Assemblies without coproducts?

[Andrej Bauer <andrej.bauer@andrej.com>]

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