Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rule-based definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for first-order calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the π-calculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxt-like rule format for open bisimulation in the π-calculus.

Abstract. Nominal techniques are based on the idea of sets with a finitelysupported atoms-permutation action. We consider the idea of nominal renaming sets, which are sets with a finitelysupported atoms-renaming action; renamings can identify atoms, permutations cannot. We show that nominal renaming sets exhibit many of the useful qualities found in (permutative) nominal sets; an elementary sets-based presentation, inductive datatypes of syntax up to binding, cartesian closure, and being a topos. Unlike is the case for nominal sets, the notion of names-abstraction coincides with functional abstraction. Thus we obtain a concrete presentation of sheaves on

Abstract. We introduce a novel variant of logical relations that maps types not merely to partial equivalence relations on values, as is commonly done, but rather to a proof-relevant generalisation thereof, namely setoids. The objects of a setoid establish that values inhabit semantic types, whilst its morphisms are understood as proofs of semantic equivalence. The transition to proof-relevance solves two well-known problems caused by the use of existential quantification over future worlds in traditional Kripke logical relations: failure of admissibility, and spurious functional dependencies. We illustrate the novel format with two applications: a direct-style validation of Pitts and Stark’s equivalences for “new ” and a denotational semantics for a region-based effect system that supports type abstraction in the sense that only externally visible effects need to be tracked; non-observable internal modifications, such as the reorganisation of a search tree or lazy initialisation, can count as ‘pure ’ or ‘read only’. This ‘fictional purity ’ allows clients of a module soundly to validate more effect-based program equivalences than would be possible with traditional effect systems. 1