A note on a free group. The decomposition of a free group functor through the category of heaps

TL;DR

This paper introduces a monadic left adjoint functor to relate groups and heaps, enabling a new decomposition of the free group functor and offering an alternative construction of free groups.

Contribution

It presents a novel adjoint functor connecting heaps and groups, providing a new perspective on free group construction via categorical decomposition.

Findings

Established a monadic adjunction between heaps and groups

Decomposed the free group functor using the adjoint

Provided a new construction method for free groups

Abstract

This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.