Internal Languages for Autonomous and ∗-Autonomous Categories

Abstract

We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and ∗-autonomous categories, in the same sense that the simply-typed lambda-calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgements are presented in the style of Gentzen's Sequent Calculus. A notable feature is the systematic treatment of naturality conditions by expressing the categorical composition, or cut in the type theory, by explicit substitution. We use let-constructs, one for each of the three type constructors tensor unit, tensor and linear function space, and a Parigot-style mu-abstraction to express the involutive negation. We show that the eight equality and three commutation congruence axioms of the ∗-autonomous type theory characterise ∗-autonomous categories exactly. More precisely we prove that there is a canonical interpretation of the (∗-autonomous) type theories in ∗-autonomous categories which is complete i.e. for any type theory, there is a model (i.e. ∗-autonomous category) whose theory is exactly that. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek: the equality of maps in any ∗-autonomous category freely generated from a discrete graph is decidable.