# The chromatic Brauer category and its linear representations

## TL;DR

This paper introduces the chromatic Brauer category, an enriched version of the Brauer category with labeled morphisms, and studies its linear representations to refine positive topological field theories.

## Contribution

It defines the chromatic Brauer category, classifies its faithful linear representations, and applies these results to enhance Banagl's positive TFT framework.

## Key findings

- Classified all faithful linear representations of the chromatic Brauer category.
- Constructed a refined positive TFT using fold line indices.
- Provided a new categorification framework for topological field theories.

## Abstract

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.