[Paper Review] On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis

Piotr Pstrągowski|arXiv (Cornell University)|Nov 25, 2014

Homotopy and Cohomology in Algebraic Topology3 references22 citations

TL;DR

This paper establishes coherence theorems for dualizable and fully dualizable objects in monoidal and symmetric monoidal bicategories, introducing coherent dual pairs and coherent fully dual pairs as property-like structures equivalent to dualizability. Using an extended diagrammatic calculus for framed surfaces, it presents the framed bordism bicategory and proves the 2-dimensional Cobordism Hypothesis by showing it is equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair.

ABSTRACT

We prove coherence theorems for dualizable objects in monoidal bicategories and for fully dualizable objects in symmetric monoidal bicategories, describing coherent dual pairs and coherent fully dual pairs. These are property-like structures one can attach to an object that are equivalent to the properties of dualizability and full dualizability. We extend diagrammatic calculus of surfaces of Christopher Schommer-Pries to the case of surfaces equipped with a framing. We present two equivalence relations on so obtained framed planar diagrams, one which can be used to model isotopy classes of framings on a fixed surface and one modelling diffeomorphism-isotopy classes of surfaces. We use the language of framed planar diagrams to derive a presentation of the framed bordism bicategory, completely classifying all two-dimensional framed topological field theories with arbitrary target. We then use it to show that the framed bordism bicategory is equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair. In lieu of our coherence theorems, this gives a new proof of the Cobordism Hypothesis in dimension two.

Motivation & Objective

• To provide a coherent, algebraic characterization of dualizable objects in monoidal bicategories.
• To extend Christopher Schommer-Pries' diagrammatic calculus to framed surfaces.
• To present a complete finite presentation of the framed bordism bicategory.
• To establish a new proof of the 2-dimensional Cobordism Hypothesis using coherence and diagrammatic methods.
• To show that the framed bordism bicategory is equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair.

Proposed method

• Introduces the concept of a coherent dual pair as a structure equivalent to dualizability in monoidal bicategories.
• Develops a framed planar diagram calculus to model isotopy and diffeomorphism-isotopy classes of framed surfaces.
• Defines two equivalence relations on framed planar diagrams: one for framings and one for surface diffeomorphism-isotopy.
• Constructs a finite presentation of the framed bordism bicategory using generators and relations from the diagram calculus.
• Applies coherence theorems to reduce the Cobordism Hypothesis to a comparison between two finitely presented symmetric monoidal bicategories.
• Uses promotion and lifting techniques in the Lack model structure on bicategories to establish equivalences between shapes and their transported structures.

Experimental results

Research questions

• RQ1 How can dualizability in monoidal bicategories be characterized by a coherent, algebraic structure?
• RQ2 What is the correct diagrammatic calculus for framed surfaces that captures isotopy and diffeomorphism-isotopy classes?
• RQ3 Can the framed bordism bicategory be completely presented using generators and relations from a framed planar diagram calculus?
• RQ4 Is the framed bordism bicategory equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair?
• RQ5 Does the coherence theorem for fully dualizable objects allow for a direct proof of the 2-dimensional Cobordism Hypothesis?

Key findings

• The forgetful homomorphism from the bicategory of coherent dual pairs to the groupoid of dualizable objects is a surjective-on-objects equivalence, proving that coherent dual pairs are property-like structures equivalent to dualizability.
• The framed bordism bicategory admits a finite presentation using framed planar diagrams with two equivalence relations: one for framings and one for diffeomorphism-isotopy.
• The framed bordism bicategory is equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair, providing a new proof of the 2-dimensional Cobordism Hypothesis.
• The promotion of equivalences and invertible 2-cells in the category of shapes ensures unique lifting of structure, enabling finite presentations of bicategories.
• The coherence theorems for fully dualizable objects in symmetric monoidal bicategories establish that coherent fully dual pairs are equivalent to full dualizability.
• The diagrammatic calculus for framed surfaces allows explicit computation and classification of 2-dimensional framed topological field theories with arbitrary target.