Simplicial Set Basics
Definition The \( n \)-simplex is the simplicial set \( \operatorname{Hom}_{\Delta}(-,[n]) \colon \Delta \rightarrow \operatorname{Set} \). #+endDefinition
\( X \) simplicial set. Let \( \pi ^\Delta _0 (X) \) be \(X_0/\sim \) with \( x \sim y \) if there is a 1-simplex \( f \in X_1 \) with \( d_0(f) = x \) and \( d_1(f) = y \).
\( X \) simplicial set. The Yoneda lemma states that \( X_n \cong \Hom_{\sSet}(\Delta ^{n},X) \). This is a natural isomorphism of functor \( (-)_{n} \cong \Hom_{\sSet}(\Delta ^{n},-) \).