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I've recently read a paper about sheaf forcing in which a sheaf of cumulative hierarchies was defined (defintion 5.3 on page 30). The same object is described in this English paper (defintion 3.1 on page 16). There was however no mention of the idea behind it's construction.

I would like to know more about this cumulative hierarchy of variable sets, but I don't quite know where to look. I suspect a more general version of the object can be found in Mac Lane and Moerdijk's "Sheafs in Geometry and Logic". Is this true and are there other treatments of this subject?

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Ick. This looks like a terrifying hybrid of the cumulative hierarchy of set theory and sheaf theory. It is indeed based on forcing techniques (more precisely, boolean-valued models) from set theory, but you won't find it in Mac Lane and Moerdijk's book. –  Zhen Lin Dec 21 '12 at 1:36

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