Proposition 11.4.0.3. 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.