On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis
Piotr Pstrągowski
TL;DR
This work proves coherence theorems for dualizable and fully dualizable objects in monoidal bicategories and symmetric monoidal bicategories, showing that coherent dual pairs and coherent fully dual pairs are property-like, i.e., determined up to equivalence by dualizability. It extends Schommer-Pries' diagrammatic calculus to framed surfaces, defines framed planar diagrams with two equivalence notions, and uses them to present the framed 2D bordism bicategory. The framed bordism bicategory is shown to be equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair, providing a direct, algebraic proof of the Cobordism Hypothesis in dimension two and yielding a complete classification of two-dimensional framed topological field theories for arbitrary targets. The paper also develops a robust framework of biased yet equivalence-stable free bicategories (unbiased semistrict and cofibrant replacements) to transfer coherence results across models, enabling explicit presentations and calculi for low-dimensional bordism and TFTs.
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.