Proposition 7.6.4.14 (03H3)

Proposition 7.6.4.14. Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories satisfying $F_0 \circ G = F_1 \circ G$. The following conditions are equivalent:

$(1)$

The resulting diagram of $\infty $-categories $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ is an equalizer diagram.

$(2)$

The commutative diagram

7.73

\begin{equation} \begin{gathered}\label{equation:homotopy-equalizer-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{G} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^-{ (F_0, F_1) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id}, \operatorname{id}) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

is a categorical pullback square (Definition 4.5.2.8).

Proof. Let us identify the pair $(F_0, F_1)$ with a functor of ordinary categories $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{QCat}$, where $\operatorname{\mathcal{J}}$ is the category described in Remark 7.6.4.2. The functor $G$ then induces a map $\operatorname{\mathcal{E}}\rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$, which can be identified with the map

\[ \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{ (\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}) } \operatorname{\mathcal{D}} \]

determined by the diagram (7.73). Proposition 7.6.4.14 now follows from the criterion of Corollary 7.5.5.8. $\square$