Peng Zhou

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====== 2025-09-05 ======

how to describe perverse schober on $(D, 0)$? it is a machine that, input a disk with stop / singular Lagrangian skeleton / holomorphic function on an open subset, and output a category; input a morphism of object, output a functor; finally, input a 2-morphisms between 1-morphism, and output a natural transformation. 

but the problem is, what is a 2-mor in the category of disk with stops? I can view disk with stops as Lagrangian in $T^*X$, or constructible sheaves in X. But these are 1-categories, hom between these objects are just set, at best chain complexes. 

How can we move these sectors? closed embedding, open restriction, and then non-char deformation. cobordism as higher morphism doesn't give me much at all.

How to use SOD? Does SOD give me SES of id -> UV -> T? Maybe we just say, 2-perv is data plus condition. this is like saying, endomorphism algebra has some generators and relations, that is to say, endomorphism algebra is the quotient of some free algebra by some 2-sided ideal. So to define action of the quotient algebra on a vector space, we just need to ask the free algebra act, and such that the relation is satisfied. Instead of saying some linear combination of the free algebra is zero, we say some chain complex of object is acyclic. The question is, how do you build that chain complex.