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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\u2019s concrete positive TFT based on fold maps into the plane.<\/jats:p>","DOI":"10.1007\/s10485-020-09619-5","type":"journal-article","created":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T06:02:51Z","timestamp":1606888971000},"page":"349-377","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Chromatic Brauer Category and Its Linear Representations"],"prefix":"10.1007","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7861-2811","authenticated-orcid":false,"given":"L. 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