In standard realizability one works with respect to an untyped universe of realizers called a
partial combinatory algebra (pca). It is well known that a pca [Ascr ] gives rise to a categorical
model of impredicative type theory via the category Asm([Ascr ]) of assemblies over
[Ascr ] or the realizability topos over [Ascr ].
Recently, John Longley introduced a typed version of pca's (Longley 1999b). The above
mentioned construction of categorical models extends to the typed case. However, in general
these are no longer impredicative. We show that for a typed pca [Tscr ] the ensuing models are
impredicative if and only if [Tscr ] has a universal type U. Such a type U can be endowed with
the structure of an untyped pca such that U and [Tscr ] induce equivalent realizability models:
in other words, a typed pca [Tscr ] with a universal type is essentially untyped. Thus, a posteriori
it turns out that nothing is lost by restricting to (untyped) pca's as far as realizability models
of impredicative type theories are concerned.
For instance, we show that for a typed pca [Tscr ] the fibred category of discrete families in
Asm([Tscr ]) is small if and only if [Tscr ] has a universal type. As the category of
¬¬-separated objects of the modified realizability topos is equivalent to
Asm([Tscr ]) for an appropriate typed pca [Tscr ] without a universal type, it follows that the
discrete families in the subcategory of ¬¬-separated objects of the modified realizability topos do
not provide a model of polymorphic λ-calculus.