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Proposition 4.5.9.17. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is exponentiable (Definition 4.5.9.10).

$(2)$

Let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which the restriction $U|_{\operatorname{\mathcal{D}}_0}$ is exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and set $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'_0 = \operatorname{\mathcal{D}}_0 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}'$. Then the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' ) \]

is also an isofibration.

$(3)$

For every isofibration $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, the induced map

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}') \]

is also an isofibration.

$(4)$

For every isofibration of $\infty $-categories $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$, the induced map

\[ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ) \]

is also an isofibration.

Proof. We first show that $(1)$ implies $(2)$. Assume that $U$ is exponentiable, let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a simplicial subset for which $U|_{\operatorname{\mathcal{D}}_0}$ is also exponentiable, let $T: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an isofibration in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}}$, and let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets which is a categorical equivalence; we wish to show that every lifting problem

4.45
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \ar [d] \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Res}_{ \operatorname{\mathcal{D}}_0/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}_0 ) \times _{ \operatorname{Res}_{\operatorname{\mathcal{D}}_0/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}'_0) } \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}' )} \end{gathered} \end{equation}

admits a solution. Note that the bottom horizontal map determines a morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$. Invoking the universal property of direct images (Proposition 4.5.9.2), we can rewrite (4.45) as a lifting problem

4.46
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun2} \xymatrix@R =50pt@C=50pt{ (A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}) \coprod _{ (A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0)} (B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 ) \ar [d]^{j} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{T} \\ B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}'. } \end{gathered} \end{equation}

Since $U$ and $U|_{\operatorname{\mathcal{D}}_0}$ are exponentiable, the horizontal maps in the diagram

4.47
\begin{equation} \begin{gathered}\label{equation:silly-diagram-for-fun} \xymatrix@R =50pt@C=50pt{ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 \ar [r] \ar [d] & B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}_0 \ar [d] \\ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar [r] & B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

are categorical equivalences. In particular, the diagram (4.47) is a categorical pushout square (Proposition 4.5.4.10). It follows that the morphism $j$ appearing in (4.46) is also a categorical equivalence (Proposition 4.5.4.11). Since $T$ is an isofibration of simplicial sets, it follows that the lifting problem (4.46) admits a solution.

The implication $(2) \Rightarrow (3)$ and $(3) \Rightarrow (4)$ are immediate. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied and suppose that we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

where both squares are pullbacks and $\overline{F}$ is a categorical equivalence; we wish to show that $F$ is also a categorical equivalence. By virtue of Exercise 3.1.7.11, there exists a monomorphism of simplicial sets $\iota : \operatorname{\mathcal{C}}'' \hookrightarrow Q$, where $Q$ is a contractible Kan complex. Replacing $\overline{F}$ by the morphism $(\iota , \overline{F}): \operatorname{\mathcal{C}}'' \hookrightarrow Q \times \operatorname{\mathcal{C}}'$ (and $F$ by the morphism $(\iota ,F): \operatorname{\mathcal{D}}'' \hookrightarrow Q \times \operatorname{\mathcal{D}}'$), we can reduce to the case where $\overline{F}$ is a monomorphism of simplicial sets, so that $F$ is also a monomorphism of simplicial sets. To show that $F$ is a categorical equivalence, it will suffice to show that if $T_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}'_0$ is an isofibration of $\infty $-categories, then every lifting problem

4.48
\begin{equation} \begin{gathered}\label{equation:exponentiable-equivalent-condition2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [d]^{F} \ar [r] & \operatorname{\mathcal{E}}_0 \ar [d]^{T_0} \\ \operatorname{\mathcal{D}}' \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}'_0 } \end{gathered} \end{equation}

admits a solution (Proposition 4.5.5.4). Invoking the universal property of direct images (Proposition 4.5.9.2), we can rewrite (4.48) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'' \ar [d]^{ \overline{F} } \ar [r] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}_0 ) \ar [d] \\ \operatorname{\mathcal{C}}' \ar [r] \ar@ {-->}[ur] & \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{E}}'_0 ). } \]

Condition $(4)$ guarantees that the right vertical map is an isofibration, so that the solution exists by virtue of our assumption that $\overline{F}$ is a categorical equivalence. $\square$