Kerodon

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

Corollary 1.1.8.17. Let $\operatorname{\mathcal{U}}$ be a full subcategory of the category $\operatorname{Set_{\Delta }}$ of simplicial sets. If $\operatorname{\mathcal{U}}$ is closed under small colimits and contains the standard $n$-simplex $\Delta ^ n$ for each $n \geq 0$, then $\operatorname{\mathcal{U}}= \operatorname{Set_{\Delta }}$.

Proof. If $\operatorname{\mathcal{U}}$ is closed under small colimits, then it satisfies conditions $(1)$ and $(2)$ of Lemma 1.1.8.15, along with condition $(3'')$ of Remark 1.1.8.16. Consequently, if it contains each of the standard simplices $\Delta ^ n$, then $\operatorname{\mathcal{U}}= \operatorname{Set_{\Delta }}$. $\square$

Alternative Proof of Corollary 1.1.8.17. Via the Yoneda embedding $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Set_{\Delta }}$, we can identify $\int ^{\operatorname{{\bf \Delta }}} S$ with the category whose objects are simplicial sets of the form $\Delta ^ n$ (for some $n \geq 0$), which are equipped with a map of simplicial sets $\Delta ^ n \rightarrow S_{\bullet }$. In particular, we have a canonical map of simplicial sets $\varinjlim _{ ([n], \sigma ) \int ^{\operatorname{{\bf \Delta }}} S } \Delta ^ n \rightarrow S_{\bullet }$. To prove Corollary 1.1.8.17, it suffices to observe that this map is an isomorphism. This is an elementary calculation which we leave to the reader (see ยง for more details). $\square$