Kerodon

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Example 5.2.5.5. In the special case $\operatorname{\mathcal{D}}= \Delta ^0$, Proposition 5.2.5.4 asserts that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \times \operatorname{\mathcal{C}}\ar [r] \ar [d] & \Delta ^{0} \ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } } \]

is a categorical pushout square: that is, that the comparison map $\operatorname{\mathcal{C}}\diamond \Delta ^{0} \rightarrow \operatorname{\mathcal{C}}\star \Delta ^{0}$ of Notation 4.5.5.3 is a categorical equivalence. This is the content of Proposition 4.5.5.12 (which is a special case of Theorem 4.5.5.8).