Remark 7.6.4.6 (Homotopy Coherent Squares). Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$. Combining Examples 1.5.2.9, 2.4.3.9, and 2.4.3.10, we see that morphisms from $\Delta ^1 \times \Delta ^1$ to $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with the following data:
- $(a)$
A collection of objects $X_{01}$, $X_{0}$, $X_{1}$, and $X$ of the category $\operatorname{\mathcal{C}}$.
- $(b)$
A collection of morphisms $f_0: X_0 \rightarrow X$, $f_1: X_1 \rightarrow X$, $g_{0}: X_{01} \rightarrow X_0$, $g_{1}: X_{01} \rightarrow X_1$.
- $(c)$
A morphism $h: X_{01} \rightarrow X$ in $\operatorname{\mathcal{C}}$ together with a pair of edges $\alpha _{0}: f_0 \circ g_0 \rightarrow h$ and $\alpha _1: f_1 \circ g_1 \rightarrow h$ in the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_{01}, X)_{\bullet }$.
We can summarize this data in a diagram
Here we can regard $(a)$ and $(b)$ as supplying a (potentially) non-commutative square diagram in the category $\operatorname{\mathcal{C}}$, and $(c)$ as supplying a witness to the fact that it commutes up to homotopy.