Weakly globular double categories and weak units

TL;DR

This paper establishes an equivalence between weakly globular double categories and fair 2-categories, clarifying their relationship in modeling weak 2-categories with strict composition but weak units.

Contribution

It provides a direct comparison and proves the equivalence of weakly globular double categories and fair 2-categories after localization, enhancing understanding of their structures.

Findings

Weakly globular double categories encode strictly associative but not strictly unital composition.

They are equivalent to fair 2-categories after localization with respect to 2-equivalences.

The comparison offers new insights into weak units in weak 2-category models.

Abstract

Weakly globular double categories are a model of weak $2$-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani $2$-categories. Fair $2$-categories, introduced by J. Kock, model weak $2$-categories with strictly associative compositions and weak unit laws. In this paper we establish a direct comparison between weakly globular double categories and fair $2$-categories and prove they are equivalent after localisation with respect to the $2$-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.