English

All Concepts are $\mathbb{C}\mathbf{at}^\#$

Category Theory 2024-05-27 v4

Abstract

We show that the double category Cat#\mathbb{C}\mathbf{at}^\# of comonoids in the category of polynomial functors (previously shown by Ahman-Uustalu and Garner to be equivalent to the double category of categories, cofunctors, and prafunctors) contains several formal settings for basic category theory and has subcategories equivalent to both the double category Org\mathbb{O}\mathbf{rg} of dynamic rewiring systems and the double category PolyE\mathbb{P}\mathbf{oly}_{\mathcal{E}} of generalized polynomials in a finite limit category E\mathcal{E}. Also serving as a natural setting for categorical database theory and generalized higher category theory, Cat#\mathbb{C}\mathbf{at}^\# at once hosts models of a wide range of concepts from the theory and applications of polynomial functors and category theory.

Keywords

Cite

@article{arxiv.2305.02571,
  title  = {All Concepts are $\mathbb{C}\mathbf{at}^\#$},
  author = {Owen Lynch and Brandon T. Shapiro and David I. Spivak},
  journal= {arXiv preprint arXiv:2305.02571},
  year   = {2024}
}

Comments

31 pages. Proofs in appendix moved into the main body, and chapter on nerves of higher categories removed to a separate submission at arXiv:2405.13157

R2 v1 2026-06-28T10:25:17.867Z