A ∞ module

Definition 19.3.4. An \(\bbA _{\infty }\)-module in \(C\) is a functor \[ F\colon \simp \catop \times [1]\longrightarrow C^{\otimes } \] over \(\LCut \) satisfying the following conditions:

(1)

The restriction to \(\simp \catop \times \{1\}\) is an \(\bbA _{\infty }\)-algebra.

(2)

Each morphism \(F([n],0)\to F([n],1)\) is inert.

(3)

If \(\alpha \colon [n]\to [m]\) is an interval inclusion with \(\alpha (n)=m\), then \(F(\alpha \catop ,\id _0)\) is inert.

We write \(\LMod ^{\bbA _{\infty }}(C)\) for the resulting full subcategory of \(\Fun _{/\oLMod ^{\otimes }}(\simp \catop \times [1],C^{\otimes })\).