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3.1.3 Exponentiation of Kan Fibrations

Let $B_{}$ and $X_{}$ be simplicial sets. In ยง1.4.3, we showed that if $X_{}$ is an $\infty $-category, then the simplicial set $\operatorname{Fun}( B_{}, X_{} )$ is an $\infty $-category (Theorem 1.4.3.7). 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

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

is a Kan fibration.

Before giving a proof of Theorem 3.1.3.1, let us note some of its consequences (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 inclusion 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.

We will deduce Theorem 3.1.3.1 from the following more refined result:

Proposition 3.1.3.5. Let $f: X_{} \rightarrow S_{}$ and $i: A_{} \hookrightarrow B_{}$ be morphisms of simplicial sets, where $i$ is a monomorphism, and let

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

be the induced map. If $f$ is a left fibration, then $\theta $ is a left fibration. If $f$ is a right fibration, then $\theta $ is a right fibration.

Proof. We will show that if $f$ is a left fibration, then $\theta $ is a left fibration; the corresponding assertion for right fibrations follows by passing to opposite simplicial sets. Set $X'_{} = \operatorname{Fun}( B_{}, X_{} )$ and $S'_{} = \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} )$; we wish to show that the evaluation map

\[ \operatorname{ev}_0: \operatorname{Fun}( \Delta ^1, X'_{} ) \rightarrow \operatorname{Fun}( \{ 0\} , X'_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S'_{} ) } \operatorname{Fun}( \Delta ^1, S'_{} ) \]

is a trivial Kan fibration. Setting $X''_{} = \operatorname{Fun}( \Delta ^1, X_{} )$ and $S''_{} = \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} )$, we can identify $\operatorname{ev}_{0}$ with the tautological map

\[ \operatorname{Fun}( B_{}, X''_{} ) \rightarrow \operatorname{Fun}( A_{}, X''_{} ) \times _{ \operatorname{Fun}( A_{}, S''_{} ) } \operatorname{Fun}( B_{}, S''_{} ). \]

By virtue of Corollary 1.4.5.5, it will suffice to show that the evaluation map $X''_{} \rightarrow S''_{}$ is a trivial Kan fibration, which is equivalent to our assumption that $f$ is a left fibration. $\square$

For later use, we record the following special case of Proposition 3.1.3.5:

Corollary 3.1.3.6. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets, let $B_{}$ be an arbitrary simplicial set, and let $\theta : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} )$ be the morphism induced by composition with $f$. If $f$ is a left fibration, then $\theta $ is a left fibration. If $f$ is a right fibration, then $\theta $ is a right fibration.