On examples and classification of Frobenius objects in Rel

Abstract

We give some new examples of Frobenius objects in the category of sets and relations R e l \mathbf {Rel} . One example is a groupoid with a twisted counit. Another example is the set of conjugacy classes of a group. We also classify Frobenius objects in R e l \mathbf {Rel} with two or three elements, and we compute the associated surface invariants using the partition functions of the corresponding topological quantum field theories (TQFT).