Explicit Substitution Internal Languages for Autonomous and *-Autonomous Categories


We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that the simplytyped -calculus with surjective pairing is the internal language for cartesian closed categories. The typing rules are given in the style of Gentzen's Sequent Calculus, and our type theory may be regarded as a term assignment system for the sequent calculus of the multiplicative ( ; ( ; ?; 1)-fragment of classical linear logic. We show that the eight equality and three commutation congruence axioms of the -autonomous type theory characterise -autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also conuent. As a immediate corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any -autonomous category freely generated from a discrete graph is decidable

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