SympSnip: The Category of modules over an \(A

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 0.0.1

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 0.0.2

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 0.0.3

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 0.0.4

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 0.0.5

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 0.0.6

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 0.0.7

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 0.0.8

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 0.0.4) 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 \(A\to B \to \cone(f)\).