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Categoriļ¬cations

The word “categorification” gained popularity in the 21st century. It was introduced by Crane [ Cra95 CF94 ] in the mid 90s. The popularity seems to be partly due to Khovanov’s discovery of a categorification of the Jones polynomial .

However, analogous ideas, without using the word categorification, already appeared long ago. For example, the first of three memos in Propp’s [ Pro ] begins with the following sentence:

A combinatorialist’s fundamental model of a non-negative integer n is a set of n points.

In combinatorics , it is important to find a bijective proof for an equality of values of two functions. For example, the binomial theorem can be proved by counting the number of subsets of a given finite set in two different ways. Inequalities can be proved by injections or surjections. Zeilberger [ Zei ] calls such proofs as combinatorializations. These are typical examples of categorifications.

I recommend the article [ BD98 ] by Baez and Dolan as a general introduction to categorifications. See also [ CY98 ] , [ Ion ] , and [ BD01 ] .

References