$A_\infty$-schemes and their derived categories

We introduce the notion of $A_\infty$-schemes and define their perfect derived categories, which provide $A_\infty$-enhancements of the perfect derived categories of the underlying schemes. This notion enables us to define a natural $A_\infty$-functor from an $A_\infty$-category whose homotopy category admits a symmetric monoidal structure—such as the Fukaya category of an SYZ fibration—to the perfect derived category of its mirror $A_\infty$-scheme. We further establish a necessary and sufficient condition for this $A_\infty$-functor to be a quasi-equivalence.