Gluing Data categories and gluing data functors

We present a novel approach to the concept of gluing in mathematics by introducing the notions of a gluing data category and a gluing data functor. Our work provides a formal categorical characterization of the notion of gluing in algebraic geometry. By using this characterization, we are able to describe gluing in a unified way that applies to a wide range of mathematical structures, including topological spaces, presheaves, sheaves, ringed topological spaces, locally ringed topological spaces, and schemes. Our results provide a fresh perspective on gluing that is both abstract and formal, offering a deeper understanding of this fundamental concept in mathematics.