\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
There is a relation between exact triangles in \(A_\infty\) categories and the \(m^3\) product operation. Given \(A\xrightarrow{f}B \xrightarrow{g}C\xrightarrow{h}A[1]\) an exact triangle, where \(f, g, h\) are chain maps, with the additional assumption that
\begin{align*}
\emprod^2(g, f)=0&& \emprod^2(f, h)=0 && \emprod^2(h, g)=0.
\end{align*}
Then we have maps between \(C\) and the twisted cone by
\begin{align*}
C\xrightarrow{h} \cone(f:A\to B)&& \cone(f:A\to B) \xrightarrow{g} C.
\end{align*}
Now, \(\emprod^2_{Tw}(h, g)=\emprod^3(g,f,h)_{\mathcal A}\), which is the identity in the homology category.