Monoid objects

Definition 5.1.1 (Monoid). Let \(C\) be an \(\infty \)-category that admits finite products. We define a monoid object in \(C\) (or \(E_1\)-monoid) to be a simplicial object \[ M\colon \simp \catop \to C, \qquad [n] \mapsto M_n := M([n]), \] satisfying the Segal condition: for every \([n] \in \simp \) the Segal map \[ (e_i^*)_{i=1}^n \colon M_n \to \prod _{i=1}^n M_1 \] induced by the \(n\) inclusion maps \(e_i \colon [1] \cong \{i-1 \leq i \} \hookrightarrow [n]\) is an isomorphism. In particular, the condition for \(n = 0\) requires \(M_0\) to be a terminal object of \(C\). We write \[ \Mon (C) \subseteq \Fun (\simp \catop ,C) \] for the full subcategory spanned by the monoids in \(C\). We refer to \(M_1\) as the underlying object of \(M\).