A Categorical Perspective on Braid Representations
We study categories whose objects are the braid representations, that is, strict monoidal functors $F : B \to \mathrm{Mat}$ from the braid category $B$ to the category of matrices $\mathrm{Mat}$. Braid representations are equivalent to solutions to the (constant) Yang--Baxter equation, so our work provides a categorical framework for their classification. Our first technical result (Theorem 2.11) implies that the assumption of strictness causes no loss of generality: any monoidal functor with source $B$ is monoidally equivalent to a strict such functor. We develop the structure of the category $\mathrm{MonFun}(B,\mathrm{Mat})$, whose objects are braid representations and whose morphisms are monoidal natural transformations. Although $\mathrm{MonFun}(B,\mathrm{Mat})$ is non-additive, we show that it does admit a rigid monoidal structure (Theorem 5.3). We introduce notions of quotient, sub- and simple objects in this setting, and prove an array of structural results such as a version of Schur's Lemma (Theorem 5.18), and a characterisation of objects that are both sub- and quotient objects (Corollary 5.20). This structural framework, for example, provides a new categorical characterisation of charge-conserving solutions. Classification is, generally, up to a suitable notion of isomorphism. So a major part of the contribution here is to introduce, compare, and contrast notions of isomorphism for braid representations. We give various properties exposing the implications of different choices of equivalence and describe some relationships among them (Theorems 7.19, 7.18, 7.9, Conjectures 7.20, 7.23). An extensive range of key examples and counterexamples illustrate the framework developed here; and we cast a number of partial classifications in categorical terms, providing proof-of-principle vindication for our methods.