Categories of partial equivalence relations as localizations

Abstract

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