Extra degeneracy argument

Proof. By assumption, there exists an augmented simplicial object with extra degeneracies \(Y\colon (\simp ^{\deg })\catop \to C\) such that \(X_n = Y_{n+1}\). We need to show that the preferred map from \(\colim _{[n] \in \simp \catop } X_n\) to \(X_{-1}\) induced by the cocone is an isomorphism.

Observe that the inclusion functor \((+1)\colon \simp \hookrightarrow \simp ^{\deg }\) admits a right adjoint given by the inclusion \(\simp ^{\deg } \hookrightarrow \simp \): for natural numbers \(n,m \geq 0\), providing a map \(\phi \colon [n] \to [m]\) of partially ordered sets is the same as providing a map \(\overline {\phi }\colon [n+1] \to [m]\) of posets satisfying the condition that \(\overline {\phi }(n+1) = m\). It follows from Corollary 21.5.7 that the functor \((+1)\colon \simp \hookrightarrow \simp ^{\deg }\) is an initial functor, and hence its opposite \((+1)\catop \colon \simp \catop \hookrightarrow (\simp ^{\deg })\catop \) is a final functor. By Theorem 21.5.1, this means that the geometric realization of \(X\) may be computed as \[ \colim _{[n] \in \simp \catop } X_n = \colim _{[n] \in \simp \catop } Y_{n+1} \simeq \colim _{[m] \in (\simp ^{\deg })\catop } Y_m. \] But since \(\simp ^{\deg }\) has an initial object given by \([0]\), \((\simp ^{\deg })\catop \) has a terminal object, and so it follows from Lemma 21.2.4 that \(\colim _{[m] \in (\simp ^{\deg })\catop } Y_m\) is isomorphic to \(Y_0 = X_{-1}\) as desired. □