Christopher J. Dean

I am a postdoctoral fellow in the Department of Mathematics and Statistics at Dalhousie University working with Peter Selinger and Julien Ross. My research involves a wide range of areas of mathematics including higher category theory, homotopy type theory, and quantum computing.

Previously, I did my doctorate at the University of Oxford under the supervision of Kobi Kremnitzer and Jamie Vicary.

Selected Papers

arxiv
Computads for weak ω-categories as an inductive type

Christopher J. Dean, Eric Finster, Ioannis Markakis, David Reutter, and Jamie Vicary

2022

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We give a new description of computads for weak globular ω-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads, and allows a new definition of ω-category that avoids the technology of globular operads. Our framework permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with variable-to-variable maps is a presheaf topos, giving a direct description of the index category. We prove that our resulting definition of ω-category agrees with that of Batanin and Leinster and that the induced notion of cofibrant replacement for ω-categories coincides with that of Garner.
@article{Dean-2022-Computads, arxiv = {2208.08719}, author = {Dean, Christopher J. and Finster, Eric and Markakis, Ioannis and Reutter, David and Vicary, Jamie}, doi = {10.48550/ARXIV.2208.08719}, keywords = {Category Theory (math.CT)}, title = {Computads for weak $ω$-categories as an inductive type}, url = {https://arxiv.org/abs/2208.08719}, year = {2022}, }
thesis
Globular Multicategories with Homomorphism Types

Christopher J. Dean

University of Oxford, 2022

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We introduce various notions of globular multicategory with homomorphism types. We develop a higher dimensional modules construction that constructs globular multicategories with strict homomorphism types. We illustrate how this construction is related to iterated enrichment. We show how various collections of “higher category-like” objects give rise to globular multicategories with homomorphism types. We show how these structures suggest a new globular approach to the semantics of (directed) homotopy type theory.
@phdthesis{Dean-2022-Thesis, author = {Dean, Christopher J.}, keywords = {Category Theory (math.CT)}, school = {University of Oxford}, title = {Globular Multicategories with Homomorphism Types}, year = {2022} }