Relative Monoidal Bondal–Orlov

In this talk I will present a relative monoidal version of the Bondal–Orlov reconstruction theorem. Concretely the goal is to put in contrast this classical result with Balmers reconstruction theorem. Using techniques of higher category theory we construct a sheaf of bifunctors over a smooth projective variety $X$ faithfully flat over a quasicompact quasiseparated scheme $S$. When every fiber over $S$ has an ample canonical bundle we observe that a tensor structure on the derived category of $X$ must under some reasonable conditions be equivalent to the one given by the usual derived tensor product. This is joint work with Artan Sheshmani.