Type Theories for Autonomous and *-Autonomous Categories: I. Type Theories and Rewrite Systems - II. Internal Languages and Coherence Theorems (1998)
| Citations: | 5 - 4 self |
BibTeX
@MISC{Koh98typetheories,
author = {T. W. Koh and C. -h. L. Ong},
title = {Type Theories for Autonomous and *-Autonomous Categories: I. Type Theories and Rewrite Systems - II. Internal Languages and Coherence Theorems},
year = {1998}
}
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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 simply-typed -calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian closed categories. The rules are presented in the style of Gentzen's Sequent Calculus. A key feature is the systematic treatment of naturality conditions by explicitly representing the categorical composition, or cut in the type theory, by explicit substitution, and the introduction of new let-constructs, one for each of the three type constructors ?;\Omega and (, and a Parigot-style ¯-abstraction to give expression to the involutive negation. The commutation congruences of these theories are precisely those imposed by the naturality conditions. In particular the type theory for -autonomous categories may be regarded as a term assignment system for the multiplicative (\Omega ; (;?;?)-fragmen...
