PreprintarXiv licencearXiv
A normal form for the Thin Flat Surfaces category
Abstract
The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.
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BibTeX
@misc{herrera2026normal,
title = {{A normal form for the Thin Flat Surfaces category}},
author = {Jaime Herrera and Carmen Rovi},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27771},
eprint = {2608.27771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27771},
}
RIS
TY - UNPB
TI - A normal form for the Thin Flat Surfaces category
AU - Jaime Herrera
AU - Carmen Rovi
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27771
UR - https://arxiv.org/abs/2608.27771
AB - The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.
ER -