Iterated realizability as a comma construction

2008 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
We show that the 2-category of partial combinatory algebras, as well as various related categories, admit a certain type of lax comma objects. This not only reveals some of the properties of such categories, but it also gives an interpretation of iterated realizability, in the following sense. Let φ : A → B be a morphism of PCAs, giving a comma object A φ B. In the realizability topos RT(B) over B, the object (A, φ) is an internal PCA, so we can construct the realizability topos over (A, φ).
more » ... s topos is equivalent to the realizability topos over the comma-PCA A φ B. This result is both an analysis and a generalization of a special case studied by Pitts in the context of the effective monad.
doi:10.1017/s0305004107000400 fatcat:7w2j6rswrfdilihredd5lu7bfa