Papers
2022
- arxivComputads for weak ω-categories as an inductive type2022
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}, } - thesisGlobular Multicategories with Homomorphism TypesChristopher J. DeanUniversity of Oxford, 2022
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} }