Example 2.4.5.7 (00LQ)

Example 2.4.5.7. The standard $2$-cube $\operatorname{\raise {0.1ex}{\square }}^2 \simeq \Delta ^1 \times \Delta ^1$ is depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] \ar [dr] & \bullet \ar [d] \\ \bullet \ar [r] & \bullet . } \]

It is a simplicial set of dimension $2$, having exactly two nondegenerate $2$-simplices (represented by the triangular regions in the preceding diagram) and five nondegenerate edges. The boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is a $1$-dimensional simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{2}$, obtained by removing the nondegenerate $2$-simplices along with the “internal” edge to obtain the directed graph depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] & \bullet \ar [d] \\ \bullet \ar [r] & \bullet . } \]

Each of the hollow $2$-cubes $\boldsymbol {\sqcap }^{2}_{1}, \boldsymbol {\sqcap }^{2}_{2}, \boldsymbol {\sqcup }^{2}_{1}, \boldsymbol {\sqcup }^{2}_{2}$ can be obtained from $\operatorname{\partial \raise {0.1ex}{\square }}^{2}$ by further deletion of a single edge, represented in the diagrams

\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] \ar@ {}[dr]|{\boldsymbol {\sqcap }^{2}_{1}} & \bullet & \bullet \ar [r] \ar [d] \ar@ {}[dr]|{\boldsymbol {\sqcap }^{2}_{2}} & \bullet \ar [d] \\ \bullet \ar [r] & \bullet & \bullet & \bullet \\ \bullet \ar [r] \ar@ {}[dr]|{\boldsymbol {\sqcup }^{2}_{1}} & \bullet \ar [d] & \bullet \ar [d] \ar@ {}[dr]|{\boldsymbol {\sqcup }^{2}_{2}} & \bullet \ar [d] \\ \bullet \ar [r] & \bullet & \bullet \ar [r] & \bullet . } \]