Kerodon

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Example 1.4.2.9 (Square Diagrams in an $\infty $-Category). Let $I$ denote the partially ordered set $[1] \times [1]$. The simplicial set $\operatorname{N}_{\bullet }(I) \simeq \Delta ^1 \times \Delta ^1$ has four vertices (given by the elements of $I$), five nondegenerate edges, and two nondegenerate $2$-simplices. Unwinding the definitions, we see that an $I$-indexed diagram in an $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to the following data:

  • A collection of objects $\{ C_{ij} \} _{0 \leq i,j \leq 1}$ in $\operatorname{\mathcal{C}}$.

  • A collection of morphisms $f: C_{00} \rightarrow C_{01}$, $g: C_{00} \rightarrow C_{10}$, $f': C_{10} \rightarrow C_{11}$, $g': C_{01} \rightarrow C_{11}$, and $h: C_{00} \rightarrow C_{11}$.

  • A $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ which witnesses $h$ as a composition of $f$ with $g'$, and a $2$-simplex $\tau $ of $\operatorname{\mathcal{C}}$ which witnesses $h$ as a composition of $g$ with $f'$.

This data can be depicted graphically as follows:

\[ \xymatrix { C_{00} \ar [rrrr]^{f} \ar [dddd]^{g} \ar [ddddrrrr]^{h} & & & & C_{01} \ar [dddd]^{g'} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ C_{10} \ar [rrrr]^{f'} & & & & C_{11}. } \]

Beware that such a diagram is usually not determined by its restriction to the simplicial subset $K_{\bullet } \subseteq \operatorname{N}_{\bullet }(I)$ of Example 1.4.2.8.