Browsing by Author "Mgani, Damas Karmel"

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    Exploring some categorical aspects of foundational concepts in algebraic geometry
    (Stellenbosch : Stellenbosch University, 2024-03) Mgani, Damas Karmel; Marques, Sophie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    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.