We show that every geometric morphism between realizability toposes satisfies the condition that its inverse image commutes with the `constant object' functors, which was assumed by John Longley in his pioneering study of such morphisms. We also provide the answer to something which was stated as an open problem on Jaap van Oosten's book on realizability toposes: if a subtopos of a realizability topos is (co)complete, it must be either the topos of sets or the degenerate topos. And we present a new and simpler condition equivalent to the notion of computational density for applicative morphisms of Schonfinkel algebras.
Keywords: realizability topos, geometric morphism, applicative morphism
2010 MSC: Primary 18B25, secondary 03D75
Theory and Applications of Categories, Vol. 28, 2013, No. 9, pp 241-249.