The syntactic dual of autonomous categories enriched over generalised metric spaces

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear {\lambda}-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear {\lambda}-theories based on this V-equational system form a category that is equivalent to a category of autonomous categories enriched over 'generalised metric spaces'. In other words, we prove the existence of a so-called syntax-semantics duality between both structures which are parametrised by V: if we choose linear {\lambda}-calculus based on inequations, we obtain a correspondence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get a correspondence with autonomous categories enriched over (ultra)metric spaces. We also show that this correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.