Kerodon

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

Remark 9.1.5.15. The proof of Proposition 9.1.5.14 shows that every $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{C}}$ admits a (canonical) presentation as the colimit of a $\kappa $-filtered diagram

\[ \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{QC}}\quad \quad (k \in \operatorname{\mathcal{K}}) \mapsto \operatorname{\mathcal{C}}_{k}, \]

where each of the $\infty $-categories $\operatorname{\mathcal{C}}_{k}$ admits a final object. In ยง9.1.8, we will prove a more refined version of this statement:

  • We can arrange that $\operatorname{\mathcal{K}}$ is (the nerve of) a $\kappa $-directed partially ordered set.

  • We can arrange that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{k}$ has the form $\operatorname{\mathcal{D}}^{\triangleright }$, for some $\infty $-category $\operatorname{\mathcal{D}}$.

  • The sizes of the $\infty $-category $\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{C}}_{k}$ can be controlled in terms of the size of $\operatorname{\mathcal{C}}$.

See Proposition 9.1.8.5. Beware that this more refined presentation is not canonical.