Kerodon

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

Remark 1.1.1.13. Since the category of sets has all (small) limits and colimits, the category of (semi)simplicial sets also has all (small) limits and colimits. Moreover, these limits and colimits are computed levelwise: for any functor

\[ S_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto S_{\bullet }(C), \]

and any nonnegative integer $n$, we have canonical bijections

\[ (\varinjlim _{C \in \operatorname{\mathcal{C}}} S(C))_{n} \simeq \varinjlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ) \quad \quad (\varprojlim _{C \in \operatorname{\mathcal{C}}} S(C))_{n}\simeq \varprojlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ). \]