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The cobordism hypothesis

David Ayala, John Francis

TL;DR

This work shows that the cobordism hypothesis for fully extended framed quantum field theories follows from a conjectural theory of factorization homology with adjoints. By developing a robust framework of stratified, vari-framed manifolds and exit-path infinity-categories, the authors formulate and prove the tangle hypothesis, which identifies k-endomorphisms in a pointed infinity-category with spaces of functors from a tangle category. Delooping these tangle categories yields a cobordism category Bord_n^fr, and the cobordism hypothesis asserts an equivalence between symmetric monoidal functors Bord_n^fr to X and the object space of X for any X with adjoints and duals. The paper also develops two enrichments (Cartesian and geometric) of the theory, clarifying how higher-category structures interact with product and geometric constructions, and outlines a path to generalizations to other tangential structures beyond framings. Together these results provide a deep link between moduli of stratifications, factorization homology, and fully extended TQFTs with potential implications for invertible theories and beyond.

Abstract

Assuming a conjecture about factorization homology with adjoints, we prove the cobordism hypothesis, after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie.

The cobordism hypothesis

TL;DR

This work shows that the cobordism hypothesis for fully extended framed quantum field theories follows from a conjectural theory of factorization homology with adjoints. By developing a robust framework of stratified, vari-framed manifolds and exit-path infinity-categories, the authors formulate and prove the tangle hypothesis, which identifies k-endomorphisms in a pointed infinity-category with spaces of functors from a tangle category. Delooping these tangle categories yields a cobordism category Bord_n^fr, and the cobordism hypothesis asserts an equivalence between symmetric monoidal functors Bord_n^fr to X and the object space of X for any X with adjoints and duals. The paper also develops two enrichments (Cartesian and geometric) of the theory, clarifying how higher-category structures interact with product and geometric constructions, and outlines a path to generalizations to other tangential structures beyond framings. Together these results provide a deep link between moduli of stratifications, factorization homology, and fully extended TQFTs with potential implications for invertible theories and beyond.

Abstract

Assuming a conjecture about factorization homology with adjoints, we prove the cobordism hypothesis, after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie.

Paper Structure

This paper contains 23 sections, 30 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Programmatic overview
  4. Preliminaries
  5. Stratified spaces
  6. Framings
  7. Flagged higher categories and Segal sheaves
  8. Factorization homology
  9. Factorization homology with adjoints
  10. The tangle hypothesis
  11. The $\mathop{\mathrm{\infty}}\nolimits$-category of tangles $\mathop{\mathrm{\frak T{\sf ang}}}\nolimits_{n\subset n+k}^{\mathop{\mathrm{\sf fr}}\nolimits}$
  12. Constructible bundles $X\rightarrow \Delta^1$ whose special fiber $X_0$ is a manifold
  13. Proof of the tangle hypothesis
  14. The cobordism hypothesis
  15. Delooping
  16. ...and 8 more sections

Key Result

Theorem 1.1

There is a fully faithful functor, factorization homology, from $(\mathop{\mathrm{\infty}}\nolimits,n)$-categories into space-valued functors on compact vari-framed stratified $n$-manifolds.

Theorems & Definitions (94)

  • Theorem 1.1: emb1a
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: $\mathop{\mathrm{\mathcal{B}\mathsf{un}}}\nolimits$ and $\mathop{\mathrm{\boldsymbol{\mathcal{E}}{\sf xit}}}\nolimits$
  • Theorem 2.4: striat
  • Remark 2.5
  • Theorem 2.6
  • ...and 84 more