A Categorical Perspective on Gluing
arxiv(2024)
摘要
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.
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