Frobenius and commutative pseudomonoids in the bicategory of spans

TL;DR

This paper characterizes Frobenius and commutative pseudomonoids in the bicategory of spans of sets using higher Segal objects, connecting algebraic structures with simplicial and higher categorical frameworks.

Contribution

It extends previous characterizations of algebra objects in spans of sets to the bicategory setting, linking them to paracyclic and Gamma-sets satisfying 2-Segal conditions.

Findings

Frobenius pseudomonoids correspond to paracyclic sets with 2-Segal properties.

Commutative pseudomonoids correspond to Gamma-sets with 2-Segal properties.

Results have applications in symplectic geometry and topological field theory.

Abstract

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in $\operatorname{Span}$ correspond, respectively, to paracyclic sets and $\Gamma$-sets satisfying the $2$-Segal conditions. These results connect closely with work of the third author on $A_\infty$ algebras in $\infty$-categories of spans, as well as the growing body of work on higher Segal objects. Because our motivation comes from symplectic geometry and topological field theory, we emphasize the direct and computational nature of the classifications and their proofs.