A Categorical Perspective on Braid Representations
P. P. Martin, E. C. Rowell, F. Torzewska

TL;DR
This paper explores the categorical structure of braid representations, introduces new notions of isomorphism, and analyzes how different targets and equivalences influence classification, with implications for solutions to the Yang-Baxter equation.
Contribution
It introduces a categorical framework for braid representations, compares notions of isomorphism, and studies subobjects, quotients, and universal properties within this context.
Findings
Category of braid functors is monoidal (Theorem 5.3)
Objects with target Match^N have sub and quotient objects
Every monoidal functor is equivalent to a strict one in a free setting
Abstract
We study categories whose objects are the braid representations, i.e. strict monoidal functors from the braid category to the category of matrices . Braid representations are equivalent to solutions to the (constant) Yang-Baxter equation. A major part of the contribution here is to introduce, compare, and contrast suitable notions of isomorphism of representations. A significant contribution here is an extensive range of key examples and counterexamples. Our approach is mainly motivated by the recent classification of charge conserving Yang-Baxter operators, in which the target is replaced by the subcategory . One objective is to understand from the categorical perspective how the classification was facilitated by this change (with the aim of generalising). Progress is made here by observing that the category of functors…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Geometric and Algebraic Topology
