On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis

TL;DR

This paper establishes coherence theorems for dualizable objects in monoidal bicategories, extends diagrammatic calculus to framed surfaces, and provides a new proof of the 2D Cobordism Hypothesis using framed bordism bicategories.

Contribution

It introduces coherence theorems for dualizable objects, extends surface calculus to framed surfaces, and classifies 2D framed topological field theories via bicategory equivalences.

Findings

Coherence theorems for dualizable and fully dualizable objects.

Diagrammatic calculus extended to framed surfaces.

Classification of 2D framed topological field theories.

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.