A note on the $(\infty,n)$-category of cobordisms
Damien Calaque, Claudia Scheimbauer
TL;DR
This work provides a detailed, explicit construction of the (∞,n)-category of n-bordisms using complete n-fold Segal spaces, forming a robust model for fully extended n-dimensional TFTs. It develops two complementary symmetric monoidal frameworks (Gamma-object and delooping-tower) and shows their equivalence, enabling a flexible approach to monoidal structures on bordisms. The paper also clarifies variants (e.g., oriented/framed bordisms), compares to Lurie’s definitions, and lays groundwork for concrete TFT realizations (to be pursued in subsequent work). The construction hinges on precise interval/bordism data (Int structures, boxing, and PBord/Bord spaces) and uses completion to achieve the desired (∞,n)-categorical completeness while preserving operadic and monoidal coherence. Overall, it provides a concrete, computable bridge between higher-categorical formalism and the geometry of bordisms, with implications for fully extended TFT classifications.
Abstract
In this extended note we give a precise definition of fully extended topological field theories à 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.