Frobenius and commutative pseudomonoids in the bicategory of spans

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.