On examples and classification of Frobenius objects in \textbf{Rel} (Q6607604)

A basic result in topological quantum field theory (TQFT) is the correspondance between 2-dimensional (oriented) TQFTs and commutative Frobenius algebras [\textit{L. Abrams}, J. Knot Theory Ramifications 5, No. 5, 569--587 (1996; Zbl 0897.57015); \textit{R. H. Dijkgraaf}, A geometrical approach to two-dimensional Conformal Field Theory, Diss. (1989; \url{https://dspace.library.uu.nl/handle/1874/210872})], which can be placed in a more general framework by defining a (commutative) Frobenius object in a (symmetric) monoidal category \(\mathcal{C}\). The upshot is the the study of Frobenius objects in any symmetric monoidal category \(\mathcal{C}\) has topological significance.\N\NThis paper studies Frobenius objects in the category \(\boldsymbol{Rel}\) whose objects are sets and whose morphisms are relations of sets [\textit{C. Heunen} et al., J. Pure Appl. Algebra 217, No. 1, 114--124 (2013; Zbl 1271.18004)], where it was shown that special dagger Frobenius objects in \(\boldsymbol{Rel} \) are in correspondence with groupoids. This result was extended in [Zbl 07915444], where it was shown that a Frobenius object in \(\boldsymbol{Rel}\) is to be encoded in a simplicial set equipped with an automorphism of the set of 1-simplicies, abiding by certain properties.\N\NThe principal results of the paper go as follows.\N\N\begin{enumerate}\N\item[(1)] New examples of Frobenius objects in \(\boldsymbol{Rel}\) are presented. One is a generalization of the groupoid example allowing for a twist that can spoil the special and dagger properties. Another example is the set of conjugacy classes of a group. These examples can have multivalued multiplication relations.\N\N\item[(2)] Frobenius objects in \(\boldsymbol{Rel}\) with two or three elements are completely classified. It turns out that, up to isomorphism, there are 5 Frobenius objects in \(\boldsymbol{Rel}\) with two elements, and there are 25 Frobenius objects in \(\boldsymbol{Rel}\) \ with three elements.\N\N\item[(3)] For all of the two- and three-element Frobenius objects in \(\boldsymbol{Rel}\), the authors compute the associated topological invariants for closed oriented surfaces.\N\end{enumerate}\N\NFor the entire collection see [Zbl 1545.18002].