  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.
