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The Copenhagen interpretation posits a boundary in the World between the observer and the non-observer (that is the rest of the World). There is knowledge (Observables measured) associated with each boundary.

But one could rerun the same experiment with the boundary elsewhere in the world. In fact one should perhaps take into account all such boundaries. It should be possible to turn this into a topology on the space. Each Open (that is Boundary) will have associated Knowledge which will then vary in some particular way.

This very much reminds me of sheaf-theoretic techniques in Geometry. Is then a sheaf-theoretic interpretation possible or advisable? Has any work been done in this direction?

Of course sheaf-theory has an alternative incarnation as an etale bundle. So if the first is possible then a bundle interpretation should be possible.

The real intent of this question isn't mathematical, but philosophical - that is a certain experiment is run and a priviliged observer is given ontological status. This appears to me not quite right. All possible observers should be given the same status. The question is then how do these local observers become a global one. The global patching of local data is what sheaf-theory is designed to accomplish.

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Most definitely, sheafification occurs all the time in quantum analyses. The classic Consistent Histories approaches are all either directly based on sheaf interpretations or they occur in topoi with sheaf interpretations. In fact, "the global patching of local data" is the point of consistent histories interpretations!

Additionally, any interpretation in monoidal categories (linear logic approaches, operationalist approaches, etc.) will see sheaves take much the same role. See the works of Fotini Markopoulou, Bob Coecke, C. J. Isham, Isar Stubb, Sonja Smets, and company.

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