Symmetric monoidal equivalences of topological quantum field theories in dimension two and Frobenius algebras

TL;DR

This paper proves that the well-known correspondence between 2D topological quantum field theories and commutative Frobenius algebras is a symmetric monoidal equivalence, clarifying the structure of these theories.

Contribution

It establishes that the categorical equivalences between 2D TQFTs and Frobenius algebras are symmetric monoidal, extending the classical correspondence to a stronger equivalence.

Findings

The equivalence between 2D TQFTs and Frobenius algebras is symmetric monoidal.

The invariant of 2D manifolds factors as a product of invariants from Frobenius algebras.

The result applies to both oriented and unoriented 2D TQFTs.

Abstract

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras.