Categorification has been a central topic in the past two decades in several areas of mathematics including representation theory and low dimensional topology. In broad terms, categorification introduces richer structure by replacing algebraic structures (groups, rings, etc) by categories, in a way that lifts the important properties of the original structure.
In this talk I will explain the general idea of categorification and illustrate it with an example from low-dimensional topology: Khovanov's link homology. We will discuss advantages and applications of categorification in different settings, and open problems in the area.
This is an overview talk and appropriate for a general mathematics audience.