Kerodon

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

Proposition 5.4.5.5. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be a $\kappa $-small simplicial set. Then there exists an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is a $\kappa $-small $\infty $-category. In particular, $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small.

Proof. Without loss of generality, we may assume that $\kappa $ is the least uncountable cardinal for which $\operatorname{\mathcal{C}}$ is $\kappa $-small, so that $\kappa $ is regular (Remark 5.4.4.7). We proceed as in the proof of Proposition 4.1.3.2. We will construct $\operatorname{\mathcal{D}}$ as the colimit of a diagram of inner anodyne morphisms

\[ \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}(0) \hookrightarrow \operatorname{\mathcal{C}}(1) \hookrightarrow \operatorname{\mathcal{C}}(2) \hookrightarrow \operatorname{\mathcal{C}}(3) \hookrightarrow \cdots \]

where each transition map fits into a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \coprod _{s \in S(n)} \Lambda ^{n_ s}_{i_ s} \ar [d] \ar [r]^-{ \{ u_ s \} _{s \in S(n)} } & \operatorname{\mathcal{C}}(n) \ar [d] \\ \coprod _{s \in S(n)} \Delta ^{n_ s} \ar [r] & \operatorname{\mathcal{C}}(n+1); } \]

here the coproducts are indexed by the collection $\{ u_{s}: \Lambda ^{n_ s}_{i_ s} \rightarrow \operatorname{\mathcal{C}}(n) \} _{s \in S(n)}$ of all inner horns in the simplicial set $\operatorname{\mathcal{C}}(n)$. Note that if the simplicial set $\operatorname{\mathcal{C}}(n)$ is $\kappa $-small, then the set $S(n)$ is also $\kappa $-small (Proposition 5.4.4.9), so that $\operatorname{\mathcal{C}}(n+1)$ is also $\kappa $-small. Since $\kappa $ is regular and uncountable, it follows that the colimit $\operatorname{\mathcal{C}}= \varinjlim \operatorname{\mathcal{C}}(n)$ is $\kappa $-small (Remark 5.4.4.6). $\square$