An alternative description of braided monoidal categories

TL;DR

This paper introduces an alternative framework for braided monoidal categories using a single constraint, the b-structure, simplifying the coherence conditions and extending to higher dimensions, with applications to key equations in mathematical physics.

Contribution

It presents a new presentation of braided monoidal categories via the b-structure, unifies coherence conditions, and extends the concept to bicategories, linking to important equations like Yang-Baxter.

Findings

b-structures simplify coherence conditions

b-structures relate to Yang-Baxter and tetrahedron equations

b-cohomology parallels abelian group cohomology

Abstract

We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to be equivalent to the usual associativity, unit and braiding axioms. We also discuss the next dimensional version, that is, b-structures on bicategories. As an application, we show how special b-categories result in the Yang-Baxter equation, and how special b-bicategories produce Zamolodchikov's tetrahedron equation. Finally, we define a cohomology theory (the b-cohomology) which plays a role analogous to the one abelian group cohomology has for braided monoidal categories.