Fibred categorical theory of obstruction and classification of morphisms

在导出范畴中关于映射的变形理论及其码域固定提升

Deformation theory for a morphism in the derived category with fixed lift of the codomain

研究在导出范畴中具有固定码域提升的复形映射的变形-阻挠计算，证明半正交分解在光滑正则方案族中唯一变形。

Descent Data and Absolute Kan Extensions

The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms create all Kan extensions that are preserved by certain morphisms. As a consequence, in the case $\mathfrak{A} = \mathsf{Cat} $, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, (naturally isomorphic to) a functor that forgets descent data. In particular, within the cla…

组合性的阻碍

Obstructions to Compositionality

本研究针对组合性概念在应用范畴论各领域含义不一的问题，引入第零、第一同伦偏序集作为范畴不变量，通过分类（余）松函子成为强函子的阻碍来定性描述组合性失效，并结合图论与量子理论的例子说明其利弊。