Perverse Schobers

TL;DR

This paper proposes a categorical generalization of perverse sheaves called perverse Schobers, replacing vector spaces with triangulated categories, and explores their basic examples and properties.

Contribution

It introduces the concept of perverse Schobers as a new categorical framework generalizing perverse sheaves, with initial examples and definitions.

Findings

Perverse Schobers are modeled by spherical functors in simple cases.

The paper provides initial examples and a conceptual framework for these objects.

It suggests a new direction for categorifying perverse sheaves.

Abstract

We suggest a possibility for a categorical generalization of the concept of a perverse sheaf, in which vector spaces are replaced by triangulated categories. We call such hypothetical objects perverse Schobers and consider several examples, giving a natural but ad hoc definition in each case. In the simplest case (perverse sheaves on a disk with one possible singular point) we propose, as a categorical analog, the data of a spherical functor.