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 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).
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 \(\operatorname{Tw}(\mathcal C)\).References
[Sei08] | Paul Seidel. Fukaya categories and Picard-Lefschetz theory, volume 10. European Mathematical Society, 2008. |