\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
definition 0.0.1
Let \(M, N\) be \(A_\infty\) modules. A pre-morphism of \(A_\infty\) modules of degree \(d\) is a collection of maps for every set of objects \(A_0, \ldots, A_{k-1}\in \mathcal A\),
\[f^{1|k-1}: M(A_{k-1k)\otimes \hom(A_{k-2}, A_{k-1})\otimes \cdots \otimes \hom(A_0, A_1)\to N(A_0)[d-k+1].\]
The set of all \(A_\infty\) pre-morphisms \(\hom^\bullet_{\text{mod}-\mathcal A}(M, N)\) is a cochain complex whose differential is
\begin{align*}
(m^1_{\hom_{\text{mod}-\mathcal A}}(f))^{1|k-1}=&\sum_{j+j_2=k}(-1)^\diamondsuitm_{N}^{1|j_2}(f^{1|j-1}\tensor \id^{\tensor j_2})\\
&+\sum_{j_1+j+j_2=k, j_1>0}(-1)^\diamondsuit f^{1|j_1+j_2-1}(\id^{\tensor j_1}\tensor m^j_{\mathcal A}\tensor \id^{\tensor j_2})\\
&+ \sum_{j_1+j+j_2=k, j_1=0}(-1)^\diamondsuit f^{1|j_1+j_2-1}( m^{1|j-1}_{M}\tensor \id^{\tensor j_2})
\end{align*}
where \(\diamondsuit_j=-k+j+1\sum_{i={j-1}}^{k}|x_i|\).