Logical systems I: Lambda calculi through discreteness

Abstract

This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently (co)complete non-degenerate categories. As a simple corollary, we obtain a variant of Freyd theorem for categories internal to any tensored category. Also, with help of introduced concept of an associated category, we prove a representation theorem relating our internal models with well-studied fibrational models for polymorphism.

Logical systems II: Free semantics

“…Note however, that in such a case g need not be uniquely determined by f . Just like in [1] we provided an elementary description of pointwise Kan extensions, we shall now give a similar characterisation of absolute Kan liftings. Let us extend the diagram of a Yoneda triangle η : y ⊲ f, g , by taking generalised…”

“…There is another, more abstract, road to Day convolution for internal categories. Recall from [1] (Section 4, Definition 18) that if F : C → W is a functor from a 1-category C to a 2-category W, then the F -externalisation fam F (A) of an object A ∈ W is defined to be the functor: hom W (F (−), A) : C op → Cat For example, in Theorem 1, fam(A) is an F -externalisation of an object (i.e. internal category) A ∈ cat(C).…”

“…In some cases, however, the definition of an adjunction is too restrictive. The following example is taken from [1].…”

“…Let us consider a category consisting of two objects -the set of natural numbers N, together with partially recursive functions, and a singleton 1, with all singleton maps 1 → N. This category is cartesian with binary products given by any effective pairing. It is not, however, cartesian closed -the evaluation cannot be partially recursive -was it, one could test for equality of two partially recursive functions by checking the equality of their corresponding natural numbers 1 . On the other hand, it is "closed" in the category of Π 0 2 functions in the sense of arithmetic hierarchy.…”

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This paper is a sequel to [1]. It provides a general 2-categorical setting for extensional calculi and shows how intensional and extensional calculi can be related in logical systems.

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