Effective Operations of Type 2 in Pcas

We exhibit a way of “forcing a functional to be an effective operation” for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new partial functions can be viewed as computable relative to that functional. It is shown that this generalizes a notion of computation relative to a functional as defined by Kleene for the natural numbers. The construction can be used to study subtoposes of the Effective Topos. We will do this for a particular functional that forces every arithmetical set to be decidable. In this paper we also prove the convenient result that the two definitions of a pca that are common in the literature are essentially the same.