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Extensional Σ-Spaces in Type Theory


Synthetic Domain Theory provides a setting for denotational semantics following Dana Scott's slogan ‘domains as sets’ in which all functions are continuous. Several approaches can be found in the literature, but they are either model-dependent or if they use an axiomatic setting then not uniformly and not explicitly. We present a completely logical approach to Synthetic Domain Theory (SDT), axiomatizing (complete) Extensional PERs. On these grounds some basic domain theory is developed. Special attention is devoted to admissibility. The axiomatic approach is advantageous since it allows for easy formalization and comparison to other axiomatic settings.

The consistency of the theory is shown by providing an appropriate realizability model. It is discussed how to get from this ‘special kind’ of SDT ‘{à la Scott}’ to a more general form which unifies several approaches.

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Reus, B. Extensional Σ-Spaces in Type Theory. Applied Categorical Structures 7, 159–183 (1999).

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  • domain theory
  • synthetic domain theory
  • type theory
  • constructive logic
  • realizability