Doubly weak double categories
Category Theory
2026-05-25 v2
Abstract
We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes -- equipped with all possible composition operations, coherently. We also characterize them using "implicit" double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a "tidiness" condition to the double bicategories of Verity, or to the cubical bicategories of Garner.
Cite
@article{arxiv.2506.23651,
title = {Doubly weak double categories},
author = {Aaron David Fairbanks and Michael Shulman},
journal= {arXiv preprint arXiv:2506.23651},
year = {2026}
}