A partial analysis of modified realizability

In [3], the following semantical picture of modified realizability is sketched: Consider the topos Set, whose objects are functions between sets, and whose arrows are commutative squares. The logic of Set corresponds to “Kripke forcing over a 2-element linear order”. Over Set, one can define the standard realizability topos Eff→ (not to be confused with Eff!). The modified realizability topos, Mod , then is a closed subtopos of Eff→; more precisely, Mod is the closed complement of Eff in Eff→. This means, that there is a subobject U of 1 in Eff→, or a global section 1 u → Ω, such that the internal topology in Eff→ whose sheaves form Mod , is given by: j(x) = x ∨ u