2-representation theory and categorification

虽然在数学中，通过简化和忘记数据将复杂的问题分解为不太复杂的问题通常是有帮助的，但匡威也经常被证明是有用的。分类一词是指寻找更复杂结构的过程，在忘记一些信息后，这些结构会重现人们想要研究的原始问题。更复杂的结构往往使我们能够推断出以前无法获得的有用信息。例如，一个整数解一个特定的方程可能是先验的，但如果我们发现这个特定的数，实际上，描述了一个集合中元素的数量表示论中的范畴化通常用某些称为2-范畴的结构来表述，这些结构编码其他范畴的广义对称性，或者换句话说，2-表示2-类别。发展2-范畴的2-表示理论（从有趣的例子中抽象出某些很好的性质），并将它们应用于激发其定义的一些原始问题，这是本提案的目的。

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.