We have no expectation that the geometric Fukaya category \(\Fuk(X)\) is triangulated: indeed, it is only guaranteed that the category is pre-additive. Never-the-less, we would like to say that \(\Fuk(X\)) can be enhanced to a triangulated category. There are general constructions which enlarge an \(A_\infty\) category \(\mathcal C\) to a triangulated category. These constructions rely on the fact that the \(A_\infty\) structure on \(\mathcal C\) already knows what its exact triangles should be. We give two enlargements: the category of modules over \(\mathcal C\), or the category of twisted complexes. These notes are based on Chapter 3 of [Sei08].

1: The Category of modules over an \(A_\infty\) category

Given \(\mathcal A\) an \(A_\infty\) category, there is a category of \(\text{Mod}- \mathcal A\) (dg category of \(A_\infty\)-modules) which is triangulated. This means describing the objects, morphisms, and compositions of this category.

definition 1.0.1

Let \(\mathcal A\) be an \(A_\infty\) category; denote the product structure by \(m^k_\mathcal A\). A right \(\mathcal A\) module (denoted by \(M\in \text{Mod}-\mathcal A\)) is a

assignment \(M(A)\) of a graded module to each \(A\in \text{Ob}(\mathcal A\)) and,
for each sequence \(A_1, \ldots, A_k\) of objects in \(\mathcal A\), a composition map \[ m^{1|{k-1}}_{M|\mathcal A}:M(A_{k-1})\otimes \hom(A_{k-2}, A_{k-1})\otimes \cdots \otimes \hom(A_1, A_0)\to M(A_k)[2-k]. \]

These are required to satisfy the quadratic \(A_\infty\) relationships: for every sequence \(A_0, \ldots, A_k\) of objects in \(A\), \begin{align*} 0=&\sum_{\substack{ j_1+j+j_2=k\\j_1=0 }} (-1)^{\clubsuit_k} m_{M|\mathcal A}^{1|j_2} \circ (m_{A|M}^{1|j}\otimes \operatorname{id}_A^{\otimes j_2})\\ &+\sum_{\substack{ j_1+j+j_2=k\\j_1>0 }} (-1)^{\clubsuit_k} m_{M|{\mathcal A}}^{1|k-j}\circ ( \operatorname{id}_M\otimes \operatorname{id}_A^{\otimes j_1}\otimes m^{j}_A \otimes \operatorname{id}^{\otimes j_2}) \end{align*} The sign \(\clubsuit\) follows the \(A_\infty\) algebra sign convention: \[\clubsuit(\underline x,k_1):= k_1+\sum_{j=1}^{k_1} \deg(x_j).\]

example 1.0.2

The name module comes from the simplest example. Let \(R\) be a ring. Now consider the \(A_\infty\) category \(\mathcal A\) which only contains one object \(A\), and \(\hom(A, A)=R\). Let \(M\) be an \(R\)-module. We obtain a \(\text{mod}-\mathcal A\) with the assignment \(M(A)=M\), and whose product \(m^{1|k}:M(A)\tensor A^{\tensor k}\to M(A)\) is \[ m^{1|1}(x,r)=x\cdot r \] and vanishes if \(k\neq 1\). The \(A_\infty\) module relations state \[m^{1|1}(m^{1|1}(x, r_1),r_2)-m^{1|1}(x, m^2(r_1, r_2))=(x\cdot r_1)\cdot r_2 - x\cdot (r_1\cdot r_2)=0\] which is guaranteed by associativity of the product. Given any chain complex of \(R\)-modules \(M^\bullet\), we similarly obtain a right \(\mathcal A\) module by taking \(M^\bullet(A)=M^A\); the \(A_\infty\) module relations now state that \(m^{1|0}\circ m^{1|0}= d_M\circ d_M=0\), and that \(d_M\) is a morphism of \(R\)-modules. There are right \(\mathcal A\)-modules beyond chain complexes. However, given any right \(\mathcal A\)-module, the homology of the complex \(H^\bullet(M(A))\) is a graded \(R\)-module.

example 1.0.3

Let \(\mathcal A\) be an \(A_\infty\) category. We can define the zero module \(M\) which has the property that for all \(A\in \mathcal A\), our module assigns \(M(A)=0\). As a result, the composition maps \(m^{1|k}\) all vanish. This trivially satisfies the quadratic \(A_\infty\) module relations. While this appears to be a artificial example, it is generally desirable to have a zero object in your category, and there is no reason a priori for your original \(A_\infty\) category \(\mathcal A\) to have a zero object. For the example we are interested--- the Fukaya category--- there is no ``zero'' Lagrangian submanifold.

example 1.0.4

Let \(\mathcal A\) be an \(A_\infty\) category. Let \(A\in \mathcal A\) be an object. We can associate to \(A\) the Yoneda module, \(\mathcal Y_A\), which on every object \(B\in \mathcal A\) assigns the chain complex \[\mathcal Y_A(B):=\hom(B, A),\] and whose product maps are defined by \[m^{1|k-1}(m,a_{k-1}, \ldots, a_0):= m^k(m,a_{k-1}, \ldots, a_0).\] Note that the \(A_\infty\) module relations for \(Y_A(B)\) are exactly the \(A_\infty\) product relations for \(\mathcal A\).

definition 1.0.5

Let \(M, N\) be \(A_\infty\) modules. A pre-morphism of \(A_\infty\) modules of degree \(d\) is a collection of maps \(f^{1|k-1}: A^{\otimes k-1}\otimes M\to N[d-k+1]\). The set of pre-morphisms, \(\hom^\bullet(M, N)\) is a cochain complex whose differential is (up to sign) \begin{align*} (m^1_{\hom^\bullet(M, N)} f)^{1|k-1}:=&\sum_{j+j_2=k}(-1)^\diamondsuit m^{1|j_2}_{N|\mathcal A} (f^{1|j-1}\tensor \id^{\tensor j_2})\\&+ \sum_{\substack{j_1+j+j_2=k\\j_1>0}} (-1)^\diamondsuit f^{1|j_1+j_2}(\id^{\tensor j_1}\tensor m^j_{\mathcal A}\tensor \id^{\tensor j_1})\\ &+ \sum_{\substack{j_1+j+j_2=k\\j_1=0}} (-1)^\diamondsuit f^{1|j_2}(m^{1|j}_{M|\mathcal A}\tensor \id^{\tensor j_1}). \end{align*}

definition 1.0.6

Let \(\mathcal A\) be an \(A_\infty\) category. The category of right \(\mathcal A\)- modules, denoted by \(\text{Mod}-\mathcal A\), is the differential graded category whose:

Objects are right \(\mathcal A\) modules and
chain complexes of morphisms are the chain complexes of \(\mathcal A\) module pre-morphism,
composition is given by \[m^2_{\text{mod}-\mathcal A}(f, g)=\sum_{j+j_2=k}-(-1)^\diamondsuit f^{1|j_2}(g^{1|j-1}\tensor \id^{j^2})\]
higher product vanishing for \(k\geq 3.\)

proposition 1.0.7

Let \(\mathcal A\) be an \(A_\infty\) category. The category of modules over \(\mathcal A\) has the structure of a dg-category. One observes that \(\text{mod}-\mathcal A\) is in general a ``nicer'' category than \(\mathcal A\), as it inherits many of the properties of the category of chain complexes.

proposition 1.0.8

Let \(\mathcal A\) be an \(A_\infty\) category. Then \(H^0(\text{mod}-\mathcal A)\) is a triangulated category. We only describe the exact triangles in the category. Given \(f\in \hom(M, N)\) a morphism of right \(A_\infty\) modules, we define the cone module to be \[\cone(f)(A):=M(A)\oplus N(A)[1]\] whose \(A_\infty\) module structure is given by \[ m^{1|k}_{\cone(f)|\mathcal A}:=m^{1|k}_{M|\mathcal A}\oplus (f^{1|k}+ m^{1|k}_{N|\mathcal A}).\] The Yoneda module construction (example 1.0.4) gives a fully faithful functor \(\mathcal A \to \text{mod}-\mathcal A\). As the category \(\text{mod}-\mathcal A\) has mapping cones, this gives a triangulated envelope for \(\mathcal A\). We therefore say that \(A\to B \to C\) is an exact triangle in \(\mathcal C\) if the image under Yoneda embedding is isomorphic to \(A\to B \to \cone(f)\).

2: the category of twisted complexes

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 2.0.1

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 2.0.2

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 2.0.3

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 2.0.4

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.

3: modules or twisted complexes?

Both definition 1.0.1 and definition 2.0.1 provide a method for identifying the exact triangles of \(\mathcal C\), an \(A_\infty\) category. We now highlight some of the differences between these two constructions. These differences are most easily seen by reducing to a simple setting. Consider \(R\) a field. Then the category of twisted complexes over \(R\) will be the category of finite dimensional graded vector spaces with a choice of basis, while \(\text{mod} R\) will be the category of \(R\)-vector spaces. Given another ring \(S\), and a ring homomorphism \(R\to S\), we obtain a map from \(\Tw(R)\to \Tw(S)\) and a map \(\text{mod}-S\to \text{mod} R\). Also observe that the category of \(R\)-vector space is much larger than the category of finite-dimensional graded vector spaces. The construction of twisted complexes is a functor on the category of \(A_\infty\) categories, \[\Tw:A_\infty-\text{cat}\to A_\infty-\text{cat}.\] The \(\text{mod}-\mathcal C\) construction gives us a contravariant functor on the category of \(A_\infty\) categories, \[\text{mod}-(-):A_\infty-\text{cat}\to (A_\infty-\text{cat})^{\text{op}}.\] For the purposes of computations in symplectic geometry, it is usually unimportant if we consider enlarging the Fukaya category by looking at modules or at twisted complexes, as both structure give us access to the exact triangles in \(\Fuk(X)\). Many proofs become cleaner to write when using the viewpoint of \(\text{mod}-\mathcal A\), while it can be notationally easier to perform computations using twisted complexes. However, if we wish to compare the Fukaya category of a symplectic manifold to some other category (as in the setting of homological mirror symmetry) the choice of triangulated envelope becomes important. In mirror symmetry, we compare the Fukaya category of a symplectic manifold \(X^A\) to the derived category of coherent sheaves on a mirror space \(X^B\). When \(X^B\) is a compact smooth complex variety, this is the same as the category of perfect complexes -- which are precisely the sheaves which can build out line bundles via the operations of taking mapping cones. For this reason, it is more common to see that Fukaya category defined to be the twisted complexes on the geometric Fukaya category in papers related to mirror symmetry.

References

[Sei08]

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