Frobenius objects in the category of relations

Convolution algebras for Relational Groupoids and Reduction

“…In [28], it has been proven that special dagger Frobenius objects in Rel (the category of sets and relations) are in one-to-one correspondence with groupoids. Recently [32], this result has been generalized using a characterization of Frobenius objects in Rel using simplicial sets. Relational convolution algebras are natural examples of Frobenius objects in Rel, in the same way that group algebras are a special class of Frobenius algebras.…”

“…Another motivation to this paper is the connection between groupoids and Frobenius objects in a dagger monoidal category. For instance, a representative example of a relational convolution algebra is the relational group algebra, a version up to equivalence, of the group algebra of a group G. Group algebras are particular cases of Frobenius algebras, so relational convolution algebras provide a new class of examples of Frobenius objects in the category of sets and relations, which are also in correspondence with groupoids [28,32]. In a work in preparation [21], we study Frobenius objects arising from groupoids in the category of spans, via simplicial sets.…”