Brandon Shapiro | Research

My research uses combinatorial and categorical techniques to study a wide range of objects in topology, geometry, algebra, and the sciences in a visual way that supports generalizations and connections between different subjects.

I am interested in many different kinds of higher categorical structures, how they relate to each other, and constructions like enrichment that many of them support. 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.

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 a wide range of objects with two different ways of mapping a smaller object to a bigger one.

Double categories together with 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, and double categories of decorated/structured cospans model scientific phenomena in a way that captures compositional structure.

Hyperbolic and Euclidean tilings of the plane provide beautiful examples of spaces built up out of cells of fixed shapes, similar to the decompositions of spaces common in geometric and algebraic topology. The categorical models of combinatorial spaces used in algebraic topology can also model tilings, but doing so requires choices of directions and labels that form a rich theory accessible to undergraduate researchers.

Papers

• A combinatorial construction of homology via ACGW categories. Joint with Maru Sarazola and Inna Zakharevich. [arXiv]


• 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]


• A Gillet-Waldhausen Theorem for chain complexes of sets. Joint with Maru Sarazola. [arXiv]


• Enrichment of Algebraic Higher Categories. [arXiv]


• Familial Monads as Higher Category Theories. [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]


• Video Research Statement. eCHT Employment Network, October 2022. [Video]
• 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

Projects I worked on as an undergraduate.

• 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]