2-representation theory and categorification

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While in mathematics it is very often helpful to break complicated problems down into less complicated ones by simplifying and forgetting data, the converse has often proved useful. The term categorification refers to the process of finding more complicated structures which, upon forgetting some information, reproduce the original problem that one wants to study. The more complicated structures often allow us to deduce useful information that was previously inaccessible. For example, an integer number solving a certain equation might a priori be anything, but if we then discover that this certain number, in fact, describes the number of elements in a set (e.g. cats in a household), it cannot be negative.Categorification in representation theory is usually formulated in terms of certain structures called 2-categories encoding generalised symmetries of other categories, or in other words, 2-representations of 2-categories. To develop the theory of 2-representations of 2-categories (with certain nice properties abstracted from interesting examples) and to apply them to some of the original problems that inspired their definition is the aim of this proposal.