Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1). Combining Remark 4.6.6.6) with Example 4.6.5.12, we obtain the following:

  • An object $Y \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible (that is, there exists a $1$-morphism from $Y$ to $Z$ and for every pair of morphisms $f,g: X \rightarrow Y$, there is a unique isomorphism $\gamma : f \xRightarrow {\sim } g$).

  • An object $Y \in \operatorname{\mathcal{C}}$ is final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible.