Remark 4.6.2.6 (046S)

Remark 4.6.2.6. Suppose we are given a commutative diagram of $\infty $-categories

4.54

\begin{equation} \begin{gathered}\label{equation:morphism-space-categorical-pullback-square} \xymatrix { \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

Combining Remark 4.6.1.14 with Corollary 3.4.1.6, we see that the following conditions are equivalent:

$(1)$

The diagram (4.54) induces a fully faithful functor from $\operatorname{\mathcal{C}}_{01}$ to the homotopy fiber product $\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.

$(2)$

For every object $X_{01} \in \operatorname{\mathcal{C}}_{01}$ having images $X_0 \in \operatorname{\mathcal{C}}_0$, $X_1 \in \operatorname{\mathcal{C}}_1$, $X \in \operatorname{\mathcal{C}}$ and every object $Y_{01} \in \operatorname{\mathcal{C}}_{01}$ having images $Y_0 \in \operatorname{\mathcal{C}}_0$, $Y_1 \in \operatorname{\mathcal{C}}_1$, $Y \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

\[ \xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}_{01}}( X_{01}, Y_{01} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}_0}( X_0, Y_1 ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_1}( X_1, Y_1 ) \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y) } \]

is a homotopy pullback square.

In particular, if (4.54) is a categorical pullback diagram, then it satisfies condition $(2)$.