  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.
