TITLE: Set-theoretic forcing over sites?
QUESTION [18 upvotes]: 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?

REPLY [6 votes]: At  http://www.mathematik.tu-darmstadt.de/~streicher/forcizf.pdf  one can find a paper describing a forcing semantics for IZF in Grothendieck toposes Sh(C,J)
in terms of the site (C,J) where C is a Grothendieck topology on a small category C.
As mentioned in the paper for the case of presheaf toposes this was done already by D. Scott before. In my forcing interpretation the "names" for sets
are coming from the cumulative hierarchy in the presheaf topos. Sheafification is built into the clauses of the forcing semantics. Instantiating C by a poset gives the usual "names" and when taking for J the
double negation topology on C we get Cohen forcing.
In this context it might be worthwhile to mention that boolean Grothendieck
toposes are precisely as subtoposes of presheaf toposes via the double negation topology. This is due to Freyd and is also described in Blass and Scedrov's AMS Memoir. In this booklet they relate Freyd's models refuting AC
to the more traditional symmetric boolean valued models as employed by Cohen for refuting (even countable) AC. Freyd's first model coincides with the one devised by Cohen but Freyd's second modelwasn't considered before in the set-theoretic literature.
It might be worth mentioning in this context the famous result of Peter Freyd that every Grothendieck topos appears as an exponential variety within
a topos boolean over the Schanuel topos (the generic permutation model used
by A. Pitts in his work on "Nominal Sets"). So Grothendieck toposes are not that different from models of set theory...
Thomas
PS My above mentioned article has appeared in the Festschrift for my good old friend Mamuka Jibladze.