On some categorical-algebraic conditions in S-protomodular categories

摘要

In the context of protomodular categories, several additional conditions havebeen considered in order to obtain a closer group-like behavior. Among them arelocally algebraic cartesian closedness and algebraic coherence. The recentnotion of S-protomodular category, whose main examples are the category ofmonoids and, more generally, categories of monoids with operations andJo'{o}nsson-Tarski varieties, raises a similar question: how to get adescription of S-protomodular categories with a strong monoid-like behavior. Inthis paper we consider relative versions of the conditions mentioned above, inorder to exhibit the parallelism with the "absolute" protomodular context andto obtain a hierarchy among S-protomodular categories.