Call-by-push-value is a ”semantic machine code”, providing a set of simple primitives from which both the call-by-value and call-by-name paradigms are built. We present its operational semantics as a stack machine, suggesting a term judgement of stacks. We then see that CBPV, incorporating these stack terms, has a simple categorical semantics based on an adjunction between values and stacks. There are no coherence requirements. We describe this semantics incrementally. First, we introduce locally indexed categories and the opGrothendieck construction, and use these to give the basic structure for interpreting the three judgements: values, stacks and computations. Then we look at the universal property required to interpret each type constructor. We define a model to be a strong adjunction with countable coproducts, countable products and exponentials. We see a wide range of instances of this structure: we give examples for divergence, storage, erratic choice, continuations, possible worlds and games (with or without a bracketing condition), in each case resolving the strong monad from the literature into a strong adjunction. And we give ways of constructing models from other models. Finally, we see that call-by-value and call-by-name are interpreted within the Kleisli and co-Kleisli parts, respectively, of a call-by-push-value adjunction.

We consider the natural combinations of algebraic computational effects such as side-effects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.’s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubuc’s characterisation of strong monads in terms of enriched large Lawvere theories.

Abstract. This paper presents a unified framework for dealing with a deduction system and a denotational semantics of exceptions. It is based on the fact that handling exceptions can be seen as a kind of generalized case distinction. This point of view on exceptions has been introduced in 2004, it is based on the notion of diagrammatic logic, which assumes some familiarity with category theory. Extensive sums of types can be used for dealing with case distinctions. The aim of this new paper is to focus on the role of a generalized extensivity property for dealing with exceptions. Moreover, the presentation of this paper makes only a