Extra degeneracies

Definition 5.2.3 (Extra degeneracies). We define the subcategory \(\simp ^{\deg } \subseteq \simp \) as the subcategory consisting of all objects \([n]\), but with only those morphisms \(\phi \colon [n] \to [m]\) satisfying \(\phi (n) = m\). An augmented simplicial object with extra degeneracies in an \(\infty \)-category \(C\) is a functor \(Y\colon (\simp ^{\deg })\catop \to C\).

There exists a functor \((+1)\colon \simp _+ \hookrightarrow \simp ^{\deg }\) given on objects by \([n] \mapsto [n+1]\), and on morphisms by sending \(\phi \colon [n] \to [m]\) to the map \[ [n+1] \to [m+1], \qquad i \mapsto \begin {cases} \phi (i) & 0 \leq i \leq n \\ m+1 & i = n+1. \end {cases} \] Given an augmented simplicial object with extra degeneracies \(Y\colon (\simp ^{\deg })\catop \to C\), precomposition with \((+1)\) yields the underlying augmented simplicial object of \(Y\). Conversely, we say that an augmented simplicial object \(X\colon \simp _+\catop \to C\) admits extra degeneracies if there exists an augmented simplicial object with extra degeneracies \(Y\) such that \(X\) is isomorphic to the underlying augmented simplicial object of \(Y\).