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Aspects of internal stuctures in algebraic categories by Nadja EGNER -
Lundi 6 juillet 2026 à 15h00 - Auditoire CYCL01 - Bâtiment de Hemptinne - Chemin du Cyclotron, 2 - 1348 Louvain-la-Neuve -
This doctoral dissertation investigates internal groupoids, n-groupoids and n-fold groupoids in various algebraic contexts, using categorical Galois theory and torsion theories. Such internal structures are well-behaved in Mal’tsev categories, of which the categories of groups, rings, associative and Lie algebras are examples. In this context, any reflexive graph admits at most one internal category structure, and the categories of internal categories and of internal groupoids are isomorphic.
We identify a wide range of new Galois structures in the (quasi-)abelian context. More precisely, we construct a family of torsion-free Birkhoff subcategories of a quasi-abelian functor category, and characterize the corresponding (higher) central extensions and compute generalized Hopf formulae for homology. This framework captures, in the quasi-abelian setting, the category of internal n-fold groupoids together with its subcategory of internal n-groupoids.
In the broader context of regular Mal’tsev categories with coequalizers, we provide an explicit description of the reflection of the category of internal double groupoids into its Birkhoff subcategory of internal 2-groupoids. We further analyze this adjunction when the base category is in addition an exact naturally Mal’tsev category. Moreover, we prove that the category of internal n-groupoids is semi-abelian, action representable, algebraically coherent, and has normalizers whenever the base category has all these properties.
In the last part of this thesis, we establish a syntactic characterization of the varieties of universal algebras that are weakly Mal’tsev categories. While reflexive graphs in the weakly Mal’tsev context still admit at most one internal category structure, the categories of internal categories and of internal groupoids are no longer necessarily isomorphic.
Jury members :
Prof. Marino Gran (UCLouvain) (Supervisor)
Dr. Pierre-Alain Jacqmin (Royal Military Academy) (Supervisor)
Prof. Pierre-Emmanuel Caprace (UCLouvain) (Chairperson)
Prof. Tim Van der Linden (UCLouvain) (Secretary)
Prof. Alan Cigoli (University of Turin)
Prof. Diana Rodelo (University of the Algarve)
Pay attention : the public defense of Nadja EGNER will also take place in the form of a videoconference
