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On the Classification of Topological Field Theories

Jacob Lurie

TL;DR

The paper sketches a proof of the Baez-Dolan cobordism hypothesis, positing that framed extended topological field theories are classified by fully dualizable objects in a symmetric monoidal $(\infty,n)$-category. It develops a modeling framework via complete Segal spaces to treat bordism categories as $(\infty,1)$- or $(\infty,n)$-categories, defines dualizability in this higher setting, and states the framed cobordism hypothesis as a universal property: Fun^{\otimes}( Bord_{n}^{\mathrm{fr}}, \mathcal{C}) ≃ {\mathcal{C}}^{\sim} with the equivalence given by evaluation at a point. The proof strategy proceeds by induction on $n$, employing Morse theory (Igusa) and an index filtration to build $n$-morphisms from disks and gluing data, while reducing to unoriented cases via $G$-structures and homotopy fixed points. This higher-categorical lens also informs connections to the Mumford conjecture and the stable cohomology of diffeomorphism groups, recasting classical geometric questions in a universal, functorial framework with potential computational leverage.

Abstract

This paper provides an informal sketch of a proof of the Baez-Dolan cobordism hypothesis, which provides a classification for extended topological quantum field theories.

On the Classification of Topological Field Theories

TL;DR

The paper sketches a proof of the Baez-Dolan cobordism hypothesis, positing that framed extended topological field theories are classified by fully dualizable objects in a symmetric monoidal -category. It develops a modeling framework via complete Segal spaces to treat bordism categories as - or -categories, defines dualizability in this higher setting, and states the framed cobordism hypothesis as a universal property: Fun^{\otimes}( Bord_{n}^{\mathrm{fr}}, \mathcal{C}) ≃ {\mathcal{C}}^{\sim} with the equivalence given by evaluation at a point. The proof strategy proceeds by induction on , employing Morse theory (Igusa) and an index filtration to build -morphisms from disks and gluing data, while reducing to unoriented cases via -structures and homotopy fixed points. This higher-categorical lens also informs connections to the Mumford conjecture and the stable cohomology of diffeomorphism groups, recasting classical geometric questions in a universal, functorial framework with potential computational leverage.

Abstract

This paper provides an informal sketch of a proof of the Baez-Dolan cobordism hypothesis, which provides a classification for extended topological quantum field theories.

Paper Structure

This paper contains 22 sections, 51 theorems, 159 equations.

Table of Contents

  1. Topological Field Theories and Higher Categories
  2. Classical Definitions
  3. Extending Down: Lower Dimensional Manifolds
  4. Higher Category Theory
  5. Extending Up: Diffeomorphism Groups
  6. Formulation of the Cobordism Hypothesis
  7. Complete Segal Spaces
  8. Bordism Categories as Segal Spaces
  9. Fully Dualizable Objects
  10. The Cobordism Hypothesis
  11. The Mumford Conjecture
  12. Proof of the Cobordism Hypothesis
  13. Inductive Formulation
  14. Reduction to the Unoriented Case
  15. Unfolding of Higher Categories
  16. ...and 7 more sections

Key Result

Proposition 1.1.8

Let $Z$ be a topological field theory of dimension $n$. Then for every closed $(n-1)$-manifold $M$, the vector space $Z(M)$ is finite dimensional, and the pairing $Z( \overline{M}) \otimes Z(M) \rightarrow k$ is perfect: that is, it induces an isomorphism $\alpha$ from $Z( \overline{M} )$ to the dua

Theorems & Definitions (301)

  • Definition 1.1.1
  • Remark 1.1.2
  • Example 1.1.3
  • Example 1.1.4
  • Definition 1.1.5: Atiyah
  • Remark 1.1.6
  • Remark 1.1.7
  • Proposition 1.1.8
  • Example 1.1.9: Field Theories in Dimension 1
  • Remark 1.1.10
  • ...and 291 more