One viewpoint on mapping cones of cochain complexes is that they give deformations of (direct sums of) objects of our categories. Given a map of cochain complexes \(f: A\to B\), the differential on \(\cone(f)\) has the form \[d_{\cone(f)}=\begin{pmatrix} d_A & 0 \\ 0 & d_B \end{pmatrix} + \begin{pmatrix} 0 & 0\\ f & 0\end{pmatrix}\] where the first term is the differential on \(A\oplus B[1]\), and the second term ``deforms'' the differential on this chain complex. Twisted complexes extend this story in several directions: firstly, we expand the set of deformations so that the objects we consider are chain complexes up to homotopy, and we allow deformations of the product (and not only differential) structure.

definition 0.0.1

Let \(\mathcal C\) be an \(A_\infty\) category. A twisted complex \((E, \delta_E)\) consists of:

\(E\), a formal direct sum of shifts of objects \[E:=\bigoplus_{i=1}^N E_i[k_i]\] where \(E_{i}\in Ob(\mathcal C)\), and \(k_i\in \ZZ\).
A differential \(\delta_E\), which can be written as a matrix of degree 1 maps \[\delta^{ij}_E: E_i[k_i]\to E_j[k_j +1] .\] These maps must satisfy the following conditions:
- the matrix \(\delta_E\) is strictly upper triangular and;
- They satisfy the Maurer-Cartan relation: \[\sum_{k\geq 1} m^k (\delta_E\otimes\cdots \otimes \delta_E) =0.\]

The condition that the matrix \(\delta_E\) is strictly upper triangular is to ensure that the sum in the Maurer-Cartan relation converges. One can also ask that there exists a filtration on \(E\), the formal direct sum of shifts of objects, and that the differential \(\delta_E\) respects the filtration (see Section 31 of [Sei08]). From this perspective, the twisted complex looks more like a formal deformation of the direct sum. 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 \((-1)^k a\).

definition 0.0.2

Let \((E, \delta_E)\) and \((F, \delta_F)\) be two twisted complexes. A morphism of twisted complexes is a collection of morphisms of \(\mathcal C\) \[f_{ij}:E_i[k_i]\to E_j[k_j].\] The set of morphisms can therefore be written as \(\hom^d((E, \delta_E), (F, \delta_F))=\bigoplus_{i, j}\hom^{d+k^F_j-k^E_i}(E_i, F_j).\) Given a sequence \(\{(E_i, \delta_i)\}_{i=0}^k\) of twisted complexes, and \(a_i\in\hom^d((E_{i-1}, \delta_{E_{i-1}}), (E_{i}, \delta_{E_{i}}))\), we have a composition \[m^k_{\operatorname{Tw}}(a_{k}\otimes \cdots \otimes a_{1} ):=\sum_{j_0, \ldots, j_k\geq 0} m^k(\delta_k^{\otimes j_k}\otimes a_{k}\otimes \delta_{k-1}^{\otimes j_{k-1}}\otimes a_{k-1}\otimes \cdots \otimes a_1\otimes \delta_0^{\otimes j_{0}}).\]

proposition 0.0.3

Let \(\mathcal C\) be an \(A_\infty\) category. The category of twisted complexes, \(\operatorname{Tw}(\mathcal C)\), is the \(A_\infty\) category whose objects are twisted complexes, morphisms are morphisms of twisted complexes, and \(A_\infty\) compositions are given by \(m^k_{\operatorname{Tw}}\).

theorem 0.0.4

Let \(\mathcal C\) be an \(A_\infty\) category. The category of twisted complexes, \(\operatorname{Tw}(\mathcal C)\) is a triangulated category. There is a fully faithful inclusion \(\mathcal A\to \operatorname{Tw}(\mathcal C)\). Furthermore, the image of \(\mathcal A\) generated \(\operatorname{Tw}(\mathcal C)\). To give twisted complexes the structure of a triangulated category, we must specify what the exact triangles are. Given a morphism \(f: (E, \delta_E)\to (F, \delta_F)\), we can define the cone of \(f\) to be the twisted complex \((E[1]\oplus F, \delta')\) where \(\delta'\) is the matrix \[ \left(\begin{array}{c|c} \delta_E &0 \\ \hline f^\delta_F & \delta_F\end{array}\right). \] There exists an inclusion functor \(i:\mathcal C\to \Tw(\mathcal C)\). We can therefore declare that the triangle \(A\to B\to C\) is exact in \(\mathcal A\) is if \(C\) is quasi-isomorphic to \(\cone(A\to B)\) in the category of twisted complexes.

References

[Sei08]

Paul Seidel. Fukaya categories and Picard-Lefschetz theory, volume 10. European Mathematical Society, 2008.