# Kerodon

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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). $\square$