Example 4.3.2.15 (016F): The Universal Property of a Cone

Example 4.3.2.15 (The Universal Property of a Cone). Let $\operatorname{\mathcal{C}}$ be a category. Applying Remark 4.3.2.14 in the special case $\operatorname{\mathcal{D}}= [0]$, we obtain a pushout diagram of categories

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

where the bottom horizontal map carries the unique object of $[0]$ to the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. This is essentially a reformulation of Examples 4.3.2.11 and 4.3.2.12. Stated more informally, the right cone $\operatorname{\mathcal{C}}^{\triangleright }$ is obtained from the product $[1] \times \operatorname{\mathcal{C}}$ by “collapsing” the full subcategory $\{ 1\} \times \operatorname{\mathcal{C}}$ to the cone point. Similarly, the left cone of a category $\operatorname{\mathcal{D}}$ is characterized by the existence of a pushout diagram

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