The simplex category is generated by faces and degeneracies

Observation 1.6.2. Every morphism in \(\simp \) can be written as a finite composite of face maps \(d_i\colon [n-1] \to [n]\) and degeneracy maps \(s_i\colon [n+1] \to [n]\) (see Notation 1.2.10). The relations these maps satisfy are known as the simplicial identities.