A Categorical Perspective on Gluing

This paper introduces the concept of gluing in a general category, enabling us to define categories that admit glued-up objects. To achieve this, we introduce the notion of a gluing index category. Subsequently, we provide an entirely abstract definition of a gluing data functor requiring only the given category to admit pushouts. We explore various characterizations of cones and limits over these functors. We introduce the concept of refined gluing, which in turn enables us to combine different gluing data effectively. Furthermore, we demonstrate that several categories of topological spaces admit glued-up objects. This, in turn, allows us to establish a concept of gluing covering and to prove that the collection of those coverings forms a Grothendieck topology.