Monoidal characterisation of groupoids and connectors

TL;DR

This paper explores how internal structures like groupoids and connectors in regular categories can be characterized using monoidal methods, specifically through Frobenius monoids and ternary structures, generalizing existing relationships.

Contribution

It introduces a monoidal framework for describing groupoids and connectors as Frobenius structures, extending the understanding of their relationships in regular categories.

Findings

Groupoids correspond to dagger Frobenius monoids in the monoidal category of relations.

Connectors are characterized as Frobenius structures with ternary multiplication.

The study generalizes the relationship between connectors and groupoids through ternary Frobenius structures.

Abstract

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.