\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
The name module comes from the simplest example. Let \(R\) be a ring. Now consider the \(A_\infty\) category \(\mathcal A\) which only contains one object \(A\), and \(\hom(A, A)=R\).
Let \(M\) be an \(R\)-module. We obtain a \(\text{mod}-\mathcal A\) with the assignment \(M(A)=M\), and whose product \(m^{1|k}:M(A)\tensor A^{\tensor k}\to M(A)\) is
\[
m^{1|1}(x,r)=x\cdot r
\]
and vanishes if \(k\neq 1\). The \(A_\infty\) module relations state
\[m^{1|1}(m^{1|1}(x, r_1),r_2)-m^{1|1}(x, m^2(r_1, r_2))=(x\cdot r_1)\cdot r_2 - x\cdot (r_1\cdot r_2)=0\]
which is guaranteed by associativity of the product.
Given any chain complex of \(R\)-modules \(M^\bullet\), we similarly obtain a right \(\mathcal A\) module by taking \(M^\bullet(A)=M^A\); the \(A_\infty\) module relations now state that \(m^{1|0}\circ m^{1|0}= d_M\circ d_M=0\), and that \(d_M\) is a morphism of \(R\)-modules.
There are right \(\mathcal A\)-modules beyond chain complexes. However, given any right \(\mathcal A\)-module, the homology of the complex \(H^\bullet(M(A))\) is a graded \(R\)-module.