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

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 $n\mathrm{Cob}$ and the cobordism bicategory $n{\mathrm{Cob}}^{ext}$ from it.

Final version; 75 pages, many pictures. Accepted for publication in Algebraic & Geometric Topology

References in corpus (3)

• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
• Rigidification of higher categorical structures
• From fractions to complete Segal spaces

Cited by in corpus (25)

• Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• Reflection positivity and invertible topological phases
• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
• Gapped boundary theories in three dimensions
• Condensations in higher categories
• Tori Detect Invertibility of Topological Field Theories
• Extended quantum field theory, index theory and the parity anomaly
• Smooth one-dimensional topological field theories are vector bundles with connection
• 2-Segal objects and the Waldhausen construction
• Nakayama categories and groupoid quantization
• Rigidification of higher categorical structures
• Quasi-Categories vs. Segal Spaces: Cartesian Edition
• Reflection Structures and Spin Statistics in Low Dimensions
• Truncated affine Rozansky--Witten models as extended TQFTs
• The Little Bundles Operad
• The Smith Fiber Sequence and Invertible Field Theories
• Segalification and the Boardman-Vogt tensor product
• The space of traces in symmetric monoidal infinity categories
• The equifibered approach to $\infty$-properads
• Two-Dimensional Extended Homotopy Field Theories
• Lorentzian bordisms in algebraic quantum field theory
• A model structure for locally constant factorization algebras
• Semisimple Field Theories Detect Stable Diffeomorphism
• A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories