Weak vertical composition

TL;DR

This paper introduces a new class of semi-strict tricategories with weak vertical composition, demonstrating they can generate all braided monoidal categories through a specific construction.

Contribution

It constructs semi-strict tricategories with weak vertical composition as categories enriched in bicategories, linking them to all braided monoidal categories.

Findings

Doubly-degenerate semi-strict tricategories are braided monoidal categories.

Any braided monoidal category can be realized from a vertically weak semi-strict tricategory.

The construction provides a new framework for understanding braided monoidal categories.

Abstract

We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category $B$ there is a doubly-degenerate ``vertically weak'' semi-strict tricategory whose associated braided monoidal category is braided monoidal equivalent to $B$.