Higher representation theory

在数学中，为了理解一个给定的事物，我们会给它添加额外的结构，这是一种常见的现象--表面上看起来让它变得更复杂，但实际上却更容易理解它。一种语言中所有可能的单词的想法似乎是压倒性的，但是将它们按词典顺序排列并将它们写在字典中，我们有一种简单的方法来获得关于任何给定单词的信息，尽管通过强加额外的结构（知道一个单词何时出现在另一个单词之前）似乎使事情变得复杂。用数学术语来说，我们把单词变成了一个类别，类别通常是通过它们的表征来研究的，这意味着对更容易接近的事物的操作，就像用某种语言写的书一样：不是所有的语言都能在给定的书中看到，但是我们读的书越多，我们对语言的了解就越好。对于某些称为代数的重要范畴，有一个非常发达的表示理论。近年来，人们发现，为了研究关于范畴的问题，增加更多的结构以获得2-范畴，例如“所有范畴的范畴”，往往是有用的。对这类2-范畴表示的例子的研究，在纯数学中的一些长期问题上取得了令人兴奋的突破。我们的项目是将代数表示理论扩展到具有与代数类似性质的2-范畴表示，目标是为已经观察到的例子提供一个结构化的框架，从而促进未来使用这些成功的概念。

It is a common occurrence in mathematics that in order to understand a given thing, we add extra structure to it - seemingly making it more complicated, but in fact making it easier to obtain insight on it. An analogy would be ordering things in the real world. The thought of all possible words in a language seems quite overwhelming, but ordering them lexicographically and writing them in a dictionary, we have an easy way to obtain information about any given one, despite seemingly having complicated things by imposing extra structure (knowing when a word comes before another). In mathematical terms, we have turned words into a category.Categories are commonly studied through their representations, meaning actions on things that are more easily accessible, not unlike books written in a certain language: Not all of the language will be visible in a given book, but the more books we read, the better we know the language. For certain important categories called algebras, there is a very well-developed theory of such representations.In recent years, it has been found that in order to study questions about categories, it is often useful to add even more structure to obtain 2-categories, e.g. the "category of all categories". The study of examples of representations of such 2-categories has led to some exciting breakthroughs on long-standing problems in pure mathematics. Our project is to extend the theory of representations of algebras to representations of 2-categories with analogous properties to those of algebras, with the goal of providing a structured framework for the examples that have been observed, hence facilitating future use of these successful concepts.