\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
definition 0.0.1
Let \((A, m^k_A)\) be an \(A_\infty\) algebra. A \(A-\text{Mod}\) \((M, m^{k|1}_{A|M})\) is a free graded \(\mathbb{k}\)-module \(M^\bullet\) equipped with multilinear products
\[
m^{k|1}:(A^\bullet)^{\otimes k}\otimes M\to (M^{\bullet+1-k})
\]
for each \(k\geq 0\). These are required to satisfy the quadratic \(A_\infty\) relationships:
\begin{align*}
0=&\sum_{\substack{ j_1+j+j_2=k_1+1\\ j_1+j\leq k_1 }} m_{A|M}^{k_1-j+1|1}\circ ( \operatorname{id}_A^{\otimes j_1}\otimes m^{j}_A \otimes \operatorname{id}^{\otimes k_1-j_1-j}\otimes \operatorname{id}_M})\\
&+\sum_{\substack{ j_1+j+j_2=k_1+1\\j_1\leq k_1\leq j_1+j-1}} m_{A|M}^{j_1|1} \circ (\operatorname{id}_A^{\otimes j_1}\otimes m_{A|M}^{k_1-j_1|1})
\end{align*}