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Example (Objects and Morphisms of $\operatorname{\mathcal{QC}}_{\ast }$). The low-dimensional simplices of the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ are easy to describe:

  • The objects of $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a (small) $\infty $-category and $C \in \operatorname{\mathcal{C}}$ is an object (which we identify with the morphism $\Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$ taking the value $C$).

  • Let $(\operatorname{\mathcal{C}},C)$ and $(\operatorname{\mathcal{D}},D)$ be objects of $\operatorname{\mathcal{QC}}_{\ast }$. A morphism from $(\operatorname{\mathcal{C}}, C)$ to $(\operatorname{\mathcal{D}}, D)$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\ast }$ can be identified with a pair $(F, \alpha )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories and $\alpha : F(C) \rightarrow D$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.