## 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.