We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.

We formulate a typed version of call-by-value λ-calculus containing variants of Felleisen's control operators A and C that provide explicit access to continuations and logically extend the propositions-as-types correspondence to classical propositional logic. We give an equational theory for this calculus, which is shown to be sound and complete with respect to a class of categorical models based on continuation-passing-style semantics.

In this work we show how some useful reductions known from ordinary intuitionistic propositional calculus can be modified for Intuitionistic Linear Logic (without modalities). The main reductions we consider are: (1) the reduction of the depth of formulas in the sequents by addition of new variables, and (2) the elimination of linear disjunction, tensor and constant F. Both transformations preserve deducibility, that is, a transformed sequent is deducible if and only if the initial one was deducible. The size of the sequent grows linearly in case (1) and ≤ On8 in case (2).

Lambek used categories with indeterminates to capture explicit variables in simply typed λ-calculus. He observed that such categories with indeterminates can be described as Kleisli categories for suitable comonads. They account for ‘functional completeness’ for Cartesian (closed) categories.

Here we refine this analysis, by distinguishing ‘contextual’ and ‘functional’ completeness, and extend it to polymorphic λ-calculi. Since the latter are described as certain fibrations, we are lead to consider indeterminates, not only for ordinary categories, but also for fibred categories. Following a 2-categorical generalisation of Lambek's approach, such fibrations with indeterminates are presented as 'simple slices' in suitable 2-categories of fibrations; more precisely, as Kleisli objects. It allows us to establish contextual and functional completeness results for some polymorphic calculi.

Motivated by a desire to treat non-termination directly in the semantics of computation, the notion of approximation between programs is studied in the context of categories of partial maps. In particular, contextual approximation and specialisation are considered and shown to coincide. Moreover, after exhibiting the approximation between total maps as a primitive notion, from an arbitrary (or axiomatic) approximation order on total maps a computationally natural approximation order on partial maps is derived. The main technical contribution is a characterisation of when this approximation order between partial maps is domain-theoretic (in the sense that the category of partial maps Cpo-enriches) provided that the approximation order between total maps is also.