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The Brauer category is a symmetric strict monoidal category that arises as acategorification of the Brauer algebras in the context of Banagl's framework ofpositive topological field theories (TFTs). We introduce the chromatic Brauercategory as an enrichment of the Brauer category in which the morphisms arecomponent-wise labeled. Linear representations of the (chromatic) Brauercategory are symmetric strict monoidal functors into the category of realvector spaces and linear maps equipped with the Schauenburg tensor product. Westudy representation theory of the (chromatic) Brauer category, and classifyall its faithful linear representations. As an application, we use indices offold lines to construct a refinement of Banagl's concrete positive TFT based onfold maps into the plane.

The chromatic Brauer category and its linear representations