\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
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 first term \(m^1(a) = d_A a + a d_A\). The vanishing of this term states that \(a\) is a chain map;
- The vanishing of the second term \(m^2(a, a)\) tells us that \(a\) squares to zero (so that it gives a differential).
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 \((-1)^k a\).