Trying to get my head around the interplay between the locality and gluing axioms in a sheaf. In brief, and given a metaphorical association of the “more global” with the “above” and of the “more local” with the “below”:
The locality axiom means that “the below determines the (identity of the) above”: whenever two sections over an open set U are indistinguishable based on their restrictions to the sections over any open cover of U, they are the same. There is no way for data that are more-locally the same to correspond to data that are more-globally different. Our view can be enriched as we move from the global to the local, but not the other way around.
The gluing axiom means that “the above determines the (coherence of the) below”: each compatible-in-pairs way of gluing together the sections over an open cover of U has a representative among the sections over U, of which the sections in the glued assemblage are the restrictions. There is no coherent more-local assemblage that does not have such a more-global representation. The global provides the local with its law, indexing its coherence.
A theme of postmodernism, and particularly of Lyotard’s treatment of the postmodern, was “incommensurability”. Between distinct local practices – language games – there is no common measure, no universal metalanguage into which, and by means of which, every local language can be translated. The image of thought given by sheaves does not contradict this, but it complicates it. The passage from the local to the global draws out transcendental structure; the passage from the global to the local is one of torsion, enrichment, discrimination. The logics of ascent and descent are linked: we cannot “go down” into the local without spinning a web of coherence along the way, and we cannot “come up” into the global without obeying a strict rule of material entailment.