Kerodon

Notation 4.6.7.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often use the symbol $\emptyset _{\operatorname{\mathcal{C}}}$ to denote an initial object of $\operatorname{\mathcal{C}}$, provided that such an object exists. In this case, we will sometimes abuse terminology by referring to $\emptyset _{\operatorname{\mathcal{C}}}$ as the initial object of $\operatorname{\mathcal{C}}$. This abuse is justified by Corollary 4.6.7.14, which guarantees that $\emptyset _{\operatorname{\mathcal{C}}}$ is uniquely determined up to a contractible space of choices (in particular, it is well-defined up to isomorphism). Similarly, we will often use the symbol ${\bf 1}_{\operatorname{\mathcal{C}}}$ to denote a final object of $\operatorname{\mathcal{C}}$, provided that such an object exists, and will sometimes abuse terminology by referring to ${\bf 1}_{\operatorname{\mathcal{C}}}$ as the final object of $\operatorname{\mathcal{C}}$. When it is unlikely to cause confusion, we will sometimes omit the subscripts and denote the objects $\emptyset _{\operatorname{\mathcal{C}}}$ and ${\bf 1}_{\operatorname{\mathcal{C}}}$ by $\emptyset $ and ${\bf 1}$, respectively.