Recall that, if $\operatorname{\mathcal{C}}$ is an $\infty $-category and $B$ is an arbitrary simplicial set, then the simplicial set $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ is also an $\infty $-category (Theorem 1.5.3.7). We now record a relative version of this result.
Proposition 4.1.4.1. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then the restriction map is also an inner fibration of simplicial sets.
\[ \rho : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]
Proof. By virtue of Proposition 4.1.3.1, it will suffice to show that every lifting problem
admits a solution, provided that $i'$ is inner anodyne. Equivalently, we must show that every lifting problem
admits a solution. This follows from Proposition 4.1.3.1, since the left vertical map is inner anodyne (Lemma 1.5.7.5) and $q$ is an inner fibration. $\square$
Corollary 4.1.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then the restriction functor $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A,\operatorname{\mathcal{C}})$ is an inner fibration.
Proof. Apply Proposition 4.1.4.1 in the special case $S = \Delta ^0$. $\square$
Corollary 4.1.4.3. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $B$ be an arbitrary simplicial set. Then composition with $q$ induces an inner fibration $\operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(B,S)$.
Proof. Apply Proposition 4.1.4.1 in the special case $A = \emptyset $. $\square$
We now record an analogous generalization of Proposition 1.5.7.6.
Proposition 4.1.4.4. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, and let $i: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. Then the restriction map is a trivial Kan fibration.
\[ \rho : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]
Proof. We wish to show that every lifting problem
admits a solution, provided that $i'$ is a monomorphism of simplicial sets. Equivalently, we must show that every lifting problem
admits a solution. This follows from Proposition 4.1.3.1, since the left vertical map is inner anodyne (Lemma 1.5.7.5) and $q$ is an inner fibration. $\square$
Proposition 4.1.4.4 admits the following converse (generalizing Theorem 1.5.6.1):
Proposition 4.1.4.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if the induced map is a trivial Kan fibration.
\[ \rho : \operatorname{Fun}(\Delta ^2,X) \rightarrow \operatorname{Fun}(\Lambda ^{2}_{1},X) \times _{ \operatorname{Fun}(\Lambda ^{2}_{1},S) } \operatorname{Fun}(\Delta ^2,S) \]
Proof. The “only if” direction follows from Proposition 4.1.4.4. For the converse, we observe that $\rho $ is a trivial Kan fibration if and only if $q$ is weakly right orthogonal to the inclusion map
for every nonnegative integer $m$. Since the collection of inner anodyne morphisms is generated (as a weakly saturated class) by such inclusions (Lemma 1.5.6.9), it follows that $q$ is weakly right orthogonal to all inner anodyne morphisms (Proposition 1.5.4.13) and is therefore an inner fibration (Proposition 4.1.3.1). $\square$
Proposition 4.1.4.6. Suppose we are given a commutative diagram of simplicial sets, where $i$ is a monomorphism and $q$ is an inner fibration. Then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ of Construction 3.1.3.7 is an $\infty $-category. Moreover, if $i$ is inner anodyne, then $\operatorname{Fun}_{A/ \, /S}(B, X)$ is a contractible Kan complex.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S } \]
Proof. By virtue of Remark 3.1.3.11, the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with a fiber of the restriction map
Proposition 4.1.4.1 asserts that $\theta $ is an inner fibration of simplicial sets, so its fibers are $\infty $-categories (Remark 4.1.1.6). If $i$ is inner anodyne, then Proposition 4.1.4.4 guarantees that $\theta $ is a trivial Kan fibration, so its fibers are contractible Kan complexes. $\square$
Corollary 4.1.4.7. Let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the simplicial set $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$ is an $\infty $-category. Moreover, if the inclusion $A \hookrightarrow B$ is inner anodyne, then $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$ is a contractible Kan complex.
Proof. Apply Proposition 4.1.4.6 in the special case $S = \Delta ^{0}$. $\square$
Corollary 4.1.4.8. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $g: B \rightarrow S$ be any morphism of simplicial sets. Then the simplicial set $\operatorname{Fun}_{/S}(B,X)$ is an $\infty $-category.
Proof. Apply Proposition 4.1.4.6 in the special case $A = \emptyset $. $\square$