String structures via higher gauge theory

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The project can be described as `categorified geometry'. Generally, the objects studied in mathematics are sets with extra structure, for example given two elements of a set we might be able to add them and get a new element of the set. However, in a set we only have limited scope to compare elements: we have the notion of equality. The notion of category is a generalization of the notion of a set; we have objects, but we also have morphisms between them with which we can compare different objects. We can picture a set as a collection of dots and we can picture a category as a collection of dots together with arrows connecting different dots. For a category, the analog of the notion of an equation between elements of a set, is the notion of an isomorphism between objects. This is a much richer notion than the notion of equality, two objects can be isomorphic in many different ways. There is a process from which we can condense a category down to a set: we can form the set of isomorphism classes of objects in the category. So if we think of a category as a collection of dots with arrows joining them, then we remove the arrows so that we have just have the collection of dots but moreover we regard two dots as the same if there was an isomorphism connecting them. This process is sometimes called `decategorification'. For example we could take the category of finite sets and functions between them. Two sets are isomorphic if they have exactly the same number of elements. The decategorification of this category is the set of natural numbers. Decategorification is a process which loses information. Categorification is a process which attempts to recover this lost information, as a result this is generally quite difficult. Categorification has been implicit in much progress in modern mathematics. It is particularly important in algebraic topology, where one wants to distinguish between different spaces by assigning `invariants' to a space in the form of algebraic information. If two spaces have different invariants then the spaces must be different. The crudest invariant one could assign to a space is a number: for instance one could count the number of holes in a surface - a donut shape has one hole while a sphere has no holes. This crude invariant was quickly categorified, the number associated to the surface was recognized as the dimension of a certain vector space which could be assigned to the surface. A more sophisticated invariant would be the assignment of a category to the surface. In this project we will study categorified geometry. We will take the usual objects studied in differential geometry at a set theoretic level (for instance bundles and Lie groups) and define and study category theoretic versions of them. In fact, we will not just be interested in the notion of a mere category, but a higher dimensional generalizations of this notion in which we have not just morphisms between objects but 2-morphisms between morphisms, 3-morphisms between 2-morphisms and so on. These structures are playing an ever more prominent role in mathematics. We will lay some foundation stones for a theory of differential geometry in this higher dimensional context. We will use this theory to develop and illuminate some aspects of the notion of `string structure' which is important in homotopy theory and string theory.