LINEAR LOGIC AUTONOMOUS CATEGORIES 1 AND COFREE COALGEBRAS RAG Seely ABSTRACT A brief outline of the categorical characterisation of Girards linear logic is given analagous to the relationship b etween cartesian closed cat egories and typed calculus The linear structure amounts to a autonomous category a closed symmetric monoidal category G with nite pro ducts and a closed involution Girards exp onential op erator is a cotriple on G which carries the canonical comonoid structure on A with resp ect to cartesian pro duct to a comonoid structure on A with resp ect to tensor pro duct This makes the Kleisli category for cartesian closed INTRODUCTION In Linear logic JeanYves Girard introduced a logical system he describ ed as a logic b ehind logic Linear logic was a consequence of his analysis of the structure of qualitative domains Girard he noticed that the interpretation of the usual conditional could b e decomp osed into two more primitive notions a linear conditional and a unary op erator called of course which is formally rather like an interior op erator X Y X Y The purp ose of this note is to answer two questions and p erhaps p ose some others First if linear category means the structure making valid the prop ortion 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 b efore b eing put namely by Barr 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 p oint out some similarities with work of Lambek see these pro ceedings and to app eal for a change in some of the notation of Girard Second what is the meaning of Girards exp onential op erator Since Girard has in fact oered several variants of in and another in Girard and Lafont one cannot b e to o dogmatic here but some certainty as to the minimal demands makes is p ossible in particular we show that ought to b e a cotriple and its Kleisli category ought to b e cartesian closed in order to capture the initial motivation of the exp onential This is already implicit in equation 1 Mathematics Subject Classication Revision G A Partially supp orted by grants from Le Fonds FCAR Queb ec RAG Seely Acknowledgement This note should b e regarded as a gloss on Girard 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 b e made clearer Others have come to similar conclusions elsewhere in this volume De Paiva considers these matters giving a fuller discussion of the interpretation of A as the cofree commutative comonoid over A in the context of Dialectica categories I would like to thank Michael Barr for p ointing out that he had considered the essence of linear categories in thus giving further evidence of the unreasonable inuence of category theory in mathematics LINEAR LOGIC There are several variations in the style Girard uses to present linear logic eg one sided sequents in and traditional sequents in Girard and Lafont I think the essence of the structure esp ecially its symmetry is clearest when sequents in the style of Szab os and Lambeks p olycategories Szabo are used here a sequent has the form A A A B B B 1 2 n 1 2 m Of course formally this is just an ordered pair of nite sequences actually sets would do of formulas The commas on the left should b e thought of as some kind of conjunction those on the right disjunction Better think of the A on the left as data i each to b e used exactly once and of the B on the right as p ossible alternate resp onses j Definition A prop ositional linear logic consists of formulas and sequents For mulas are generated by the binary connectives and and by the unary op eration from a set of constants including I and and from variables Sequents consist of ordered pairs of nite sequences of formulas as ab ove actually nite sets of formulas would b e b etter in view of perm b elow but let us pass over this p oint The sequents are generated by the following rules from initial sequents ie axioms which include the following Greek capitals represent nite sequences sets of formulas Axioms id A A A IR I L R L 1 d A A d A A Rules perm for any p ermutations A A cut Linear Logic and Categories A B v ar B A R IL I A B A B L R A B A B A B A B R L A B A B A B A B L R A B A B A B A B L R A B A B A B A B A B L R A B A B A B Remarks Concerning notation In Girard a somewhat dierent notation is used I have made changes so as to use wherever p ossible notation that is standard from a categorical viewp oint This table summarises the changes ? Girard A P or t or u Here A I Symbols not changed I b elieve it is more imp ortant to pair the connectives by de Morgan duality with with than by distributivity considerations as would justify Girards t with u Furthermore and seem to really b e cartesian pro duct and categorical sum so those symbols seem more appropriate than Girards particularly his I must confess to b eing unable to nd an entirely satisfactory notation for the de Morgan dual to tensor pro duct either in words dual tensor seems preferable to cotensor or tensor sum or to Girards par or in symbols has b een chosen for its neutrality might have b een b etter were it not already so widely in use elsewhere The following sequents may b e derived m A B A B A B A B AB AB e A A B B AB 1 1 A B A A A B AB AB 2 2 B A B A B B AB AB s A B B A AB 1 a A B C A B C a A B C A B C AB C AB C 0 0 and similar sequents s a for RAG Seely A B B A AB It is easy to see that m is equivalent to R w to L e AB AB AB to L to L to R in the presence of cut Indeed the AB AB rules amount to building in the required amount of cut to allow cutelimination to go through As for symmetry and asso ciativity these follow from the rule perm and the implicit asso ciativity of concatenation We give a as an illustration AB C (m) (m) B C B C A B C A B C cut per m A B C A B C L C A B A B C C A B A B C per m L A B C A B C (a) A B C A B C As for it is given by AB (e) A A B B v ar per m R per m B A B A A B B A If we are to characterize the notion of a linear category we must complete the description of linear logic as a deductive system in the sense of Lambek and Scott First we must add the equations b etween derivations of sequents needed to get the structure of a p olycategory Szabo these equations essentially make cut into a p olycomp osition of p olyarrows which is asso ciative partially commutative and has units id Analagous equations for multicategories may b e found in this A volume in Lambek for this reason I will not go into detail here for these or the remaining equations Next we must account for the monoidal structure of I and their duals by adding equations which make sequents A A B B 1 n 1 m equivalent to sequents A A B B Clearly there are maps using 1 n 1 m the evident hom notation A A B B A A B B 1 n 1 m 1 n 1 m given by the rules L R L R cut the p oint is that these maps b e isomorphisms and inverse to each other Similarly it is likely that we want the structure to b e symmetric monoidal closed and have nite pro ducts and copro ducts each of these adds to the list of equations in 1 the evident way For instance a and a must b e inverse as must s and s AB BA This last could b e weakened if we only want a braided monoidal category as in Joyal and Street However this would complicate the rest of the structure so we shall not pursue this further Moreover R should give a bijection Linear Logic and Categories A B A B whose inverse is (e) A B A A B B cutperm A B Finally it seems that is a contravariant functor in view of v ar that it is strong in view of and that it is an involution in view of d which thus must AB 1 b e inverse to d These yield further equations including the following if we are to have a autonomous category as dened in Barr for any A B these derivations 1 of the sequent A B A B are equal d d Here :B :A AB 1 d d is a case of the schema (f ) (g ) C A B D (f g ) A B C D given by (e) (g ) A A B B B D cut (f ) C A A A B D cut R C A B D A B C D and is a case of cut viz in general :B :A AB (f ) (g ) A B B C cut (g )(f ) A C Since the required equations may b e easily generated from the ab ove recip e and are in essence to b e found in the references given for the most part and since this pro cess is familiar for instance to that of Lambek and Scott for calculus I shall avoid the messy notational baggage needed to make all the details explicit by stating b oldly and without discussion Definition A linear category G is a autonomous category with nite pro ducts Remarks For a fuller discussion of autonomous categories see Barr Here just let me say that G is a closed symmetric monoidal category G with an involution op G G given by a dualising ob ject in our notation this means A A and the canonical arrow A A is an isomorphism Barr uses for our In such a category the existence of nite copro ducts follows from nite pro ducts by de