03 juillet 2020
will take place in the form of a video conference Teams
Galois theory for reflexive graphs and simplicial objects by Arnaud DUVIEUSART
Pour l’obtention du grade de Docteur en sciences
Categorical Galois Theory was introduced by Janelidze as a way to unify, among others, Magid’s generalization of Galois theory for commutative rings and the theory of coverings of locally connected spaces. Later, Janelidze showed that central extensions of groups could also be understood as a special case of this theory, and with Kelly he used this analogy to define central extensions relative to an admissible subcategory of a given category.
In this thesis, we explore new examples of Galois structures and the link between centrality and generalized commutator conditions. First, for any exact Mal’tsev category with coequalizers, we construct a Galois structure for the category of pairs of equivalence relations on the same object, and its subcategory of pairs with trivial Smith-Pedicchio commutator. We give a centrality criterion for this context, generalizing the one previously known for centrality with respect to abelian objects. This also allows us to characterize central extensions of reflexive graphs over a fixed base with relative to internal groupoids.
In semi-abelian categories, internal graphs and groupoids correspond to precrossed and crossed modules. We use our previous results to characterize central extensions of precrossed modules with respect to crossed modules by the triviality of an internal Peiffer commutator. This allows us to give a Hopf-type formula for the homology of precrossed modules using Peiffer commutators, thus generalizing previous results of Conduché and Ellis.
In the last chapter, inspired by Brown and Janelidze, we study a Galois structure based on the internal nerve functor between groupoids and simplicial objects in exact Mal’tsev categories. We construct a left adjoint to this functor, thus proving that internal groupoids form a Birkhoff subcategoy of the category of simplicial objects. We prove that the central extensions for the induced Galois structures coincide with regular epimorphic exact fibrations in the sense of Glenn.
Jury members :
- Prof. Marino Gran (UCLouvain), supervisor
- Prof. Michel Willem (UCLouvain), chairperson
- Prof. Tim Van der Linden (UCLouvain), secretary
- Prof. Enrico Vitale (UCLouvain)
- Prof. Wendy Lowen (UAntwerpen, Belgium)
- Prof. Sandra Mantovani (Universita degli Studi di Milano, Italy)
Pay attention :
The public defense of Arnaud Duvieusart scheduled for Friday 03 July at 05:30 p.m will indeed take place in the form of a video conference Teams.