Doubly weak double categories

TL;DR

This paper introduces a new definition of double categories where composition is weak in both directions, using structures called double computads and characterizations via implicit double categories, extending existing concepts with a coherence-based approach.

Contribution

It defines doubly weak double categories through double computads and implicit structures, providing a new framework for weak double categorical compositions.

Findings

Characterization of doubly weak double categories using implicit double categories.

Extension of existing double bicategories with a 'tidiness' condition.

A coherence-based approach to weak double categorical structures.

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.