Exploring some categorical aspects of foundational concepts in algebraic geometry

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dc.contributor.advisorMarques, Sophieen_ZA
dc.contributor.authorMgani, Damas Karmelen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2024-02-26T23:43:48Z
dc.date.accessioned2024-04-26T08:11:09Z
dc.date.available2024-02-26T23:43:48Z
dc.date.available2024-04-26T08:11:09Z
dc.date.issued2024-03
dc.descriptionThesis (PhD)--Stellenbosch University, 2024. en_ZA
dc.description.abstractENGLISH ABSTRACT: This thesis examines the foundational concepts of algebraic geometry, with a partic- ular emphasis on elucidating its categorical aspects. Our primary contribution lies in the comprehensive exploration of the gluing property across diverse categories. Throughout this exploration, we introduce a categorical framework for gluing, fea- turing two pivotal constructs: the gluing index category and the gluing data functor. This framework not only provides a unified methodology applicable to (pre)sheaves on sites, (locally) ringed topological spaces and schemes but also paves the way for potential future extensions into new categories. Furthermore, our research focuses on the separation property of (pre)sheaves, presenting a categorical description of separafication through the introduction of stalk sheaves associated with a presheaf. We also investigate the concept of sheafi- fication, aiming to understand if a sheaf can be defined as a composition of limits within the category of (pre)sheaves. While we successfully achieve this goal at a local level, it presents captivating prospects for further inquiry. In addition to these original contributions, this thesis presents an extensive and meticulous exploration of the fundamental principles of algebraic geometry, with a central emphasis on category theory. This part includes intricate details and formalities not readily accessible in existing literature on algebraic geometry.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: Hierdie tesis ondersoek die grondliggende konsepte van algebra¨ıese meetkunde, met ′n besondere klem op die toeligting van die kategoriese aspekte daarvan. Ons primeˆre bydrae leˆ in die omvattende verkenning van die gom-eiendom oor diverse kategoriee¨ heen. Dwarsdeur hierdie verkenning stel ons ′n kategoriese raamwerk vir gom be- kend wat twee spilkonstrukte bevat: die gomindekskategorie en die gomdatafunk- tor. Hierdie raamwerk verskaf nie net ′n verenigde metodologie van toepassing op (voor)gerwe op terreine, (plaaslik) geringde topologiese ruimtes en skemas nie, maar leˆ ook die weg vir potensie¨le toekomstige uitbreidings na nuwe kategoriee¨. Verder fokus ons navorsing op die skeidingseienskap van (voor)gerwe, en bied ′n kategoriese beskrywing van skeiding deur die bekendstelling van stronkgerwe wat met ′n voorgerf geassosieer word. Ons ondersoek ook die konsep van gerf, met die doel om te gerf gedefinieer kan word as ′n samestelling van limiete binne die kategorie van (voor)gerwe. Alhoewel ons hierdie doelwit suksesvol op plaaslike vlak bereik, bied dit boeiende vooruitsigte vir verdere ondersoek. Benewens hierdie oorspronklike bydraes, bied hierdie tesis ′n uitgebreide en nou- keurige verkenning van die fundamentele beginsels van algebra¨ıese meetkunde, met ′n sentrale klem op kategorie-teorie. Hierdie deel sluit ingewikkelde besonderhede en formaliteite in wat nie geredelik beskikbaar is in bestaande literatuur oor algebra¨ıese meetkunde nie.af_ZA
dc.description.versionDoctorateen_ZA
dc.format.extentxvii, 223 pagesen_ZA
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/130179
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.rights.holderStellenbosch Universityen_ZA
dc.subject.lcshGeometry, Algebraic -- Mathematical modelsen_ZA
dc.subject.lcshTopological spaces -- Mathematical modelsen_ZA
dc.subject.lcshSheaves (Algebraic topology) -- Data processingen_ZA
dc.subject.lcshSheaf theoryen_ZA
dc.subject.lcshGluingen_ZA
dc.subject.lcshCategories (Mathematics)en_ZA
dc.subject.nameUCTDen_ZA
dc.titleExploring some categorical aspects of foundational concepts in algebraic geometryen_ZA
dc.typeThesisen_ZA
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