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Exploring some categorical aspects of foundational concepts in algebraic geometry
Thesis (PhD)--Stellenbosch University, 2024.
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| Format: | Thesis |
| Language: | en_ZA en_ZA |
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Stellenbosch : Stellenbosch University
2024
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| _version_ | 1867613899331731457 |
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| access_status_str | Open Access |
| author | Mgani, Damas Karmel |
| author2 | Marques, Sophie |
| author_browse | Marques, Sophie Mgani, Damas Karmel |
| author_facet | Marques, Sophie Mgani, Damas Karmel |
| author_sort | Mgani, Damas Karmel |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/130179 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:43:28.625Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/130179 Exploring some categorical aspects of foundational concepts in algebraic geometry Mgani, Damas Karmel Marques, Sophie Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Geometry, Algebraic -- Mathematical models Topological spaces -- Mathematical models Sheaves (Algebraic topology) -- Data processing Sheaf theory Gluing Categories (Mathematics) UCTD Thesis (PhD)--Stellenbosch University, 2024. ENGLISH 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. AFRIKAANSE 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. Doctorate 2024-02-26T23:43:48Z 2024-04-26T08:11:09Z 2024-02-26T23:43:48Z 2024-04-26T08:11:09Z 2024-03 Thesis https://scholar.sun.ac.za/handle/10019.1/130179 en_ZA en_ZA Stellenbosch University xvii, 223 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Geometry, Algebraic -- Mathematical models Topological spaces -- Mathematical models Sheaves (Algebraic topology) -- Data processing Sheaf theory Gluing Categories (Mathematics) UCTD Mgani, Damas Karmel Exploring some categorical aspects of foundational concepts in algebraic geometry |
| title | Exploring some categorical aspects of foundational concepts in algebraic geometry |
| title_full | Exploring some categorical aspects of foundational concepts in algebraic geometry |
| title_fullStr | Exploring some categorical aspects of foundational concepts in algebraic geometry |
| title_full_unstemmed | Exploring some categorical aspects of foundational concepts in algebraic geometry |
| title_short | Exploring some categorical aspects of foundational concepts in algebraic geometry |
| title_sort | exploring some categorical aspects of foundational concepts in algebraic geometry |
| topic | Geometry, Algebraic -- Mathematical models Topological spaces -- Mathematical models Sheaves (Algebraic topology) -- Data processing Sheaf theory Gluing Categories (Mathematics) UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/130179 |
| work_keys_str_mv | AT mganidamaskarmel exploringsomecategoricalaspectsoffoundationalconceptsinalgebraicgeometry |