Kerodon

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Example 1.4.2.4 (Non-Commuting Squares). Let $K_{\bullet }$ denote the boundary of the product $\Delta ^1 \times \Delta ^1$: that is, the simplicial subset of $\Delta ^1 \times \Delta ^1$ given by the union of the simplicial subsets $\operatorname{\partial \Delta }^1 \times \Delta ^1$ and $\Delta ^1 \times \operatorname{\partial \Delta }^1$. Then $K_{\bullet }$ is a $1$-dimensional simplicial set, corresponding to a directed graph which we can depict as

\[ \xymatrix@R =40pt@C=40pt{ \bullet \ar [r] \ar [d] & \bullet \ar [d] \\ \bullet \ar [r] & \bullet .} \]

We can then display a $K_{\bullet }$-indexed diagram in an $\infty $-category $\operatorname{\mathcal{C}}$ pictorially

\[ \xymatrix@R =40pt@C=40pt{ C_{00} \ar [r]^{f} \ar [d]^{g} & C_{01} \ar [d]^{g'} \\ C_{10} \ar [r]^{ f'} & C_{11}, } \]

where each $C_{ij}$ is an object of $\operatorname{\mathcal{C}}$, $f$ is a morphism in $\operatorname{\mathcal{C}}$ from $C_{00}$ to $C_{01}$, $g$ is a morphism in $\operatorname{\mathcal{C}}$ from $C_{00}$ to $C_{10}$, $f'$ is a morphism in $\operatorname{\mathcal{C}}$ from $C_{10}$ to $C_{11}$, and $g'$ is a morphism in $\operatorname{\mathcal{C}}$ from $C_{01}$ to $C_{11}$.