Horns

Definition 1.6.5 (Horns). For \(n \geq 1\) and \(0 \leq k \leq n\), we define the \(k\)-horn \(\Lambda ^n_k\) as the subsimplicial set \[ \Lambda ^n_k \subseteq \Delta ^n \] whose \(m\)-simplices are those maps \(\phi \colon [m] \to [n]\) in \(\simp \) such that there is some element \(i \in [n] \setminus \{k\}\) which is not in the image of \(\phi \). Equivalently, it is the union inside \(\Delta ^n\) of all of its faces except for the \(k\)-th face.