# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.02056