Public Thesis defense - IRMP


21 août 2020



Auditoire CYCL01 - Chemin du Cyclotron, 2

Selected Topics in Goursat Categories by Idriss TCHOFFO NGUEFEU

Pour l’obtention du grade de Docteur en sciences

Over the last thirty years, many interesting results have been discovered in the categorical extension of 2-permutable varieties, called Mal’tsev categories. Many of these results still hold in regular categories satisfying the strictly weaker property of 3-permutability, called Goursat categories. A nice feature of regular Mal’tsev and Goursat categories is that Gumm’s Shifting Lemma holds in these categories, a property which allowed, for example, to develop commutator theory in universal algebras. The aim of this thesis is twofold: on the one hand, we extend to Goursat categories the main results obtained for the theory of projective covers and internal structures in Mal’tsev categories. In particular, we give some characterizations of the categories which are the projective covers of Goursat categories. Then, we show that the structure of internal connector is stable under quotients in any Goursat category. As a consequence, the category of internal connectors in a Goursat category is again a Goursat category. This implies that Goursat categories can be characterized in terms of a simple property of internal groupoids and internal categories. On the other hand, we study the Shifting Lemma in regular Mal’tsev and Goursat categories. We prove that regular Mal’tsev and Goursat categories can be characterized through suitable variations of the Shifting Lemma. We also investigate two properties related to the Shifting Lemma and called the Triangular Lemma and the Trapezoid Lemma in universal algebras. We establish some characterizations of regular Mal’tsev and Goursat categories with distributive lattice of equivalence relations through variations of the Triangular Lemma and Trapezoid Lemma.

Jury members :

  • Prof. Marino Gran (UClouvain), supervisor
  • Prof. Diana Rodelo (Univ. Algarve, Portugal), supervisor
  • Prof. Michel Willem (UCLouvain), chairperson
  • Dr. Pierre-Alain Jacqmin (UCLouvain), secretary
  • Prof. Andrea Montoli (Univ. Milan, Italy)
  • Prof. Manuela Sobral (Univ. Coimbra, Portugal)

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