A Categorical Perspective on Braid Representations

Abstract

We study categories whose objects are the braid representations, i.e. strict monoidal functors $F\colon B\rightarrow Mat$ from the braid category $B$ to the category of matrices $Mat$. 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 $Mat$ is replaced by the subcategory $Match^N$. 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 $MonFun(B,Mat)$ is itself a monoidal category (Theorem 5.3). In addition, we introduce the notion of sub and quotient objects, proving that an object that is both sub and quotient corresponds to an endomorphism in $MonFun(B,Mat)$ (Theorem 5.18). We also observe that objects with target $Match^N$ always have sub and quotient objects. This monoidal category leads us to consider monoidal subcategories whose objects share a given property, giving rise to a new way to see how group-type and involutive solutions, for example, fit into our framework. Another objective is to understand how universal such a restricted target is. Here we give various properties exposing the implications of different choices of equivalence and describe some relationships among them (Theorem 7.18, Theorem 7.9, Conj. 7.19, Conj. 7.22). A key result of this paper is Th.2.9, which shows that, in a `sufficiently free' setting including our case, every monoidal functor is equivalent to a strict one.