It is well known that, although the category of topological spaces is not cartesian closed, it possesses many cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v).We give a simple and uniform proof of cartesian closedness for many categories of topological spaces, including (ii)-(v), and implicitly (i), and we also give a self-contained proof that (vi) is cartesian closed. Our main aim, however, is to compare the categories (i)-(vi), and others like them.When restricted to Hausdorff spaces, (ii)-(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the cartesian closed structure, finite products coincide in (i)-(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)-(v). Finally, the topologies of the spaces in the categories (i)-(iv) are analysed in terms of Lawson duality.MSC (2000): 54D50, 54D55, 54C35, 06B35.Lemma 2·1. For X exponentiable, the evaluation mappingProof. The transpose of E is the identity map on [X ⇒ Y ], hence continuous, and thus E is continuous since X is exponentiable.Lemma 2·2. For X exponentiable, the exponential topology on C(X, Y ) is uniquely determined.Proof. Suppose that [X ⇒ Y ] and [X ⇒ Y ] are C(X, Y ) endowed with topologies that satisfy the properties of an exponential. By Lemma 2·1 the evaluation map E : [X ⇒ Y ] × X → Y is continuous, and then the identity function on C(X, Y ) from [X ⇒ Y ] to [X ⇒ Y ] is continuous since the latter is an exponential. Reversing the roles of [X ⇒ Y ] and [X ⇒ Y ] gives continuity of the identity in the reverse direction.In light of the preceding observation we denote by [X ⇒ Y ] the set C(X, Y ) of continuous maps endowed with the exponential topology.Lemma 2·3. For X exponentiable and g ∈ C(Y, Z), the mapis continuous by Lemma 2·1, and thus its transpose, which one sees directly to be [X ⇒ g], is continuous.It follows from the preceding that every exponentiable X gives rise to a functor [X ⇒ ·] : Top → Top defined by Y → [X ⇒ Y ] and g → [X ⇒ g]. One observes directly from the fact that [X ⇒ Y ] is an exponential that this functor is right adjoint to the functor · × X. Lemma 2·4. The product of two exponentiable spaces is exponentiable.Proof. Let X 1 and X 2 be exponentiable spaces. Then for all spaces A, one has bijections. Hence the exponential topology on C(X 1 , [X 2 ⇒ Y ]) induces an exponential topology on C(X 1 × X 2 , Y ).Given a fa...

No abstract

This paper introduces the enriched effect calculus, which extends established type theories for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations.The enriched effect calculus is implemented as an extension of a basic effect calculus without linear primitives, which is closely related to Moggi's computational metalanguage, Filinski's effect PCF and Levy's call-by-push-value. We present syntactic results showing: the fidelity of the behaviour of the linear connectives of the enriched effect calculus; the conservativity of the enriched effect calculus over its non-linear core (the effect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched effect calculus.The second half of the paper investigates models for the enriched effect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard effect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.

Abstract. According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects, and are not accounted for by the standard theory of relational parametricity. In this paper, we develop a foundational framework for extending the notion of relational parametricity to programming languages with effects.

We argue that, by supporting a mixture of "compositional" and "structural" styles of proof, sequent-based proof systems provide a useful framework for the formal verification of processes. As a worked example, we present a sequent calculus for establishing that processes from a process algebra satisfy assertions in Hennessy-Milner logic. The main novelty lies in the use of the operational semantics to derive introduction rules, on the left and right of sequents, for the operators of the process calculus. This gives a generic proof system applicable to any process algebra with an operational semantics specified in the GSOS format. Using a general algebraic notion of GSOS model, we prove a completeness theorem for the cut-free fragment of the proof system, thereby establishing the admissibility of the cut rule. Under mild (and necessary) conditions on the process algebra, an ω-completeness result, relative to the "intended" model of closed process terms, follows.