Inert active Fin pointed

Definition 17.4.1. Given a finite set \(I\), we write \(I_+ := I \sqcup *\) for the resulting finite pointed set. A morphism \(f\colon I_+ \to J_+\) of finite pointed sets is called inert if on the preimage of \(J\) it restricts to a bijection \(f^{-1}(J) \iso J\). We say that \(f\) is active if \(f^{-1}(*) = \{*\}\).

Given a finite set \(I\), the Segal map \(\rho _i\colon I_+ \to \{i\}_+\) for \(i \in I\) is defined as the map that is the identity on \(\{i\}\) and sends all other \(j \in I \setminus \{i\}\) to the basepoint.