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We present a novel approach to the concept of gluing in mathematics byintroducing the notions of a gluing index category and a generalized gluingdata functor. Our work provides a formal categorical characterization of thenotion of gluing in algebraic geometry. By using this characterization, we areable to describe gluing in a unified way that applies to a wide range ofmathematical structures, including topological spaces, presheaves, sheaves,ringed topological spaces, locally ringed topological spaces, and schemes. Ourresults provide a fresh perspective on gluing that is both abstract and formal,offering a deeper understanding of this fundamental concept in mathematics.

Gluing index category and generalized gluing data functor