Kerodon

Variant 1.1.1.8. Let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ denote the category whose objects are linearly ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \} $ (where $n$ is a nonnegative integer) and whose morphisms are strictly increasing functions $\alpha : [m] \hookrightarrow [n]$. If $\operatorname{\mathcal{C}}$ is any category, we will refer to a functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ as a semisimplicial object of $\operatorname{\mathcal{C}}$. We typically use the notation $C_{\bullet }$ to indicate a semisimplicial object of $\operatorname{\mathcal{C}}$, whose value on an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}$ we denote by $C_{n}$.