Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.3.6.8. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, let $f: K \rightarrow X$ be any morphism of simplicial sets, let $K_0$ be a simplicial subset of $K$, and set $f_0 = f|_{K_0}$. If the inclusion $K_0 \hookrightarrow K$ is left anodyne, then the restriction map $X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0)}} S_{ / (q \circ f)}$ is a trivial Kan fibration. If the inclusion $K_0 \hookrightarrow K$ is right anodyne, then the restriction map $X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/}} S_{(q \circ f)/}$ is a trivial Kan fibration.

Proof. We will prove the first assertion; the second follows by a similar argument. Assume that the inclusion $K_0 \hookrightarrow K$ is right anodyne. We wish to show that, for every monomorphism of simplicial sets $i: A \hookrightarrow A'$, every lifting problem For this, it will suffice to show that if $i: A \hookrightarrow A'$ is any monomorphism of simplicial sets, then any lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] \ar [r] & X_{/f_0} \times _{ S_{ / (q \circ f_0)}} S_{ / (q \circ f)} } \]

admits a solution. Unwinding the definitions, this is equivalent to solving an associated lifting problem

\[ \xymatrix@R =50pt@C=50pt{ (A \star K) \coprod _{ A \star K_0 } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar@ {-->}[ur] \ar [r] & S, } \]

where the left vertical morphism is the pushout-join of Construction 4.3.6.3. Since the left vertical map is inner anodyne (Proposition 4.3.6.4), the desired solution exists by virtue of our assumption that $q$ is an inner fibration (Proposition 4.1.3.1). $\square$