Groupoids and cogroupoids: an one object approach

TL;DR

This paper introduces a simplified one-object framework for defining groupoids and cogroupoids within categories, aiming to streamline related constructions and establish connections with group objects and Hopf algebras.

Contribution

It provides a novel one-object approach to internal groupoids and cogroupoids, simplifying existing constructions and linking them to group objects and Hopf algebras.

Findings

Introduces a one-object definition of internal groupoids and cogroupoids.

Establishes relations with group objects and Hopf algebras.

Simplifies constructions related to internalizations, Lie groupoids, and Hopf algebroids.

Abstract

The aim of this paper is to provide a definition of groupoid and cogroupoid internal to a category which makes use of only one object and morphisms, in contrast with the two object approach commonly found in the literature. We will give some examples and we will establish a relation with group objects (and Hopf algebras). The definitions presented here were designed to simplify some constructions related to internalizations, Lie groupoids and Hopf algebroids.