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arXiv · 2506.23651

Doubly weak double categories

Aaron David Fairbanks·Michael Shulman
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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.

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Aaron David Fairbanks, Michael Shulman. 2025-06-30. Doubly weak double categories. https://doi.org/10.1007/s10485-026-09863-1

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