\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
\(A_\infty\) category
definition 0.0.1
An \(A_\infty\) category \(\mathcal A\) is the data of
- A collection of objects \(\text{Ob}(\mathcal A)\),
- For each pair of objects \(A_0, A_1\) a graded free module \(\hom^\bullet(A_0, A_1)\),
- For each sequence of objects \(A_0, \ldots , A_k\), a composition map
\[
m^k:\hom(A_{k-1}, A_k)\otimes \cdots \otimes \hom(A_0, A_1)\to \hom(A_0, A_k)[2-k]
\]
for each \(k>1\). These are required to satisfy the quadratic filtered \(A_\infty\) relationship
\[
0=\sum_{k_1+k'+k_2=k} (-1)^{\clubsuit(\underline x, k_1)} (m^{k_1+1+k_2})\circ (\operatorname{id}^{\otimes k_1}\otimes m^{k'}\otimes id^{\otimes k_2}) (x_1, \cdots ,x_k)
\]
The sign is determined by \(\clubsuit(\underline x,k_1):= k_1+\sum_{j=1}^{k_1} \deg(x_j)\).
Connections
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