Definition 4.5.9.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets. We will say that $U$ is exponentiable if it satisfies the following condition:
- $(\ast )$
For every diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{B}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{B}}' \ar [r] & \operatorname{\mathcal{B}}} \]in which both squares are pullbacks, if $\overline{F}$ is a categorical equivalence, then $F$ is also a categorical equivalence.