The Brauer category is a symmetric strict monoidal category that arises as a categorification of the Brauer algebras in the context of Banagl's framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl's concrete positive TFT based on fold maps into the plane.

中文翻译：

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彩色布劳尔类别及其线性表示

Brauer 范畴是对称严格幺半群范畴，它是在 Banagl 的正拓扑场论 (TFTs) 框架背景下作为 Brauer 代数的分类出现的。我们引入了彩色 Brauer 类别作为 Brauer 类别的丰富，其中态射是按分量标记的。(色) Brauer 范畴的线性表示是对称严格幺半群函子进入实向量空间范畴和配备 Schauenburg 张量积的线性映射。我们研究（色）布劳尔范畴的表示理论，并对其所有忠实的线性表示进行分类。作为一个应用程序，我们使用折叠线的索引来构建基于折叠图到平面的 Banagl 的具体正 TFT 的改进。