A comparison between weakly protomodular and protomodular objects in unital categories

TL;DR

This paper investigates the differences between protomodular and weakly protomodular objects in unital categories, introducing new algebraic structures and providing examples to clarify their distinctions.

Contribution

It introduces left pseudocancellative unital magmas and characterizes weakly protomodular objects within their algebraic variety, highlighting the distinctions from protomodular objects.

Findings

Prototypical examples of weakly protomodular objects that are not protomodular.

Introduction of left pseudocancellative unital magmas.

Clarification of the relationship between the two notions in unital categories.

Abstract

We compare the concepts of protomodular and weakly protomodular objects within the context of unital categories. Our analysis demonstrates that these two notions are generally distinct. To establish this, we introduce left pseudocancellative unital magmas and characterise weakly protomodular objects within the variety of algebras they constitute. Subsequently, we present an example of a weakly protomodular object that is not protomodular in this category.