Kerodon

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

Definition 1.1.1.12. Let $\operatorname{Set}$ denote the category of sets. A simplicial set is a simplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$. A semisimplicial set is a semisimplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$. If $S_{\bullet }$ is a (semi)simplicial set, then we will refer to elements of $S_{n}$ as $n$-simplices of $S_{\bullet }$. We will also refer to the elements of $S_{0}$ as vertices of $S_{\bullet }$, and to the elements of $S_{1}$ as edges of $S_{\bullet }$.

We let $\operatorname{Set_{\Delta }}= \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set})$ denote the category of functors from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to $\operatorname{Set}$. We refer to $\operatorname{Set_{\Delta }}$ as the category of simplicial sets.