Categorifications from planar diagrammatics

TL;DR

This paper discusses diagrammatic methods for categorification, focusing on functors, natural transformations, and biadjointness, and reviews graphical descriptions of Soergel bimodules and quantum groups, based on Takagi lectures.

Contribution

It provides a comprehensive overview of diagrammatic categorification techniques and their applications to quantum groups and Soergel bimodules.

Findings

Diagrammatic presentation of functors and transformations

Graphical description of Soergel bimodules

Categorification of quantum groups

Abstract

A diagrammatic presentation of functors and natural transformations and the virtues of biadjointness are discussed. We then review a graphical description of the category of Soergel bimodules and a diagrammatic categorification of positive halves of quantum groups. These notes are a write-up of Takagi lectures given by the author in Hokkaido University in June 2009.