Kerodon

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

Remark 1.1.8.18. We can state Corollary 1.1.8.17 more informally as follows: the category $\operatorname{Set_{\Delta }}$ of simplicial sets is generated, under small colimits, by objects of the form $\Delta ^ n$. In fact, one can say more: it is freely generated (under small colimits) by the essential image of the Yoneda embedding

\[ \operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Set_{\Delta }}\quad \quad [n] \mapsto \Delta ^ n. \]

This is a general fact about presheaf categories, which we will return to in ยง.