Limits and colimits, generators and relations of partial groups

TL;DR

This paper investigates the categorical properties of partial groups, establishing their completeness and cocompleteness, and introduces free partial groups, showing every partial group can be obtained as a quotient of a free one.

Contribution

It proves that the category of partial groups is complete and cocomplete, and constructs free partial groups over a richer category, extending the understanding of their algebraic structure.

Findings

The category of partial groups is both complete and cocomplete.

Finite partial groups form a finitely complete and cocomplete subcategory.

Every partial group can be realized as a quotient of a free partial group.

Abstract

We analyse limits and colimits in the category $Part$ of partial groups, algebraic structures introduced by A. Chermak. We will prove that $Part$ is both complete and cocomplete and, in addition, that the full subcategory of finite partial groups is both finitely complete and finitely cocomplete. Cocompleteness is then used in order to define quotients of partial groups. We will also identify a category richer than $Set$ (the category of sets and set-maps) and build the free partial groups over objects is such category; this yields a larger class of free partial groups, eventually allowing to prove that every partial group is the quotient of a free partial group.