Introduction to higher order categorical logic
Introduction to higher order categorical logic
Categorical completeness results for the simply-typed lambda-calculus
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
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The λ-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of λ-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is λ-definable. These results subsume earlier ones using cartesian closed categories, as well as those employing so-called Henkin and Kripke λ-models.