What is category theory?

Category theory can be succinctly defined as the study of things and the relationships between them: we don’t really care so much which things, as long as the relationships follow some basic rules. Examples include numbers and the relationship x is less than or equal to y, points on a map and the relationship you can walk from point A to point B, or sets and the relationships given by functions between those sets. Category theory got its start by providing a necessary language for the development of algebraic topology, but since then has greatly expanded in scope and influence.

What do I actually do?

I started off doing what you might call hardcore 3-dimensional category theory; the 3 here means you study three different layers of relationships instead of just the usual one. As time has gone on, I have gotten more interested in topological, algebraic, and physical applications of this theory. I am generally interested in extra algebraic structure on categorical objects, such as monads, operads, or distinguished dualizable or invertible objects/higher cells. Current research projects include

• studying how Picard 2-categories encode the algebraic information in stable homotopy 2-types, with a particular emphasis on translating topological invariants into categorical structure;
• an exploration of the interaction between classical operads and group actions, with an application to invertible objects in different kinds of monoidal categories; and
• an analogue for quasicategories of Beck’s theorem on distributive laws, using the homotopy coherent monads of Riehl-Verity.

I’d like to get more into TQFTs and higher categories of modules.

Publications

You can find a full list of my publications here.

And what do my students do?

My past PhD students, along with thesis titles, are:

• Thomas Athorne, Coalgebraic cell complexes
• Alexander Corner, Day convolution for monoidal bicategories
• Edward Prior, Action operads and the free G-monoidal category on n invertible objects.

Some other possible projects for future students include:

• prove coherence for lax homomorphisms between tricategories or monoidal 2-categories of some type,
• investigate Leinster-style operads using profunctors instead of spans,
• investigate the 2-categorical aspects of objects like spherical or modular categories, and
• prove the Homotopy Hypothesis for iterated, weakly enriched categories.