On some categorical-algebraic conditions in S-protomodular categories

TL;DR

This paper explores conditions in S-protomodular categories to understand their group-like behavior, establishing a hierarchy and parallels with protomodular categories, especially focusing on monoids and related structures.

Contribution

It introduces relative versions of algebraic conditions in S-protomodular categories, revealing a hierarchy and parallels with classical protomodular categories.

Findings

Established a hierarchy among S-protomodular categories.

Connected conditions like algebraic coherence to monoid-like behavior.

Extended classical conditions to the relative context in categorical algebra.

Abstract

In the context of protomodular categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion of S-protomodular category, whose main examples are the category of monoids and, more generally, categories of monoids with operations and Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a description of S-protomodular categories with a strong monoid-like behavior. In this paper we consider relative versions of the conditions mentioned above, in order to exhibit the parallelism with the "absolute" protomodular context and to obtain a hierarchy among S-protomodular categories.