标题： 一个关于辫表示的范畴视角 显示英文标题

标题： A Categorical Perspective on Braid Representations

摘要： 我们研究的对象是辫表示的范畴，即从辫子范畴 $B$ 到矩阵范畴 $Mat$ 的严格单调函子。辫表示等价于常数型 Yang-Baxter 方程的解，因此它们的分类问题也是等价的。我们考虑两种范畴：范畴 $MoonFun(B,Mat)$ 其态射为自然变换，以及范畴 $MonFun(B,Mat)$ 其态射为单调自然变换。这种范畴化背景自然为问题提供了三重焦点：源范畴 $B$；目标范畴 $Mat$；以及它们之间函子间的自然变换和其他对称性。事实上，这种方法的主要动机来源于最近对守恒荷 Yang-Baxter 算子的分类，在其中 $Mat$ 被替换为某个子范畴。一个目标是从范畴论的角度理解这一改变如何促进了问题的解决（旨在推广）。另一个目标是理解适当的等价概念——目标范畴的限制引入了新的概念，例如内等价和外等价的概念。还有一个目标是理解这样一个受限目标的普适性（它是否为某种适当等价关系下的所有等价类提供了横截面？）。然后目标是以一个足够普适的目标完成分类。 关于三重焦点的第一部分，范畴$B$，我们回顾了几个特征。 已经有人做过推广到类似范畴的工作，例如环面编织范畴。 但我们在这里的重要输入之一是将$MonFun(B,-)$重新表述为同构的杨-巴克对象范畴——这一操作很大程度上依赖于来源正是$B$。 显示英文摘要

摘要： We study categories whose objects are the braid representations, that is, strict monoidal functors from the braid category $B$ to the category of matrices $Mat$. Braid representations are equivalent to solutions to the (constant) Yang-Baxter equation. So their classification problems are also equivalent. We consider both the categories $MoonFun(B,Mat)$ whose morphisms are natural transformations and the category $MonFun(B,Mat)$ whose morphisms are the monoidal natural transformations. This categorical contextualisation naturally gives a three-fold focus to the problem: the source $B$; the target $Mat$; and the natural transformations and other symmetries between functors between them. Indeed the approach is mainly motivated by the recent classification of charge conserving Yang-Baxter operators, in which $Mat$ is replaced by a certain subcategory. One objective is to understand from the categorical perspective how the solution was facilitated by this change (with the aim of generalising). Another is to understand appropriate notions of equivalence - the restriction of target introduces new notions such as that of inner and outer equivalences. And another is to understand how universal such a restricted target is (does it give a transversal of all equivalence classes for a suitable notion of equivalence?). And then the aim is to complete the classification with a suitably universal target. With regard to the first part of the triple-focus, the category $B$, we review several features. Work has already been done to generalise to similar categories such as the loop braid category. But a significant part of our input here is to recast $MonFun(B,-)$ as the isomorphic category of Yang-Baxter objects - which manoeuvre leans heavily on the source being $B$ on the nose.