Not enough points is enough

Authors:
Antonio Bucciarelli;Thomas Ehrhard;Giulio Manzonetto
Affiliations:
Laboratoire PPS, CNRS, UMR, Université Paris 7, Paris Cedex 05, France;Laboratoire PPS, CNRS, UMR, Université Paris 7, Paris Cedex 05, France and Institut de Mathématiques de Luminy, CNRS, UMR, Marseille, France;Laboratoire PPS, CNRS, UMR, Université Paris 7, Paris Cedex 05, France and Università Ca'Foscari di Venezia, Dipartimento di Informatica, Venezia, Italia
Venue:
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
Year:
2007

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Abstract

Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bunch of first-order axioms (λ-models), or as reflexive objects in cartesian closed categories (categorical models). In this paper we show that any categorical model of λ-calculus can be presented as a λ-model, even when the underlying category does not have enough points. We provide an example of an extensional model of λ-calculus in a category of sets and relations which has not enough points. Finally, we present some of its algebraic properties which make it suitable for dealing with non-deterministic extensions of λ-calculus.