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 hyperkähler geometry are sketched.

为研究 Lefschetz 纤维化的辛几何和 Tyurin 退化的代数几何的各种特征，引入了分类形式主义。这种方法是由同调镜像对称、导出的非对易几何和 Fukaya 范畴理论与反常 Schober 中的系数所决定的。主要技术成果包括（i）相对卡拉比-丘结构的概念与球函子概念的某种细化的比较，（ii）构造卡拉比-丘结构的局部到全局粘合原理，以及(iii) 在膜的某些派生模空间上构建移位辛结构和拉格朗日结构。草图勾勒出衍生的 hyperkähler 几何理论的潜在应用。