In short, my work focuses on combinatorial models of higher categories. Higher categories are a generalisation of spaces, adding a notion of “direction” to the latter: While in a space any path can be travelled along in two directions, in higher categories this need not be the case. This is relevant for the description of irreversible processes. In my PhD thesis I build a fully algebraic and computer implementable model of higher categories by observing that higher categories can be locally described by certain stratified manifolds. Higher categories can also be a foundation for all of mathematics. In future research I would like to follow this line of thought further, ultimately arriving at a foundational framework (or “type theory”) for working with higher categories. I hope that this has interesting applications to Algebraic Topology, and provides a “more computational” alternative to homotopy type theory (a recent and successful type theory which has connections to undirected spaces).
- Associative n-categories (Submitted: Dec 2018. Final: May 2019. .pdf)
- Abstract: My thesis defines topological n-cubes, a generalisation of string and manifold diagrams, as a geometric model for pasting diagrams in higher categories. It then analyses the combinatorial structure of topological n-cubes, leading to a combinatorial model called singular n-cubes. Based on singular n-cubes I give new definitions of several notions in higher category theory (including presentations of weak/semi-strict higher categories, coherent invertibility, presentations of semi-strict higher groupoids, semi-strict “associative” higher categories).
- Some remarks: The first (roughly) 100 pages are a concise summary of all ideas, and should be an easy and self-contained read for most readers familier with basic ordinary category theory. The remaining 400 pages are mostly rather detailed and concrete computations, which explore the combinatorial properties of the theory.
Unpublished and expository writing
- A datastructure for higher sesqui-categories (Sep 2016). Draft (.pdf)
- The precursor combinatorics to my thesis. They only model higher “sesqui-categories” and not full higher categories.
- Basic concepts in homotopy theory (Oct 2015). Expository notes (.pdf)
- Expository notes, attempting to give a more modern treatment of basic Algebraic Topology, and then discuss its abstraction in model categories and quasicategories
- Enriched Category Theory (May 2014). Part III essay (.pdf)
- Very basic enriched category theory, essentially adding details and computations to Kelly’s book.
- On how to continue the sequence (categories, functors, natural transformations, modifications …) (Mar 2019). K-theory seminar, CUNY, New York
- Higher categories from higher-dimensional manifolds (Feb 2019). Category theory seminar, Johns Hopkins Univesity, Baltimore
- Manifolds and higher categories (Nov 2018). McGill Category Theory and Logic Seminar, Montreal
- Combinatorial Cobordism (Oct 2018). Category Theory Octoberfest 2018, New York
- Associative n-categories (Sep 2018). New York Category Theory Seminar, CUNY, New York
- Higher-dimensional Programming (May 2018). Applied Category Theory Seminar, MIT, Cambridge MA
- Associative n-categories (Apr 2018). 103rd Peripatetic Seminar on Sheaves and Logic, Brno
- TQFTs and string diagrams (Oct 2017). Junior Algebra and Representation Theory Seminar, Oxford
- University of Oxford, PDRA in Mathematics (2019-2021)
- University of Oxford, PhD in Mathematics and Computer Science (2014-2019)
- University of Cambridge, Part III in Mathematics (2013-2014)
- ETH Zurich, BSc in Physics (2010-2013)