  A categorical formalism is introduced for studying various features of the
symplectic geometry of Lefschetz fibrations and the algebraic geometry of
Tyurin degenerations. This approach is informed by homological mirror symmetry,
derived noncommutative geometry, and the theory of Fukaya categories with
coefficients in a perverse Schober. The main technical results include (i) a
comparison between the notion of relative Calabi-Yau structures and a certain
refinement of the notion of a spherical functor, (ii) a local-to-global gluing
principle for constructing Calabi-Yau structures, and (iii) the construction of
shifted symplectic structures and Lagrangian structures on certain derived
moduli spaces of branes. Potential applications to a theory of derived
hyperk\"ahler geometry are sketched.
