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.

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