Perverse sheaves on Riemann surfaces as Milnor sheaves

TL;DR

This paper offers a new categorical framework to describe perverse sheaves on stratified Riemann surfaces using representations of the paracyclic category, providing an intrinsic, t-structure-free perspective.

Contribution

It introduces the paracyclic category as a model for perverse sheaves, bridging exit and entrance path categories on stratified surfaces.

Findings

Provides an intrinsic definition of perverse sheaves

Models hybrid exit-entrance behavior with the paracyclic category

Enables description without derived categories or t-structures

Abstract

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 category without reference to derived categories and t-structures.