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Remark The low-dimensional simplices of $\operatorname{ \pmb {\mathcal{QC}} }$ are simple to describe:

  • An object of $\operatorname{ \pmb {\mathcal{QC}} }$ is a (small) $\infty $-category $\operatorname{\mathcal{C}}$.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are objects of $\operatorname{\mathcal{QC}}$, then a morphism from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in $\operatorname{ \pmb {\mathcal{QC}} }$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

  • A $2$-simplex $\sigma $ of $\operatorname{ \pmb {\mathcal{QC}} }$ can be identified with a diagram

    \[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{D}}\ar [dr]^{G} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-60pt>^-{\mu } & \\ \operatorname{\mathcal{C}}\ar [ur]^{F} \ar [rr]_{H} & & \operatorname{\mathcal{E}}} \]

    where $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are (small) $\infty $-categories, $F$, $G$, and $H$ are functors, and $\mu : G \circ F \rightarrow H$ is a morphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. Moreover, $\sigma $ is thin if and only if $\mu $ is an isomorphism of functors (Proposition