This question is mostly for the realizabilitologists on the list. Let A be a PCA. The category of assemblies (or pers) over A has finite coproducts because any PCA contains true, false, and if-then-else. Let now A be a typed PCA (TPCA), according to John Longley. This means we have a non-empty set of types, operations * and -> on types (not necessarily freely generating the types). For each type t we have a set of values A_t. We require the K and S combinators to exist, as well as pairing and projections. We do NOT require that there be a boolean type, or a type of natural numbers. Some examples of TPCAs: - finite sets, with * and -> interpreted as cartesian product and exponential - Goedel's T - countably-based algebraic lattices - any PCA A where the type structure is then trivial and A_t = A. Assemblies over a TPCA are formed like the usual assemblies, except we have to specify underlying types. An assembly (S,t,|=) is a set S with a type t and a realizability relation |= between S and A_t. Now, do assemblies over a tpca A have binary coproducts? If A contains a type which resembles the booleans, we can do it. But I don't see how to do it in general. It's probably a trick involving higher-order functions. Andrej