Localisations of cobordism categories and invertible TFTs in dimension two

TL;DR

This paper computes localisations of 2D cobordism categories, enabling a classification of invertible 2D topological field theories across various cases, advancing understanding in geometry and quantum field theory.

Contribution

It provides a comprehensive computation of localisations of 2D cobordism categories, leading to a classification of invertible 2D topological field theories in multiple settings.

Findings

Classification of invertible 2D TFTs in orientable and non-orientable cases

Explicit descriptions of localisations of 2D cobordism categories

Unified framework for open and closed cases

Abstract

Cobordism categories have played an important role in classical geometry and more recently in mathematical treatments of quantum field theory. Here we will compute localisations of two-dimensional discrete cobordism categories. This allows us, up to equivalence, to determine the category of invertible two-dimensional topological field theories in the sense of Atiyah. We are able to treat the orientable, non-orientable, closed and open cases.