\Ff-monoid

Definition 17.3.5 (\(\Ff \)-monoid). Let \(\Ff = (F,F_L,F_R)\) be an adequate triple satisfying the conditions of Convention 17.3.1.

(1)

Given an \(\infty \)-category \(C\) with finite products, we say that a functor \(M\colon F_L\catop \to C\) satisfies the Segal condition if for objects \(I_1, \dots , I_n \in F\), the maps \(e_i\colon I_i \hookrightarrow \bigsqcup _{j=1}^n I_j\) induce an isomorphism \[ (e_i^*)_{i=1}^n \colon M(\bigsqcup _{j=1}^n I_j) \iso \prod _{i=1}^n M(I_i). \]

(2)

For \(C\) with finite products, we define an \(\Ff \)-monoid in \(C\) to be a functor \[ M\colon \Span _{L,R}(F) \to C \] such that \(M\vert _{F_L\catop }\) satisfies the Segal condition. We denote by \[ \Mon _{\Ff }(C) \quad \subseteq \quad \Fun (\Span _{L,R}(F), C) \] the full subcategory spanned by the \(\Ff \)-monoids.