An Invitation to Categorification

1. What is Categorification? Categorification is the process of promoting an algebraic object to one with more structure. All of the structure of the original object is retained and can be accessed via a decategorification procedure that forgets this new higherlevel structure. Originally this term indicated the replacement of set-based constructions with categorical notions, where equalities are replaced by explicit isomorphisms. However, over the years, the term categorification has been used in a variety of different contexts, making its precise definition somewhat nebulous. In most cases, one specifies a rigorous “decategorification” procedure that forgets some type of structure. Categorification is then the inverse process of trying to find the more sophisticated structure that would decategorify to some specified object of interest. Within this broad framework are some guiding principles

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