Kerodon

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

4.1.3 Exponentiation for Left and Right Fibrations

We now establish a stability property for left and right fibrations under exponentiation.

Proposition 4.1.3.1. Let $f: X_{} \rightarrow S_{}$ and $i: A_{} \hookrightarrow B_{}$ be morphisms of simplicial sets, where $i$ is a monomorphism, and let

\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]

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.

Corollary 4.1.3.2. 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 4.1.3.1 is essentially equivalent to the following stability property of left and right anodyne morphisms:

Proposition 4.1.3.3. Let $f: A \hookrightarrow B$ and $f': A' \hookrightarrow B'$ be monomorphisms of simplicial sets. If $f$ is left anodyne, then the induced map

\[ \theta : (A \times B') \coprod _{ A \times A'} ( B \times A') \hookrightarrow B \times B' \]

is left anodyne. If $f$ is right anodyne, then $\theta $ is right anodyne.

Proof. We will prove the second assertion (the first follows by a similar argument). We proceed as in the proof of Proposition 3.1.2.7. 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

\[ \theta _{f,f'}: (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \hookrightarrow B_{} \times B'_{} \]

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 3.1.2.8 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 $(\Lambda ^{n}_{i} \times B'_{} ) \coprod _{ \Lambda ^{n}_{i} \times A'_{} } (\Delta ^ n \times A'_{} )$, 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 1.4.5.12, 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 3.1.2.9. $\square$

Proof of Proposition 4.1.3.1. 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

\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]

is also a left fibration (the dual assertion about right fibrations follows by passing to opposite simplicial sets). By virtue of Proposition 4.1.2.5, this is equivalent to the assertion that every lifting problem

\[ \xymatrix@C =100pt{ A'_{} \ar [d]^{i'} \ar [r] & \operatorname{Fun}( B_{}, X_{} ) \ar [d]^{\rho } \\ B'_{} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) } \]

admits a solution, provided that $i'$ is left anodyne. Equivalently, we must show that every lifting problem

\[ \xymatrix@C =100pt{ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ B_{} \times B'_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]

admits a solution. This follows from Proposition 4.1.2.5, since the left vertical map is left anodyne (Proposition 4.1.3.3) and the right vertical map is a left fibration. $\square$

Proposition 4.1.3.3 has another application, which will be useful in the next section:

Proposition 4.1.3.4. Let $f: X_{} \rightarrow S_{}$ and $i: A_{} \rightarrow B_{}$ be morphisms of simplicial sets, and let

\[ \rho : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]

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.

Proof. We proceed as in the proof of Proposition 4.1.3.1. 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

\[ \xymatrix@C =100pt{ A'_{} \ar [d]^{i'} \ar [r] & \operatorname{Fun}( B_{}, X_{} ) \ar [d]^{\rho } \\ B'_{} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) } \]

admits a solution. Equivalently, we must show that every lifting problem

\[ \xymatrix@C =100pt{ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ B_{} \times B'_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]

admits a solution. This follows from Proposition 4.1.2.5, since the left vertical map is left anodyne (Proposition 4.1.3.3) and the right vertical map is a left fibration. $\square$