Public Thesis defense - IRMP

Aspects of Regular and Exact Completions by Vasileios ARAVANTINOS - SOTIROPOULOS

Pour l’obtention du grade de Docteur en sciences

Over the last 30 years, the constructions of regular and exact completions of weakly lex categories and of the exact completion of a regular category have been extensively studied in the context of Category Theory and its applications. In particular, many results have been obtained concerning characterizations of when these completions satisfy various properties of interest, leading often to characterizations of certain classes of regular and exact categories, such as varieties of Universal Algebra. In the first part of this thesis, we consider when the regular and exact completions of a weakly lex category satisfy a property called 2-star-permutability, which subsumes at the same time two important notions: that of Mal'tsev and that of subtractive category. The main result here is a characterization of those categories which occur as projective covers of such 2-star-permutable regular categories. In the second part of the thesis, we consider the exact completion of a regular category in the context of categories enriched over the cartesian closed category of partially ordered sets (posets) and monotone functions. We provide a construction of the completion in this setting by employing a type of enriched calculus of relations and prove that the expected universal property is satisfied. Furthermore, we obtain a result which characterizes the universal functor from a regular poset-enriched category into its completion and use it to deduce interesting examples, both in the realm of ordered Universal Algebra and of ordered topological spaces. In particular, we show that the exact completion in this enriched sense of the category of Priestley spaces is the category of compact ordered spaces introduced by L. Nachbin.

Jury members :

• Prof. Marino Gran (UCLouvain), supervisor
• Prof. Panagis Karazeris (Univeristy of Patras, Greece), supervisor
• Prof. Michel Willem (UCLouvain), chairperson
• Dr. Pierre-Alain Jacqmin (UCLouvain), secretary
• Prof. Sandra Mantovani (University of Milan, Italy)
• Prof. Diana Rodelo (Univerity of Algarve, Portugal)

Pay attention :

The public defense of Vasileios Aravantinos-Sotiropoulos scheduled for Friday 16 July at 4:00 p.m will take place in the form of a video conference Teams

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