Augmented simplex category

Definition 5.1.8. We define the augmented simplex category as \(\simp _+ := \simp ^{\triangleleft }\), the cone of the simplex category. We denote the cone point by \([-1]\). A functor \(\simp _+\catop \to C\) is called an augmented simplicial object. It corresponds to a simplicial object \(X_{\bullet }\colon \simp \catop \to C\) equipped with a cocone \(X_{\bullet } \Rightarrow \const _{X_{-1}}\).