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Linear Logic, Autonomous Categories and Cofree Coalgebras
 In Categories in Computer Science and Logic
, 1989
"... . A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a c ..."
Abstract

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. A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a closed involution. Girard's exponential operator, ! , is a cotriple on G which carries the canonical comonoid structure on A with respect to cartesian product to a comonoid structure on !A with respect to tensor product. This makes the Kleisli category for ! cartesian closed. 0. INTRODUCTION. In "Linear logic" [1987], JeanYves Girard introduced a logical system he described as "a logic behind logic". Linear logic was a consequence of his analysis of the structure of qualitative domains (Girard [1986]): he noticed that the interpretation of the usual conditional ")" could be decomposed into two more primitive notions, a linear conditional "\Gammaffi" and a unary operator "!" (called "of cours...