Standard simplices

Definition 1.6.4 (Simplices). For every natural number \(n\), we define the \(n\)-simplex \(\Delta ^n\) as the simplicial set represented by \([n]\): \[ \Delta ^n := \Hom _{\simp }(-, [n]) \colon \simp \catop \to \Set . \] By the Yoneda lemma this defines a fully faithful functor \[ \Delta ^{\bullet }\colon \simp \hookrightarrow \sSet . \] Again by the Yoneda lemma there is a natural bijection \(X_n \cong \Hom _{\sSet }(\Delta ^n, X)\), so that we may think of an \(n\)-simplex in \(X\) as a morphism \(\sigma \colon \Delta ^n \to X\) of simplicial sets.