Frobenius objects in the category of relations

TL;DR

This paper characterizes Frobenius objects in the category of relations using simplicial sets, extending known correspondences with groupoids and providing new examples related to cohomology classes in manifolds.

Contribution

It offers a simplicial set-based characterization of Frobenius objects in relations, generalizing previous groupoid correspondences and constructing novel examples from manifold cohomology.

Findings

Frobenius objects characterized via simplicial sets.

Extension of groupoid correspondence to broader Frobenius objects.

Construction of Frobenius objects from cohomology classes in manifolds.

Abstract

We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations are in correspondence with groupoids. As an additional example, we construct a Frobenius object in the category of relations whose elements are certain cohomology classes in a compact oriented Riemannian manifold.