Kerodon

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Example 4.3.2.5 (Cones). Let $[0]$ denote the category having a single object and a single morphism, and let $\operatorname{\mathcal{C}}$ be an arbitrary category. We let $\operatorname{\mathcal{C}}^{\triangleleft }$ denote the join $[0] \star \operatorname{\mathcal{C}}$, and $\operatorname{\mathcal{C}}^{\triangleright }$ the join $\operatorname{\mathcal{C}}\star [0]$. We refer to $\operatorname{\mathcal{C}}^{\triangleleft }$ as the left cone of $\operatorname{\mathcal{C}}$, and to $\operatorname{\mathcal{C}}^{\triangleright }$ as the right cone on $\operatorname{\mathcal{C}}$.

More informally, we can describe the left cone $\operatorname{\mathcal{C}}^{\triangleleft }$ as the category obtained from $\operatorname{\mathcal{C}}$ by adjoining a new object $X_0$ satisfying

$\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleleft }}( X_0, Y) = \ast \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleleft }}( X_0, X_0 ) = \ast \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleleft }}( Y, X_0 ) = \emptyset$

for $Y \in \operatorname{\mathcal{C}}$. Note that $X_0$ is an initial object of the category $\operatorname{\mathcal{C}}^{\triangleleft }$, which we will refer to as the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$. Similarly, the right cone $\operatorname{\mathcal{C}}^{\triangleright }$ is obtained from $\operatorname{\mathcal{C}}$ by adjoining a new object which we refer to as the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ (and which is a final object of $\operatorname{\mathcal{C}}^{\triangleright }$).