S-horn

Let \(\Delta\) be the simplex category and \([n]\) a simplex. Suppose \(S \subseteq [n]\) is a subobject, then the \(S\)-horn \(\Lambda_S^n\) is defined as simplicial set

\begin{align*} \Lambda_{S}^n: \Delta \rightarrow& \mathrm{Set} \\ [m] \mapsto& \{f \in \mathrm{Hom}_{\Delta}([m],[n]) \vert \exists i \in [n] \setminus S, i \not\in \mathrm{im}(f)\} \\ (g: [m] \rightarrow [m']) \mapsto& g^{ \rightarrow} \end{align*}

Note that \(g^{ \rightarrow}\) is welldefined, as

\begin{align*} \mathrm{im}(f \circ g) \subseteq \mathrm{im}(f) \end{align*}

hence \(i \not\in \mathrm{im}(f)\) implies \(i \not\in \mathrm{im}(f)\)