External tensor product of categories of perverse sheaves

Notes on Beilinson's "How to glue perverse sheaves"

The titular, foundational work of Beilinson not only gives a technique for gluing perverse sheaves but also implicitly contains constructions of the nearby and vanishing cycles functors of perverse sheaves. These constructions are completely elementary and show that these functors preserve perversity and respect Verdier duality on perverse sheaves. The work also defines a new, "maximal extension" functor, which is left mysterious aside from its role in the gluing theorem. In these notes, we present the complete details of all of these constructions and theorems.

Explicit models for perverse sheaves

We consider categories of generalized perverse sheaves, with relaxed constructibility conditions, by means of the process of gluing $t$-structures and we exhibit explicit abelian categories defined in terms of standard sheaves categories which are equivalent to the former ones. In particular, we are able to realize perverse sheaves categories as non full abelian subcategories of the usual bounded complexes of sheaves categories. Our methods use induction on perversities. In this paper, we restrict ourselves to the two-strata case, but our results extend to the general case.

Perverse sheaves on Riemann surfaces as Milnor sheaves

Constructible sheaves of abelian groups on a stratified space can be equivalently described in terms of representations of the exit-path category. In this work, we provide a similar presentation of the abelian category of perverse sheaves on a stratified surface in terms of representations of the so-called paracyclic category of the surface. The category models a hybrid exit-entrance behaviour with respect to chosen sectors of direction, placing it "in between" exit and entrance path categories. In particular, this perspective yields an intrinsic definition of perverse sheaves as an abelian ca…

PROBs and perverse sheaves I. Symmetric products

Algebraic structures involving both multiplications and comultiplications (such as, e.g., bialgebras or Hopf algebras) can be encoded using PROPs (categories with PROducts and Permutations) of Adams and MacLane. To encode such structures on objects of a braided monoidal category, we need PROBs (braided analogs of PROPs). Colored PROBs correspond to multi-sorted structures. In particular, we have a colored PROB B governing non-negatively graded bialgebras in braided categories. As a category, B splits into blocks B_n according to the grading. We relate B_n with the category P_n of perverse shea…

截断因子化 perverse 套 sheaves 在 Sym(C) 上

Truncated factorized perverse sheaves on Sym(C)

本文研究了Sym(C)上截断因子化 perverse sheaves 的分类与等价性，定义了不同d值的范畴并证明了其与V的等价关系，以及相关诱导 functor 的性质。

关于反常层等变导出范畴

On the Equivariant Derived Category of Perverse Sheaves

本文将 Beilinson 实现形式体系推广到伪函子与伪自然框架，证明代数群作用下反常层导出范畴与构造导出范畴等价的等变版本，并验证其他非 D-模结论的等变类比。