Kerodon

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

Remark 5.4.4.4. The low-dimensional simplices of $\operatorname{\mathcal{QC}}$ are simple to describe:

  • An object of $\operatorname{\mathcal{QC}}$ 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 $\operatorname{\mathcal{QC}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

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

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

    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 an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.