Higher-Order Behavioural Conformances via Fibrations

Coinduction is a widely used technique for establishing behavioural equivalence of programs in higher-order languages. In recent years, the rise of languages with quantitative (e.g.~probabilistic) features has led to extensions of coinductive methods to more refined types of behavioural conformances, most notably notions of behavioural distance. To guarantee soundness of coinductive reasoning, one needs to show that the behavioural conformance at hand forms a program congruence, i.e. it is suitably compatible with the operations of the language. This is usually achieved by a complex proof technique known as \emph{Howe's method}, which needs to be carefully adapted to both the specific language and the targeted notion of behavioural conformance. We develop a uniform categorical approach to Howe's method that features two orthogonal dimensions of abstraction: (1) the underlying higher-order language is modelled by an \emph{abstract higher-order specification} (AHOS), a novel and very general categorical account of operational semantics, and (2) notions of behavioural conformance (such as relations or metrics) are modelled via fibrations over the base category of an AHOS. Our main result is a fundamental congruence theorem at this level of generality: Under natural conditions on the categorical ingredients and the operational rules of a language modelled by an AHOS, the greatest behavioural (bi)conformance on its operational model forms a congruence. We illustrate our theory by deriving congruence of bisimilarity and behavioural pseudometrics for probabilistic higher-order languages.