Remark 1.1.3.12. Proposition 1.1.3.11 can be reformulated using the language of Kan extensions (see Definition 7.3.0.1): it asserts that a simplicial set $S: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ has dimension $\leq k$ if and only if it is left Kan extended from the full subcategory of $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ spanned by the objects $\{ [n] \} _{n \leq k}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$