Doubly weak double categories

Abstract: Double categories are typically weak in one direction: the horizontal (or "loose") 1-cells compose weakly associatively as in a bicategory (a.k.a. weak 2-category), whereas the vertical (or "tight") 1-cells compose strictly associatively as in a strict 2-category. There have been various attempts to define double categories with weakly associative composition in both directions. This is desirable in order to capture a broader range of examples, as well as to obtain a more symmetric theory. What is puzzling is that trying to straightforwardly generalize the usual definition leads into deadlock around coherence conditions.

We introduce a surprisingly easy definition that involves no coherence conditions. It arises as the double-categorical analogue of a surprisingly easy definition of bicategory that involves no coherence conditions. We then connect this definition to other definitions that are more analogous to the usual definition of bicategory. In particular we show our doubly weak double categories are equivalent to Verity's double bicategories satisfying a mild condition that already appears in the literature, and they are equivalent to Garner's cubical bicategories satisfying a similar condition. Joint work with Michael Shulman.

category theory

Audience: researchers in the topic

Third Virtual Workshop on Double Categories