Linear Logic, -Autonomous Categories and Cofree Coalgebras

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 nite 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], Jean-Yves 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 \?" and a unary operator \!" (called \of course"), which is formally rather like an interior operator: (1) The purpose of this note is to answer two questions (and perhaps pose some others.) First, if \linear category" means the structure making valid the proportion linear logic : linear category = typed-calculus : cartesian closed category then what is a linear category? This question is quite easy, and in true categorical spirit, one nds that it was answered long before being put, namely by Barr 1979]. Our intent here is mainly to supply a few details to make the matter more precise (though we leave many more details to the reader), to point out some similarities with work of Lambek 1987] (see these proceedings), and to appeal for a change in some of the notation of Girard 1987]. Second, what is the meaning of Girard's exponential operator ! ? Since Girard has in fact ooered several variants of ! in 1987], and another in Girard and Lafont 1987], one cannot be too dogmatic here, but some certainty as to the minimal demands ! makes is possible | in particular we show that ! ought to be a cotriple, and its Kleisli category ought to be cartesian closed, in order to capture the initial motivation of the exponential. Acknowledgement. This note should be regarded as a \gloss" on Girard 1987], providing the categorical context and terminology for that work; I think the categorical setting provides a genuine improvement, and in particular, indicates how the notation may be made clearer. Others …