A ∞ algebra

Definition 19.3.1. Let \(C\) be a monoidal \(\infty \)-category. An \(\bbA _{\infty }\)-algebra in \(C\) is a functor \[ A\colon \simp \catop \longrightarrow C^{\otimes } \] over \(\Cut \colon \simp \catop \to \Assoc ^{\otimes }\) which sends the dual of every interval inclusion \[ e_i\colon [1]\simeq \{i-1\leq i\}\hookrightarrow [n] \] to an inert morphism. We write \[ \Alg _{\bbA _{\infty }}(C) \subseteq \Fun _{/\Assoc ^{\otimes }}(\simp \catop ,C^{\otimes }) \] for the full subcategory of such functors.