\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

With regards to the first point: Suppose that we have a (not necessarily exact) sequence of chain complexes \(A\xrightarrow{f} B \xrightarrow{g} C\). The total complex of this sequence will not be a chain complex (as \(g\circ f \neq 0\)). However, to build a twisted complex from this data we will only need that \(g\circ f\) is homotopic to zero. Suppose that \(H:A\to C[1]\) is a homotopy (so that \(d_AH+Hd_C=g\circ f\)). Then \[\delta = \begin{pmatrix} 0 & 0 & 0\\ f & 0 & 0\\ H & g & 0 \end{pmatrix}\] gives us a twisted complex on \(A\oplus B[1]\oplus C[2]\). For the second point: Let \((A, m^k)\) be an \(A_\infty\) algebra. There are a particularly nice class of deformations of \(A_\infty\) governed by elements \(a\in A^1\) satisfying the Maurer-Cartan equation: \[m^1(a)+m^2(a\otimes a)+m^3(a\otimes a \otimes a)+\cdots =0.\] In order for this equation to make sense, one needs show that the sum converges. This is usually achieved by asking that \(A\) be filtered and that \(m^k(a^{\otimes k})\) lies increasingly positive filtration levels. When one can make sense of this equation, we can define a new \(A_\infty\) algebra, \((A, m^k_a)\) whose product is defined by \[m^k_a:=\sum_{n>0}\sum_{j_0+\cdots+j_k=n} m^{k+n}(a^{\otimes j_0}\otimes \id \otimes a^{\otimes j_1}\otimes \id \cdots \otimes a^{\otimes j_{k-1}}\otimes \id \otimes a^{\otimes j_k})\] Now consider the setting where \(C\) is a chain complex, and \(A=\hom(C, C)\). Then \(a\in A^1\) corresponds to a map \(a: C\to C[1]\), and the Maurer-Cartan equation has two terms: The combination of these two terms checks the condition that \((d_A+a)\circ (d_A -a)=0\); that is that we can deform the differential by \((-1)^k a\).