A fundamental problem of the embedded Fukaya category is that it lacks enough geometric objects. Morally, Fukaya category is a construction that inputs the symplectic geometry of Lagrangian branes, and outputs the representation theory of an $A_{\infty}$-category. Now the general feature of $A_{\infty}$-module categories is that one can take cones and idempotent summands, two properties which are useful for classifying such categories, and desirable for mirror symmetry. The problem is that cones and idempotent summands are not obviously represented by embedded Lagrangian objects under the Yoneda embedding. The common solution is to sideline this issue by the formal algebraic construction of twisted complexes and idempotent completions. This is not quite adequate for the Thomas-Yau conjecture. However, we will discuss how the introduction of immersed Lagrangian objects geometrizes the twisted complexes (cf. section 6.2). The geometric meaning of idempotents is an open problem.

The formal algebraic constructions are well explained in [9, section 3] and [73, section 4], to which we refer the reader for more details. Given objects $L_{1},\ldots L_{N}$ of the Fukaya category $\mathcal{A}$, a twisted complex $(L,b_{L})$ consists of

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  The formal shifted direct sum $L=\oplus_{1}^{N}L_{i}[k_{i}]$ with $k_{i}\in\mathbb{Z}$ formally keeping track of degrees (the geometric meaning of the shift $[1]$ is to add a constant $\pi$ to the Lagrangian phase, which reverses the orientation of the Lagrangian, with a corresponding twist to the spin structure),

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  and a strictly triangular differential $b_{L}\in\End(L)$, i.e. a collection of maps $b_{ij}\in\Hom^{k_{j}-k_{i}+1}(L_{i},L_{j})$ for $i>j$,⁶⁴⁶⁴ 64 In most symplectic references such as [9] the morphisms $b_{ij}$ go in the opposite direction $i<j$. This just amounts to reversing the ordering of $L_{1},\ldots,L_{N}$. We find our reversed convention a little more convenient for the Harder-Narasimhan decomposition.

satisfying the equation

$$\sum_{k\geq 1}m_{k}(b_{L},\ldots,b_{L})=0,\quad\emph{i.e.}$$

$$\sum_{k\geq 1}\sum_{i=i_{0}>\ldots>i_{k}=j}m_{k}(b_{i_{k-1}i_{k}},\ldots,b_{i_{0}i_{1}})=0.$$

Notice the strict triangularity implies the sum is finite. One can define morphisms between these twisted complexes, and assign $A_{\infty}$-structures to make twisted complexes into an $A_{\infty}$-category $Tw\mathcal{A}$, into which $\mathcal{A}$ naturally embeds fully faithfully. Using the $A_{\infty}$-structure, it makes sense to talk about closed morphisms and cohomologies, similar to the construction of Floer cohomology.

Given twisted complexes $A=(L,b_{L}),B=(L^{\prime},b_{L^{\prime}})\in Tw\mathcal{A}$, and a closed morphism $f\in\Hom^{0}((L,b_{L}),(L^{\prime},b_{L^{\prime}}))$, the abstract mapping cone of $f$ is the twisted complex

$$\text{Cone}(f)=\left(L[1]\oplus L^{\prime},\left(\begin{matrix}b_{L}&0\\ f&b_{L^{\prime}}\end{matrix}\right)\right).$$

Generally, a mapping cone of $f$ is an object of $Tw\mathcal{A}$ quasi-isomorphic to $\text{Cone}(f)$. This gives rise to a distinguished triangle $A\to B\to\text{Cone}(f)\xrightarrow{[1]}A$. This illustrates the advantage of introducing twisted complexes: $Tw\mathcal{A}$ is a triangulated category.

The cohomological category of $Tw\mathcal{A}$ is commonly denoted $D^{b}Fuk(X)$. This has the same objects as $Tw\mathcal{A}$, but the Floer cochain spaces are replaced by their $H^{0}$, namely we remember the Floer cohomology.

Under the Yoneda embedding, $Tw\mathcal{A}$ embedds into its module category. The idempotent closure $Tw^{\pi}\mathcal{A}$ is obtained by formally adding the direct summands of the Yoneda image of twisted complexes in $Tw\mathcal{A}$. The cohomological category of $Tw^{\pi}\mathcal{A}$ is commonly denoted $D^{\pi}Fuk(X)$. In the variant setting of compact $X$, it is usually $D^{\pi}Fuk(X)$ instead of $D^{b}Fuk(X)$ that shows up in mirror symmetry, since the derived category of coherent sheaves is automatically idempotent closed.