[Paper Review] On the Classification of Topological Field Theories

Jacob Lurie|ArXiv.org|May 4, 2009

Homotopy and Cohomology in Algebraic TopologyMathematics11 references89 citations

TL;DR

This paper provides a comprehensive exposition of the classification of topological field theories via the cobordism hypothesis, establishing that extended topological field theories in n dimensions are classified by fully dualizable objects in an (∞,n)-category. Using higher category theory and the theory of complete Segal spaces, the author proves that the bordism category of manifolds with tangential structures admits a universal symmetric monoidal functor to such categories, thereby unifying quantum field theory and algebraic topology through higher categorical structures.

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.

Motivation & Objective

• To provide an expository account of the classification of extended topological field theories using higher category theory.
• To formulate and prove a version of the Baez-Dolan cobordism hypothesis for (∞,n)-categories.
• To generalize the classification to manifolds with tangential structures, such as framings and orientations.
• To extend the framework to include tangles and singular manifolds, revealing universal properties of related (∞,n)-categories.
• To establish a universal characterization of topological field theories through fully dualizable objects in symmetric monoidal (∞,n)-categories.

Proposed method

• Formalizing the bordism category as a complete Segal space to model the (∞,n)-category of manifolds with tangential structures.
• Defining fully dualizable objects in an (∞,n)-category as the key data classifying extended TFTs.
• Using obstruction theory and the index filtration to reduce the classification problem to lower-dimensional cases.
• Applying an inductive formulation of the cobordism hypothesis to build functors from bordism categories to (∞,n)-categories.
• Leveraging the theory of higher categories with adjoints to model symmetric monoidal functors from bordism categories.
• Utilizing the tangle hypothesis as a generalization, showing that framed tangles classify free ribbon (∞,1)-categories.

Experimental results

Research questions

• RQ1 How can extended topological field theories be classified using higher category theory and the cobordism hypothesis?
• RQ2 What is the role of fully dualizable objects in classifying symmetric monoidal functors from bordism categories?
• RQ3 How does the cobordism hypothesis generalize to manifolds with additional tangential structures such as framings or orientations?
• RQ4 Can the tangle hypothesis be derived from the cobordism hypothesis via universal properties in higher categories?
• RQ5 What is the universal property of the (∞,n)-category of tangles with framing or orientation data?

Key findings

• The cobordism hypothesis is proven: extended topological field theories are classified by fully dualizable objects in an (∞,n)-category with adjoints.
• The bordism category of n-dimensional manifolds with tangential structures is modeled as a complete Segal space, enabling the universal property of TFTs.
• A symmetric monoidal functor from the framed bordism category to an (∞,n)-category is uniquely determined by a fully dualizable object and a choice of dualizing data.
• The tangle hypothesis is established as a consequence: the (∞,n)-category of framed tangles is the free symmetric monoidal (∞,n)-category with duals on a single generator.
• The construction of topological field theories via the cobordism hypothesis extends to singular manifolds and more general tangential structures, with universal functors arising from obstruction-theoretic lifting.
• The action of O(n) on the ∞-groupoid of (∞,n)-categories with duals leads to a characterization of ribbon structures in terms of SO(n)-fixed points, recovering classical ribbon categories in the 1-category truncation.