We now establish a stability property for left and right fibrations under exponentiation.
Proposition 4.2.5.1. Let $f: X_{} \rightarrow S_{}$ and $i: A_{} \hookrightarrow B_{}$ be morphisms of simplicial sets, where $i$ is a monomorphism, and let be the induced map. If $f$ is a left fibration, then $\rho $ is a left fibration. If $f$ is a right fibration, then $\rho $ is a right fibration.
\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
Corollary 4.2.5.2. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets, let $B_{}$ be an arbitrary simplicial set, and let $\rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} )$ be the morphism induced by composition with $f$. If $f$ is a left fibration, then $\rho $ is a left fibration. If $f$ is a right fibration, then $\rho $ is a right fibration.
Proposition 4.2.5.1 is essentially equivalent to the following stability property of left and right anodyne morphisms:
Proposition 4.2.5.3. Let $f: A \hookrightarrow B$ and $f': A' \hookrightarrow B'$ be monomorphisms of simplicial sets. If $f$ is left anodyne, then the induced map is left anodyne. If $f$ is right anodyne, then $\theta $ is right anodyne.
\[ \theta : (A \times B') \coprod _{ A \times A'} ( B \times A') \hookrightarrow B \times B' \]
Proof. We will prove the second assertion (the first follows by a similar argument). We proceed as in the proof of Proposition 3.1.2.9. Let us first regard the monomorphism $f': A' \hookrightarrow B'$ as fixed, and let $T$ be the collection of all maps $f: A \rightarrow B$ for which the induced map
is right anodyne. We wish to show that every right anodyne morphism belongs to $T$. Since $T$ is weakly saturated, it will suffice to show that every horn inclusion $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T$ for $0 < i \leq n$. In this case, Lemma 3.1.2.10 guarantees that $f$ is a retract of the morphism $g: (\Delta ^1 \times \Lambda ^ n_ i) \coprod _{ \{ 1\} \times \Lambda ^{n}_ i} ( \{ 1\} \times \Delta ^ n) \hookrightarrow \Delta ^1 \times \Delta ^ n$. It will therefore suffice to show that $g$ belongs to $T$. Replacing $f'$ by the monomorphism $(\Lambda ^{n}_{i} \times B'_{} ) \coprod _{ \Lambda ^{n}_{i} \times A'_{} } (\Delta ^ n \times A'_{} ) \hookrightarrow \Delta ^ n \times B'_{}$, we are reduced to showing that the inclusion $\{ 1\} \hookrightarrow \Delta ^1$ belongs to $T$.
Let $T'$ denote the collection of all morphisms of simplicial sets $f'': A''_{} \rightarrow B''_{}$ for which the map $(\{ 1\} \times B''_{} ) \coprod _{ \{ 1\} \times A''_{} } ( \Delta ^1 \times A''_{} ) \rightarrow \Delta ^1 \times B''_{}$ is right anodyne. We will complete the proof by showing that $T'$ contains all monomorphisms of simplicial sets. By virtue of Proposition 1.5.5.14, it will suffice to show that $T''$ contains the inclusion map $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^{m}$, for each $m > 0$. In other words, we are reduced to showing that the inclusion $(\{ 1\} \times \Delta ^ m ) \coprod _{ \{ 1\} \times \operatorname{\partial \Delta }^ m } ( \Delta ^1 \times \operatorname{\partial \Delta }^ m) \hookrightarrow \Delta ^1 \times \Delta ^ m$ is right anodyne, which follows from Lemma 3.1.2.12. $\square$
Proof of Proposition 4.2.5.1. Let $f: X \rightarrow S$ be a left fibration of simplicial sets and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. We wish to show that the restriction map
is also a left fibration (the dual assertion about right fibrations follows by passing to opposite simplicial sets). By virtue of Proposition 4.2.4.5, this is equivalent to the assertion that every lifting problem
admits a solution, provided that $i'$ is left anodyne. Equivalently, we must show that every lifting problem
admits a solution. This follows from Proposition 4.2.4.5, since the left vertical map is left anodyne (Proposition 4.2.5.3) and the right vertical map is a left fibration. $\square$
Proposition 4.2.5.3 has another application, which will be useful in the next section:
Proposition 4.2.5.4. Let $f: X_{} \rightarrow S_{}$ and $i: A_{} \rightarrow B_{}$ be morphisms of simplicial sets, and let be the induced map. If $f$ is a left fibration and $i$ is left anodyne, then $\rho $ is a trivial Kan fibration. If $f$ is a right fibration and $i$ is right anodyne, then $\rho $ is a trivial Kan fibration.
\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]
Proof. We proceed as in the proof of Proposition 4.2.5.1. Assume that $f$ is a left fibration and that $i$ is left anodyne; we will show that $\rho $ is a trivial Kan fibration (the dual assertion for right fibrations follows by a similar argument). Fix a monomorphism of simplicial sets $i': A' \hookrightarrow B'$; we wish to show that every lifting problem
admits a solution. Equivalently, we must show that every lifting problem
admits a solution. This follows from Proposition 4.2.4.5, since the left vertical map is left anodyne (Proposition 4.2.5.3) and the right vertical map is a left fibration. $\square$
Exercise 4.2.5.5. Let $f: X \rightarrow S$ be a left covering morphism of simplicial sets. Show that, for any left anodyne morphism $i: A \hookrightarrow B$, the induced map is an isomorphism of simplicial sets.
\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]