A note on the $(\infty,n)$-category of cobordisms

TL;DR

This paper constructs an $( abla,n)$-category of $n$-dimensional cobordisms with tangential structures using complete $n$-fold Segal spaces, providing a rigorous foundation for fully extended topological field theories.

Contribution

It introduces a precise $( abla,n)$-category model for cobordisms, extending the categorical framework for topological field theories with tangential structures.

Findings

Constructs an $( abla,n)$-category of cobordisms with tangential structures.

Shows how to recover classical cobordism categories from the new model.

Provides a symmetric monoidal structure compatible with the $( abla,n)$-category.

Abstract

In this extended note we give a precise definition of fully extended topological field theories \`a la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\infty,n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms $n\operatorname{Cob}$ and the cobordism bicategory $n\operatorname{Cob}^{ext}$ from it.