\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
definition 0.0.1
Let \(\mathcal A\) be an \(A_\infty\) category; denote the product structure by \(m^k_\mathcal A\). A right \(\mathcal A\) module (denoted by \(M\in \text{Mod}-\mathcal A\)) is a
- assignment \(M(A)\) of a graded module to each \(A\in \text{Ob}(\mathcal A\)) and,
- for each sequence \(A_1, \ldots, A_k\) of objects in \(\mathcal A\), a composition map
\[
m^{1|{k-1}}_{M|\mathcal A}:M(A_{k-1})\otimes \hom(A_{k-2}, A_{k-1})\otimes \cdots \otimes \hom(A_1, A_0)\to M(A_k)[2-k].
\]
These are required to satisfy the quadratic \(A_\infty\) relationships: for every sequence \(A_0, \ldots, A_k\) of objects in \(A\),
\begin{align*}
0=&\sum_{\substack{ j_1+j+j_2=k\\j_1=0 }} (-1)^{\clubsuit_k} m_{M|\mathcal A}^{1|j_2} \circ (m_{A|M}^{1|j}\otimes \operatorname{id}_A^{\otimes j_2})\\
&+\sum_{\substack{ j_1+j+j_2=k\\j_1>0 }} (-1)^{\clubsuit_k} m_{M|{\mathcal A}}^{1|k-j}\circ ( \operatorname{id}_M\otimes \operatorname{id}_A^{\otimes j_1}\otimes m^{j}_A \otimes \operatorname{id}^{\otimes j_2})
\end{align*}