Derived Braids of Decreasing Products and their Categories

TL;DR

This paper investigates derived braids from decreasing products in braided monoidal categories, establishing their equivalence and generalizing the Yang-Baxter equation, with implications for categorical structure classification.

Contribution

It proves that decreasing products formed from even-strand braids have equivalent derived braids, extending understanding of braid-based categorical coherence.

Findings

Derived braids from decreasing products are equivalent.

Equality of derived braids generalizes the Yang-Baxter equation.

Provides a foundation for classifying categorical structures using braids.

Abstract

Derived braids have been used to classify categorical structures based on the braid underlying a braided monoidal category V. With four-strand braids underlying the composition morphisms of tensor products of categories enriched over V, equality of derived braids has been seen to correspond with the results of Joyal and Street on coherence for braided monoidal categories, namely that diagrams commute when the braids underlying the legs of the diagram are equal. Equality of derived braids can then be seen as a generalization of the Yang-Baxter equation that appears in the work of Joyal and Street. The main result is a proof that decreasing products, which are braids formed from component braids with even numbers of strands, have equivalent derived braids. Plans for future work include interpreting what categorical structures correspond to decreasing products.