I am interested in many different kinds of higher categorical structures and how they relate to each other, and use the language of polynomial functors to describe common constructions across the various flavors of higher categories. When I say "higher categories" I mean any structure involving collections of cells in various shapes and algebraic operations for composing certain arrangements of those cells into a single cell. For instance, n-categories are algebraic structures on n-globular sets, and n-tuple categories are algebraic structures on diagrams of cubical cells which keep track of the different directions. So far, I have generalized enrichment of categories to any such higher category.
Algebraic K-theory describes how various types of objects are built up out of smaller pieces, where "smaller" is tracked by certain morphisms in a category. My work builds on an axiomatic framework for K-theory which uses double categories to model objects with two different ways of mapping a smaller object to a bigger one. This approach generalizes classical theorems of K-theory to non-abelian settings, such as varieties and finite sets, where chain complex models of K-theory have yet to be fully explored.
Polynomial endofunctors on the category of sets provide a rich language for describing phenomena in both category theory and real-world applications. Polynomial coalgebras model "open" dynamical systems, where the state of the system updates based on external input. Enrichment in a double category of polynomial coalgebras can be used to describe nested hierarchies of interacting dynamical systems, including examples from machine learning, strategic game theory, and prediction markets. Polynomial functors on the category of categories have even further potential to elegantly model structures in topology, continuous dynamics, and the theory of double categories.
Papers
- All Concepts are Cat#. Joint with Owen Lynch and David Spivak. [arXiv]
- A Polynomial Construction of Nerves for Higher Categories. Joint with David Spivak. [arXiv]
- Structures in Categories of Polynomials. Joint with David Spivak. [arXiv]
- A Compositional Account of Motifs, Mechanisms, and Dynamics in Biochemical Regulatory Networks. Joint with Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo Ocal, and Evan Patterson. Compositionality 6, 2, 2024. [arXiv]
- Duoidal Structures for Compositional Dependence. Joint with David Spivak. [arXiv]
- Dynamic Operads, Dynamic Categories: From Deep Learning to Prediction Markets. Joint with David Spivak. Electronic Proceedings in Theoretical Computer Science, 2022 (Applied Category Theory conference). [arXiv]
- Enrichment of Algebraic Higher Categories. [arXiv]
- Familial Monads as Higher Category Theories. [arXiv]
- A Gillet-Waldhausen Theorem for chain complexes of sets. Joint with Maru Sarazola. [arXiv]
- Partial evaluations and the compositional structure of the bar construction. Joint with Carmen Constantin, Tobias Fritz, and Paolo Perrone. Theory and Applications of Categories, Vol. 39, No. 11, 322-364, 2023. [arXiv]
- Weak cartesian properties of simplicial sets. Joint with Carmen Constantin, Tobias Fritz, and Paolo Perrone. Journal of Homotopy and Related Structures, 18, 477-520, 2023. [arXiv]
Thesis
- Shape Independent Category Theory. [PDF]
Blog Posts
- A Dynamic Monoidal Category for Strategic Games. [Post]
Talks
- Higher Category Theory in Cat#. Topos Institute Colloquium, November 2023. [Animated Slides] [Less Animated Slides]
- Finite Posets as Algebraic Expressions in Duoidal Categories. Category Theory Octoberfest, October 2023. [Animated Slides] [Less Animated Slides]
- Dynamic PROPs for Networked Learners. CALCO, Special Session on Category Theory and Machine Learning, June 2023. [Animated Slides] [Less Animated Slides]
- Polynomial Functors for Categorical Open Dynamics. Joint Math Meetings, Special Session on Applied Category Theory, January 2023. [Animated Slides] [Less Animated Slides]
- Double Presheaf Categories via Polynomial Functors. Applied Category Theory 2022, November 2022. [Notes]
- Dynamic Operads for Evolving Organizations. Applied Category Theory Conference, July 2022. [Animated Slides] [Less Animated Slides]
- Familial Monads for Higher and Lower Category Theory. Workshop on Polynomial Functors, March 2022. [Animated Slides] [Less Animated Slides]
- Compositional Structure of Partial Evaluations. Categories and Companions Symposium, June 2021. [Animated Slides] [Less Animated Slides]
- Compositional Structure of Partial Evaluations. MIT Category Seminar, September 2020. [Animated Slides] [Less Animated Slides]
- Cubical omega-Categories and Cubical Theta. Cubical Sets Seminar, MSRI, May 2020. [Notes]
- Constructing Cubes from Semicubes. Cubical Sets Seminar, MSRI, May 2020. [Notes]
- Shape Independent Category Theory and Enrichment. Category Theory OctoberFest, Johns Hopkins University, October 2019. [Animated Slides][Less Animated Slides]
Undergraduate Research
- Densities of Hyperbolic Cusp Invariants. Joint with Colin Adams, Rose Kaplan-Kelly, Michael Moore, Shruthi Sridhar, and Joshua Wakefield. Proceedings of the American Mathematical Society. Volume 146, Number 9, September 2018, Pages 4073–4089. [PDF] [arXiv]
- specgen: A Tool for Modeling Statecharts in CSP. Joint with Chris Casinghino. Nasa Formal Methods 282, 2017.
- Nonstandard Neutrino Interactions In Supernovae. Joint with James Kneller, Gail McLaughlin, Charles Stapleford, and Daavid Väänänen Physical Review D 94, 093007, 2016. [PDF] [arXiv]