Coherent span-valued 2D TQFTs

TL;DR

This paper establishes a correspondence between certain algebraic structures called commutative Frobenius pseudomonoids in spans and 2-Segal cosymmetric sets, providing a framework for 2D topological quantum field theories valued in spans.

Contribution

It introduces a novel link between Frobenius pseudomonoids and 2-Segal cosymmetric sets, and presents a construction from partial monoids to these sets.

Findings

Frobenius pseudomonoids correspond to 2-Segal cosymmetric sets.

A method to construct 2-Segal cosymmetric sets from partial monoids.

Framework for interpreting 2D TQFTs in the bicategory of spans.

Abstract

We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field theory taking values in the bicategory of spans. We also describe a construction that produces a 2-Segal cosymmetric set from any partial monoid equipped with a distinguished element.