arXiv subjects

Arnaud Duvieusart

Publications and source records attributed to Arnaud Duvieusart.

5 recordsLinked to original sources

arXiv2026

Ternary semi-direct products in semi-abelian categories

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.

math.CT↗

arXiv2026

A direction functor approach to the cohomology of small categories

We show how the direction functors can be used to develop a cohomology theory for Barr-exact and S-Maltsev categories, where S is a suitable class of split epimorphisms with a fixed section. Using the fact that, for any set B, the category of small categories with B as set of object is S-Maltsev with respect to the class of Schreier points, we show that the cohomology theory of small categories arising from the direction functors coincides with the one introduced by Hoff and Golasinski.

math.CT↗

arXiv2020

Fundamental groupoids for simplicial objects in Mal'tsev categories

We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and in fact a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).

math.CT↗

arXiv2019

Galois theory and the categorical Peiffer commutator

We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional property. We prove that this commutator also characterizes double central extensions, obtaining then some Hopf formulas for the second and third homology objects of internal precrossed modules.

math.CT↗

arXiv2018

Higher commutator conditions for extensions in Mal'tsev categories

We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed modules of Lie algebras, and compact groups.

math.CT↗