Ternary semi-direct products in semi-abelian categories
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.
Related
- Homology and higher-order intermediate categories for proper abelian subcategories
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- Transfer of abelian model structures to equivariant categories and homotopy squares
- Construction of n-cotorsion pairs in some abelian categories
- Exactness of the 2-categories of abelian and triangulated categories
BibTeX
@misc{duvieusart2026ternary,
title = {{Ternary semi-direct products in semi-abelian categories}},
author = {Arnaud Duvieusart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01671},
eprint = {2610.01671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01671},
}
RIS
TY - UNPB
TI - Ternary semi-direct products in semi-abelian categories
AU - Arnaud Duvieusart
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01671
UR - https://arxiv.org/abs/2610.01671
AB - 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.
ER -