The most traditional approach to solving mathematical problems is: start with a difficult problem and simplify it until it becomes easy enough to be solved. However, developing mathematical theories quite often goes in the opposite direction: starting with an “easy” theory one tries to “generalize” it to something more complicated which could hopefully describe a much bigger class of phenomena.

An example of the latter is the development of what is now known as Khovanov homology. The Jones polynomial is a very elementary classical combinatorial invariant appearing in the low dimensional topology. However, as most of known topological invariants, it is not an absolute one (it does not distinguish all knots). Some twelve years ago Mikhail Khovanov has developed a very advanced “refinement” of Jones polynomial which seriously increased the level of theoretical sophistication necessary to be able to define and work with it. Instead of elementary combinatorics and basic algebra, Khovanov’s definition was based on category theory and homological algebra and very soon led to the study of higher categorical structures. This “categorification” of the Jones polynomial created a new direction in topology and attracted a lot of attention from some other parts of mathematics, notably algebra and category theory. Within a few years categorification became an intensively studied subject in several mathematical areas. It completely changed the viewpoint on many long standing problems and led to several spectacular results and applications.