Let $B_{}$ and $X_{}$ be simplicial sets. In ยง1.5.3, we showed that if $X_{}$ is an $\infty $-category, then the simplicial set $\operatorname{Fun}( B_{}, X_{} )$ is an $\infty $-category (Theorem 1.5.3.7). If $X_{}$ is a Kan complex, we can say more: the simplicial set $\operatorname{Fun}( B_{}, X_{} )$ is also a Kan complex (Corollary 3.1.3.4). This is a consequence of the following stronger result:
Theorem 3.1.3.1. Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, and let $i: A_{} \hookrightarrow B_{}$ be any monomorphism of simplicial sets. Then the induced map is a Kan fibration.
\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
Proof. By virtue of Remark 3.1.2.7, it will suffice to show that if $i': A' \hookrightarrow B'$ is an anodyne morphism of simplicial sets, then every lifting problem of the form
admits a solution. Equivalently, we must show that every lifting problem
admits a solution. This follows from Remark 3.1.2.7, since the left vertical map is anodyne (Proposition 3.1.2.9) and the right vertical map is a Kan fibration. $\square$
Let us note some special cases of Theorem 3.1.3.1 (which can be obtained by taking the simplicial set $A_{}$ to be empty, the simplicial set $S_{}$ to be $\Delta ^0$, or both).
Corollary 3.1.3.2. Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets. Then, for every simplicial set $B_{}$, composition with $f$ induces a Kan fibration $\operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} )$.
Corollary 3.1.3.3. Let $X_{}$ be a Kan complex. Then, for every monomorphism of simplicial sets $i: A_{} \hookrightarrow B_{}$, the restriction map $\operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( A_{}, X_{} )$ is a Kan fibration.
Corollary 3.1.3.4. Let $X_{}$ be a Kan complex and let $B_{}$ be an arbitrary simplicial set. Then the simplicial set $\operatorname{Fun}( B_{}, X_{} )$ is a Kan complex.
Theorem 3.1.3.1 has an analogue for trivial Kan fibrations:
Theorem 3.1.3.5. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets and let $f: X_{} \rightarrow S_{}$ be a Kan fibration. Then the induced map is a trivial Kan fibration.
\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
Proof. We proceed as in the proof of Theorem 3.1.3.1. Let $i': A'_{} \hookrightarrow B'_{}$ be a monomorphism of simplicial sets; we must show that every lifting problem
admits a solution. Equivalently, we must show that every lifting problem
admits a solution. This follows from Remark 3.1.2.7, since the left vertical map is anodyne (Proposition 3.1.2.9) and the right vertical map is a Kan fibration. $\square$
Taking $S_{} = \Delta ^0$ in the statement of Theorem 3.1.3.5, we obtain the following:
Corollary 3.1.3.6. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets and let $X_{}$ be a Kan complex. Then the restriction map $\operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( A_{}, X_{} )$ is a trivial Kan fibration.
To formulate some further consequences of Theorem 3.1.3.1, it will be convenient to introduce some notation.
Construction 3.1.3.7. Let $B$ and $X$ be simplicial sets, and let $\operatorname{Fun}(B,X)$ be the simplicial set parametrizing morphisms from $B$ to $X$ (Construction 1.5.3.1).
Suppose we are given another simplicial set $A$ equipped with a pair of morphisms $i: A \rightarrow B$ and $f: A \rightarrow X$. In this case, we let $\operatorname{Fun}_{A/}(B,X) \subseteq \operatorname{Fun}(B,X)$ denote the fiber of the precomposition morphism $\operatorname{Fun}(B,X) \xrightarrow {\circ i} \operatorname{Fun}(A,X)$ over the vertex $f \in \operatorname{Fun}(A,X)$.
Suppose we are given another simplicial set $S$ equipped with a pair of morphism $g: B \rightarrow S$ and $q: X \rightarrow S$. We let $\operatorname{Fun}_{/S}(B,X) \subseteq \operatorname{Fun}(B,X)$ denote the fiber of the postcomposition morphism $\operatorname{Fun}(B,X) \xrightarrow { q \circ } \operatorname{Fun}(B,S)$ over the vertex $g \in \operatorname{Fun}(B,S)$.
Suppose we are given a commutative diagram of simplicial sets
In this case, we let $\operatorname{Fun}_{A/ \, /S}(B, X) \subseteq \operatorname{Fun}(B,X)$ denote the simplicial subset given by the intersection $\operatorname{Fun}_{A/}(B,X) \cap \operatorname{Fun}_{/S}(B,X)$.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} & S. } \]
Remark 3.1.3.8. Let $B$ and $X$ be simplicial sets, and let us identify vertices of $\operatorname{Fun}(B,X)$ with morphisms $\overline{f}: B \rightarrow X$ in the category of simplicial sets. Then:
Suppose we are given another simplicial set $A$ equipped with a pair of morphisms $i: A \rightarrow B$ and $f: A \rightarrow X$. Then vertices of the simplicial set $\operatorname{Fun}_{A/}(B,X)$ can be identified with morphisms $\overline{f}: B \rightarrow X$ satisfying $f = \overline{f} \circ i$.
Suppose we are given another simplicial set $S$ equipped with a pair of morphisms $g: B \rightarrow S$ and $q: X \rightarrow S$. Then vertices of the simplicial set $\operatorname{Fun}_{/S}(B,X)$ can be identified with morphisms $\overline{f}: B \rightarrow X$ satisfying $g = q \circ \overline{f}$.
Suppose we are given a square diagram of simplicial sets
Then vertices of the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ can be identified with solutions of the associated lifting problem: that is, morphisms of simplicial sets $\overline{f}: B \rightarrow X$ satisfying $f = \overline{f} \circ i$ and $g = q \circ \overline{f}$.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S. } \]
Remark 3.1.3.9. Suppose we are given a diagram of simplicial sets which does not commute. Then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{A/}(B,X) \cap \operatorname{Fun}_{/S}(B,X)$ of Construction 3.1.3.7 can still be defined, but is automatically empty.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S } \]
Remark 3.1.3.10. Suppose we are given a commutative diagram of simplicial sets Then:
If $S \simeq \Delta ^{0}$ is a final object of the category of simplicial sets, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{A/}(B,X)$ (as simplicial subsets of $\operatorname{Fun}(B,X)$).
If $A \simeq \emptyset $ is an initial object of the category of simplicial sets, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{/S}(B,X)$ (as simplicial subsets of $\operatorname{Fun}(B,X)$).
If $S \simeq \Delta ^{0}$ and $A \simeq \emptyset $ are final and initial objects, respectively, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}(B,X)$.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S. } \]
Remark 3.1.3.11. Suppose we are given a commutative diagram of simplicial sets Then the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with the fiber of the induced map over the vertex given by the pair $(f,g)$.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]_-{i} & X \ar [d]^-{q} \\ B \ar [r]_-{g} \ar@ {-->}[ur]^{ \overline{f} } & S. } \]
\[ \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]
Example 3.1.3.12. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then, for each vertex $s \in S$, the simplicial set $\operatorname{Fun}_{/S}( \{ s\} , X)$ can be identified with the fiber $X_{s} = \{ s\} \times _{S} X$.
Proposition 3.1.3.13. Suppose we are given a commutative diagram of simplicial sets where $i$ is a monomorphism and $q$ is a Kan fibration. Then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ is a Kan complex. If $i$ is anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B,X)$ is contractible.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S, } \]
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
Theorem 3.1.3.1 guarantees that $\theta $ is a Kan fibration, so its fibers are Kan complexes by virtue of Remark 3.1.1.9. If $i$ is anodyne, then $\theta $ is a trivial Kan fibration (Theorem 3.1.3.5), so its fibers are contractible Kan complexes (Remark 1.5.5.10). $\square$
Corollary 3.1.3.14. Let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and let $f: A \rightarrow X$ be a morphism of simplicial sets. If $X$ is a Kan complex, then the simplicial set $\operatorname{Fun}_{A/}(B, X)$ is a Kan complex. If the inclusion $A \hookrightarrow B$ is anodyne, then the Kan complex $\operatorname{Fun}_{A/}(B, X)$ is contractible.
Proof. Apply Proposition 3.1.3.13 in the special case $S = \Delta ^{0}$. $\square$
Corollary 3.1.3.15. Let $q: X \rightarrow S$ be a Kan 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 a Kan complex.
Proof. Apply Proposition 3.1.3.13 in the special case $A = \emptyset $. $\square$