Research and publications

What is higher category theory?

Category theory might be defined as the mathematical study of relationships. In a category, we have some particular things that we want to study, and we have ways of relating them to each other. One example might be a category which has numbers in it, and the relationships between these numbers are given by whether or not some number is at least as big as another. The things we want to compare are called objects, and the relationships between them go by the name of morphisms. Another example of a category might have objects which are the cities of the world, and a morphism from one city to another is just a route from the beginning city to the ending city that you could drive by car.

In our first example, the relationships were quite basic: the number 7 is larger than the number 3, and there is nothing more to say. In the second example, the relationships were more complicated, as there are many ways to drive between pairs of cities, and there are also many pairs of cities for which you cannot drive between them at all. Additionally, the morphisms in our second example (the routes between cities) themselves have relationships between them of various kinds. One example of such a relationship is that one route between a pair of cities might be longer than another. This introduces the notion of relationships between relationships, and then relationships between relationships between relationships, and so on. It is this concept that gets referred to as the dimension of a category: how many layers of relationship we can examine between a pair of objects.

The study of higher category theory can be divided into two parts. First, we might just be interested in how to organize all these layers of relationships in a coherent and sensible way. Depending on what kinds of relationships you want to study, this can be a very easy or very hard task. Second, you might study the consequences of such a system. Ordinary category theory shows how many different mathematical objects can exhibit similar properties based merely on the kinds of relationships that these objects have with those around them; higher category theory does the same, only now everything gets a little more complicated.

Higher category theory is also intimately connected with homotopy theory and algebraic topology, both as a language and as a complementary theory. Spaces can be seen as special kinds of higher categories called groupoids or n-groupoids, and every category or higher category has a classifying space which encodes its geometric information. Further, there is the growing field of (∞, n)-categories which are part higher category and part homotopy theory.



Current projects

Here are some projects I am currently working on.

  • Higher distributive laws: (With James Cranch) We are trying to give a unified approach to higher dimensional distributive laws with the particular goal to apply this theory to structures appearing in stable homotopy theory.
  • The stable homotopy hypothesis in dimension 2: (With Niles Johnson and Angélica M. Osorno) We study Picard 2-categories which are symmetric monoidal 2-categories in which objects, 1-cells, and 2-cells are all appropriately invertible. The aim of the project is to use these to model stable homotopy 2-types, and moreover to understand topological invariants categorically.
  • Group actions on invertible objects: (With Ed Prior) Invertible objects in a symmetric monoidal category carry a canonical action of the cyclic group of order 2, a kind of universal multiplication by -1. We investigate the same phenomenon in other kinds of monoidal categories such as braided ones.

Contact Details

Dr Nick Gurski
Room J7, Hicks Building
Telephone: 0114 2223838 (internal extension: 23838)

Last updated: 16 October 2015