Categories of partial equivalence relations as localizations

We construct a category of fibrant objects C〈P〉 in the sense K. Brown from any indexed frame (a kind poset generalizing triposes) P, and show that its homotopy is Barr-exact C[P] partial equivalence relations compatible functional relations. In particular this gives presentation realizability toposes as categories. give criteria for existence left right derived functors to C〈Φ〉:C〈P〉→C〈Q〉 induced by finite-meet-preserving transformations Φ:P→Q between frames.