Kerodon

Proof. Combine Proposition 5.4.3.1 with Proposition 5.4.2.8. $\square$

Proof. By definition, the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is the fiber over $Y$ of the projection map $\pi : \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$. Since $\pi $ is an interior fibration (Proposition 5.4.3.1), each of its fibers is an $\infty $-category (Remark 5.4.2.5). A similar argument shows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is an $\infty $-category. $\square$

Proof. Unwinding the definitions, we can identify the diagram (5.38) with a morphism of simplicial sets

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

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

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 5.4.3.8. Note that the composite maps

can be amalgamated to a morphism of simplicial sets

whose source can be identified with the horn $\Lambda ^{m+1+n}_{i} \subseteq \Delta ^{m+1+n}$ (Lemma 4.3.6.16). We wish to show that $\tau _0$ can be extended to a map

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

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

is left-degenerate. $\square$

Proof of Proposition 5.4.3.1. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will show that the projection map $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration; the analogous assertion for the coslice simplicial set $\operatorname{\mathcal{C}}_{f/}$ follows by a similar argument. Let $m \geq 2$ and suppose that we are given a lifting problem

We wish to show that a solution exists under any of the following additional assumptions:

$(a)$

The integer $i$ is equal to zero and the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ 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 the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ m-1 < m \} ) }$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

In cases $(a)$ and $(b)$, this follows immediately from Proposition 5.4.3.8. In case $(c)$, we observe that for every vertex $x \in K$, the composite map

is a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, this degenerate $2$-simplex is thin, so that existence of the desired extension again follows from Proposition 5.4.3.8. $\square$

Proof. Arguing as in the proof of Proposition 5.4.3.8, we can reduce to the case where $K = \Delta ^ n$ is a standard simplex and $K_0 = \operatorname{\partial \Delta }^ n$ is its boundary. In this case, the lifting problem (5.39) determines a morphism of simplicial sets

whose source can be identified with the horn $\Lambda ^{n+2}_{1} \subseteq \Delta ^{n+2}$ (Lemma 4.3.6.16), and we wish to extend $\tau $ to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$. If $n > 0$, then the desired extension exists because $\tau _0$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (by virtue of assumption $(\ast _0)$). If $n = 0$, then our assumption that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category allows us to extend $\tau _0$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. $\square$