Frobenius algebras can be given a category-theoretic definition in terms of the category of vector spaces. This leads to a definition of Frobenius object in any monoidal category. In this talk, I will describe Frobenius objects in categories where the objects are sets and the morphisms are spans or relations. These categories can be viewed as toy models for the symplectic category. The main result is that, in both cases, it is possible to construct a simplicial set that encodes the data of the Frobenius structure.
This work is part of a bigger program aimed at better understanding the relationship between Poisson geometry and two-dimensional topological field theory, and I will do my best to explain this bigger-picture motivation. This is based on work with Ruoqi Zhang and work in progress with Ivan Contreras and Molly Keller.