Relative Frobenius algebras are groupoids

TL;DR

This paper characterizes groupoids using dagger Frobenius algebras in sets and relations, extends the framework to non-unital cases with H*-algebras, and explores the connection to semigroupoids.

Contribution

It provides a functorial characterization of groupoids via dagger Frobenius algebras and establishes a new link to locally cancellative regular semigroupoids.

Findings

Groupoids are characterized as special dagger Frobenius algebras.

A non-unital generalization relates H*-algebras to semigroupoids.

A universal passage connects these algebraic structures.

Abstract

We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets and relations, and locally cancellative regular semigroupoids. Finally, we study a universal passage from the former setting to the latter.