The Classification of Two-Dimensional Extended Topological Field Theories

TL;DR

This paper classifies 2-dimensional extended topological field theories by providing generators and relations for the associated bordism bicategories, and applies this to classify theories with algebraic targets, including Frobenius algebras.

Contribution

It offers a complete generators and relations presentation of 2D extended bordism bicategories and classifies extended topological field theories with various target bicategories.

Findings

Complete presentation of 2D extended bordism bicategories.

Classification of extended TFTs with algebraic targets.

Equivalence of oriented extended TFTs to non-commutative separable symmetric Frobenius algebras.

Abstract

We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a (non-commutative) separable symmetric Frobenius algebra. The text is divided into three chapters. The first develops a variant of higher Morse theory and uses it to obtain a combinatorial description of surfaces suitable for the higher categorical language used later. The second chapter is an extensive treatment of the theory of symmetric monoidal bicategories. We introduce several stricter variants on the notion of symmetric monoidal bicategory, and give a very general treatment of the notion of presentation by generators and relations. Finally we provide a host of strictification and cohernece results for symmetric monoidal bicategories. The final chapter focuses on extended tqfts. We give a precise treatment of the extended bordism bicategory equipped with additional structure (such as framings or orientations). We apply the results of the previous two chapters to obtain a simple presentation of both the oriented and unoriented bordism bicategories, and describe the general method to obtain such classifications for other choices of structure. We examine the consequences of our classification when the target is the bicategory of algebras, bimodules, and maps, over a fixed commutative ground ring.