The Sierpinski Object in the Scott Realizability Topos

Jaap van Oosten, Tom de Jong

Research output: Contribution to journalArticleAcademicpeer-review


We study the Sierpinski object Σ in the realizability topos based on Scott's graph model of the λ-calculus. Our starting observation is that the object of realizers in this topos is the exponential ΣN, where N is the natural numbers object. We define order-discrete objects by orthogonality to Σ. We show that the order-discrete objects form a reflective subcategory of the topos, and that many fundamental objects in higher-type arithmetic are order-discrete. Building on work by Lietz, we give some new results regarding the internal logic of the topos. Then we consider Σ as a dominance; we explicitly construct the lift functor and characterize Σ-subobjects. Contrary to our expectations the dominance Σ is not closed under unions. In the last section we build a model for homotopy theory, where the order-discrete objects are exactly those objects which only have constant paths.
Original languageEnglish
Pages (from-to)1-16
JournalLogical Methods in Computer Science
Issue number3
Publication statusPublished - 20 Aug 2020


  • Computer Science
  • Logic in Computer Science,Mathematics
  • Logic,68Q05, 18B25


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