the desired extension exists because assumption $(a)$ guarantees that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category and the $2$-simplex

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 < m+1+n\} ) \subseteq \Lambda ^{m+1+n}_{i} \xrightarrow {\tau _0} \operatorname{\mathcal{C}} \]

is left-degenerate. $\square$