A comparison between weakly protomodular and protomodular objects in unital categories
Xabier García-Martínez, Andrea Montoli, Diana Rodelo, Tim Van der Linden
TL;DR
The paper demonstrates that protomodular and weakly protomodular objects do not coincide in all unital categories. It introduces left pseudocancellative unital magmas and analyzes weak protomodularity within the associated category $\mathsf{LPM}$, showing that nontrivial distinctions arise. A key result (Theorem wprot) characterizes weakly protomodular objects in $\mathsf{LPM}$ via a compositional left-division condition $x_1\backslash( x_2 \backslash ( \cdots \backslash (x_n \backslash x)\cdots )) = e$, with left loops satisfying the condition. Concrete examples based on $\mathbb{Z}$ yield weakly protomodular but not protomodular objects, and the work establishes strict inclusions $\{\text{left loops}\} \subsetneq \{\text{protomodular objects}\} \subsetneq \{\text{weakly protomodular objects}\}$, highlighting the nuanced landscape of these 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.