Iterated realizability as a comma construction

• Autores: Pieter J. W. Hofstra
• Localización: Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, Vol. 144, Nº 1, 2008, págs. 39-51
• Idioma: inglés
• DOI: 10.1017/s0305004107000400
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• Resumen

  • 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 f: A ? B be a morphism of PCAs, giving a comma object A f B. In the realizability topos RT(B) over B, the object (A, f) is an internal PCA, so we can construct the realizability topos over (A, f). This topos is equivalent to the realizability topos over the comma-PCA A f B. This result is both an analysis and a generalization of a special case studied by Pitts in the context of the effective monad.