It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.

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Abstract. We introduce the notion of effectoid as a way of axiomatising the notion of “computational effect”. Guided by classical algebra, we define several effectoids equationally and explore their relationship with each other. We demonstrate their computational relevance by applying them to global exceptions, partiality, continuations, and global state. 1

We investigate the duality between call-by-name recursion and call-by-value iteration on the -calculi. The duality between call-by-name and call-by-value was first studied by Filinski, and Selinger has studied the category-theoretic duality on the models of the call-by-name -calculus and the call-by-value one. We extend the call-by-name -calculus and the call-by-value one with a fixed-point operator and an iteration operator, respectively. We show that the dual translations constructed by Selinger can be expanded into our extended -calculi, and we also discuss their implications to practical applications.

We propose an axiomatization of fixpoint operators in typed call-by-value programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform T-fixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinski's fixpoint operator derived from an iterator (infinite loop constructor) in the presence of firstclass continuations, provided that we define the uniformity principle on such an iterator via a notion of effect-freeness (centrality). We then explain how these two results are related in terms of the underlying categorical structures.

We investigate continuations in the context of idealized call-by-value programming languages. On the semantic side, we analyze the categorical structures that arise from continuation models of call-by-value languages. On the syntactic side, we study the call-by-value continuation-passing transformation as a translation between equational theories. Among the novelties are an unusually simple axiomatization of control operators and a strengthened completeness result with a proof based on a delaying transform.