Internal Languages for Autonomous and *-Autonomous Categories
نویسندگان
چکیده
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-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 >; and (, to witness the left-introduction rules, and a Parigot-style-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 connuent. As a corollary, we solve a Coherence Problem a la Lambek 11]: the equality of maps in any-autonomous category freely generated from a discrete graph is decidable.
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ورودعنوان ژورنال:
- Electr. Notes Theor. Comput. Sci.
دوره 29 شماره
صفحات -
تاریخ انتشار 1999