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arXiv · 2610.01671

Ternary semi-direct products in semi-abelian categories

Arnaud Duvieusart
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Abstract

We study the construction of ternary semi-direct products in semi-abelian categories, following a definition of higher semi-direct product previously introduced by Carrasco and Cegarra for groups and Lie algebras. We show that these objects are determined by actions and equivariant morphisms satisfying a certain compatibility condition in the ambient category, and that actions by semi-direct products are a special case of this construction. We also show how their structure can be further simplified in algebraically coherent or locally algebraically cartesian closed categories. We also give a concrete description of the structure of ternary semi-direct products in concrete categories, and how our categorical interpretation of the necessary structure relates to the group and Lie-theoretic versions given by Carrasco and Cegarra.

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Arnaud Duvieusart. 2026-10-01. Ternary semi-direct products in semi-abelian categories. https://arxiv.org/abs/2610.01671

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