Exploring some categorical aspects of foundational concepts in algebraic geometry.
After presenting the outline of the entire thesis, we will shift our focus to a categorical perspective on gluing. We will guide you through the process of extracting the gluing index category and the concept of a gluing data functor from the conventional gluing topological data. Additionally, we will introduce the notion of a refinement morphism for gluing data functors, allowing us to combine and “compose” these functors. Through a concrete example, we will illustrate how these gluing processes can be applied. We will then discuss all the categories in which we have demonstrated the existence of glued-up objects. Finally, we will conclude the talk by demonstrating how gluing data functors define coverings that establish a Grothendieck topology on the category of topological spaces.