We show that such structures can be systematically produced in any number of dimensions by using the geometry of secondary polytopes, esp. their factorization properties. In particular, in 2 dimensions, we produce, out of a polyhedral “coefficient system”, a dg-category R with a semi-orthogonal decomposition and an L∞-algebra g. We show that g is quasi-isomorphic to the ordered Hochschild complex of R, governing deformations preserving the semi-orthogonal decomposition.
This allows us to give a more precise mathematical formulation of the (conjectural) alternative description of the Fukaya–Seidel category of a Kahler manifold endowed with a holomorphic Morse function.