Classifying anima of a monoid

Construction 5.1.14 (Classifying anima of a monoid). Let \(C\) be an \(\infty \)-category admitting finite products and geometric realizations. Given a monoid \(M\), we define its classifying anima \(\bB M\) as \[ \bB M \quad := \quad \abs {M} \quad = \quad \colim _{[n] \in \simp \catop } M_n \qin C. \] We may turn this into a pointed object in \(C\) via the map \(* = M_0 \to \colim _{[n] \in \simp \catop } M_n\). This construction is functorial in \(M\), resulting in a functor \(\bB \colon \Mon (C) \to C_*\).