Galois theory for reflexive graphs and simplicial objects
(2020)
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- Authors
- Supervisors
- Gran, Marino
- Abstract
- Categorical Galois Theory was introduced by Janelidze as a way to unify, among others, Magid’s generalization of Galois Theory for commutative rings, coverings of locally connected spaces, and central extensions of groups. He and Kelly then used it to define central extensions relative to an admissible subcategory. In this thesis, we explore new examples of Galois structures and the links between centrality and generalized commutators. We construct a Galois structure for the category of pairs of equivalence relations in an exact Mal’tsev category with coequalizers, and the subcategory of pairs with trivial Smith-Pedicchio commutator, and characterize the corresponding central extensions by a suitable commutator condition. This also yields a centrality criterion for extensions of reflexive graphs over a fixed base relative to internal groupoids. In semi-abelian categories, we use our previous results to characterize central extensions of precrossed B-modules relative to crossed B-modules by the triviality of an internal Peiffer commutator. This allows us to give a Hopf-type formula for the homology of precrossed B-modules using Peiffer commutators, thus generalizing previous results of Conduché and Ellis. Inspired by Brown and Janelidze, we also study a Galois structure based on the 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, and characterize the central extensions in this context.
- Affiliations
Citations
Duvieusart, A. (2020). Galois theory for reflexive graphs and simplicial objects. https://hdl.handle.net/2078.5/96316