All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation models, and symmetric models. However, the topos-theoretic analogues of these notions—namely topoi of sheaves on locales, continuous actions of groups, and combinations of the two—are all special cases of one notion, namely the topos of sheaves on a site. Is there anywhere to be found a direct construction, in the classical world of membership-based set theory, of a "forcing model" relative to an arbitrary site?



Set-theoretic forcing over sites?
To the best of my knowledge, this has never been "officially" described in the set theoretic literature. This has been described by Blass and Scedrov in *Freyd's models for the independence of the axiom of choice* (Mem. Amer. Math. Soc. 79, 1989). (It is of course implicit and sometimes explicit in the topoi literature, for example Mac Lane and Moerdijk do a fair bit of the translation in *Sheaves in Geometry and Logic*.) There are certainly a handful of set theorists that are well aware of the generalization and its potential, but I've only seen a few instances of crossover. In my humble opinion, the lack of such crossovers is a serious problem (for both parties). To be fair, there are some important obstructions beyond the obvious linguistic differences. Foremost is the fact that classical set theory is very much a classical theory, which means that the double-negation topology on a site is, to a certain extent, the only one that makes sense for use classical set theory. On the other hand, although very important, the double-negation topology is not often a focal point in topos theory.




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Thanks to the comments by Joel Hamkins, it appears that there is an even more serious obstruction. In view of the main results of Grigorieff in
*Intermediate submodels and generic extensions in set theory*, Ann. Math. (2) 101 (1975), it looks like the forcing posets are, up to equivalence, precisely the small sites (with the double-negation topology) that preserve the axiom of choice in the generic extension.


