  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.
