Categorical-algebraic conditions in semi-abelian categories
(2025)
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- Abstract
- This thesis explores and studies several categorical conditions within the semi-abelian framework, introducing new concepts and deepening the understanding of existing ones. We examine the hierarchy of intermediate conditions separating semi-abelian and abelian categories, highlighting distinctions through examples and counterexamples. A key contribution is the notion of representability of representations, which is shown to characterise Lie algebras within varieties of non-associative algebras. Similarly, the associativity of the cosmash product is studied, showing that it uniquely characterises commutative associative algebras within the same framework. Arithmetical categories, traditionally studied in Barr-exact Mal’tsev contexts, are revisited in the semi-abelian setting. Protomodularity appears as an important component in characterising arithmetical semi-abelian categories as those without non-trivial abelian objects. This perspective allows for new insights into the behaviour of Higgins commutators in these categories. Lastly, the category of Heyting semilattices is investigated. Its categorical-algebraic features, such as commuting subobjects or normal subobjects, are analysed to establish conditions like algebraic coherence or action accessibility. Serving as an arithmetical category satisfying transitivity of normality—a new condition introduced in this thesis—the category of Heyting semilattices exemplifies the application of results from this work.
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Vienne, C. (2025). Categorical-algebraic conditions in semi-abelian categories. https://hdl.handle.net/2078.5/244318