Fo ck Space A Mo del of Linear Exp onential Typ es y Prakash Panangaden RF Blute Scho ol of Computer Science Department of Mathematics McGill University University of Ottawa Montreal Canada HA A Ottawa Canada KN N prakashcsmcgillca rblutemathstatuottawaca z RAG Seely Department of Mathematics McGill University Montreal Canada HA K ragsmathmcgillca Abstract It has b een observed by several p eople that in certain contexts the free symmetric algebra construction can provide a mo del of the linear mo dality This construction arose indep en dently in quantum physics where it is considered as a canonical mo del of quantum eld theory In this context the construction is known as bosonic Fock space Fo ck space is used to analyze such quantum phenomena as the annihilation and creation of particles There is a strong intu itive connection to the principle of renewable resource which is the philosophical interpretation of the linear mo dalities In this pap er we examine Fo ck space in several categories of vector spaces We rst consider vector spaces where the Fo ck construction induces a mo del of the ; ; fragment in the cat egory of symmetric algebras When considering Banach spaces the Fo ck construction provides a mo del of a weakening cotriple in the sense of Jacobs While the mo dels so obtained mo del a smaller fragment it is closer in practice to the structures considered by physicists In this case Fo ck space has a natural interpretation as a space of holomorphic functions This suggests that the nonlinear functions we arrive at via Fo ck space are not merely continuous but analytic Finally we also consider fermionic Fo ck space which corresp onds algebraically to skew sym metric algebras By considering fermionic Fo ck space on the category of nitedimensional vec tor spaces we obtain a mo del of full prop ositional linear logic although the mo del is somewhat degenerate in that certain connectives are equated Intro duction Linear logic was intro duced by Girard G as a consequence of his decomp osition of the traditional connectives of logic into more primitive connectives The resulting logic is more resource sensitive this is achieved by placing strict control over the structural rules of contraction and weakening Research supp orted in part by FCAR of Queb ec y Research supp orted in part by an NSERC Research Grant z Research supp orted in part by an NSERC Research Grant and by FCAR of Queb ec intro ducing a new mo dal op erator of course denoted to indicate when a formula may b e used in a resourceinsensitive mannerie when a resource is renewable Without the op erator the essence of linear logic is carried by the multiplicative connectives at its most basic level linear logic is a logic of monoidalclosed categories in much the same way that intuitionistic logic is a logic of cartesianclosed categories In mo delling linear logic one b egins with a monoidalclosed category and then adds appropriate structure to mo del linear logics additional features To mo del linear negation one passes to the autonomous categories of Barr B To mo del the additive connectives one then adds pro ducts and copro ducts Finally to mo del the exp onentials and so regain the expressive strength of traditional logic one adds a triple and cotriple satisfying prop erties to b e outlined b elow This program was rst outlined by Seely in Se Linear logic b ears strong resemblance to linear algebra from which it derives its name but one signicant dierence is the diculty in mo delling The category of vector spaces over an arbitrary eld is a symmetric monoidal closed category indeed in some sense the prototypical monoidal category and as such provides a mo del of the intuitionistic variant of multiplicative linear logic Furthermore this category has nite pro ducts and copro ducts with which to mo del the additive connectives It thus makes sense to lo ok for mo dels of various fragments of linear logic in categories of vector spaces However mo delling the exp onentials is more problematic It is the primary purp ose of this pap er to present metho ds of mo delling exp onential typ es in categories arising from linear algebra We study mo dels of the exp onential connectives in categories of linear spaces which have monoidal but generally not monoidalclosed structure We shall also include a mo del in nitedimensional vector spaces which is closed The construction which will b e used to mo del the exp onential formulas although standard in algebra arose indep endently in quantum eld theory and is known as Fock space It was designed as a framework in which to consider manyparticle states The key p oint of departure for quantum eld theory was the realization that elementary particles are created and destoyed in physical pro cesses and that the mathematical formalism of ordinary quantum mechanics needs to b e revised to take this into account The physical intuitions b ehind the Fo ck construction will also b e familiar to mathematicians in that it corresp onds to the free symmetric algebra on a space As such it induces a pair of adjoint functors and hence a cotriple in the algebra category It is this cotriple which will b e used to mo del It should b e noted that this category of algebras inherits the monoidal structure from the underlying category of spaces but there is no hop e that this category could have a monoidalclosed structure While Fo ck space has an abstract representation in terms of an innite direct sum physicists such as Ashtekar Bargmann Segal and others see AMA Ba S have analyzed concrete representations of Fo ck space as certain classes of holomorphic functions on the base space Thus these mo dels further the intuition that the exp onentials corresp ond to the analytic prop erties of the space In fact there is a clear sense in which morphisms in the Kleisli category for the cotriple can b e viewed as generalized holomorphic functions Thus there should b e an analogy to coherence spaces where the Kleisli category corresp onds to the stable maps Fo ck space also has two additional features which corresp ond to additional structure not ex pressible in the syntax of linear logic These are the annihilation and creation op erators which are used to mo del the annihilation and creation of particles in a eld These may give a tighter control of resources not expressible in the pure linear logic Thus these mo dels may b e closer to the b ounded linear logic of Girard Scedrov and Scott GSS A p ossible application of this work is that the rened connectives of linear logic may lend insight into certain asp ects of quantum eld theory For example there are two distinct metho ds of combining particle states One can sup erimp ose two states onto a single particle or one can have two particles co existing The former seems to corresp ond to additive conjunction and the latter to the multiplicative This physical imagery is missing in quantum mechanics which was sp ecially designed to handle a single particle it only shows up in quantum eld theory In this pap er we b egin by reviewing the categorical structure necessary to mo del linear logic and sp ecically exp onential typ es We then describ e the Fo ck construction on vector spaces and explain the prop erties of the resulting mo del We next consider Fo ck space on normed vector spaces While the mo del so obtained has weaker prop erties this case is closer to that considered by physicists In fact in this case the Fo ck construction gives a mo del of a weakening cotriple in the sense of Jacobs J We next describ e the interpretation of Fo ck space as a space of Holomorphic functions Finally the physical meaning of the Fo ck construction is discussed We wish to p oint out that this pap er corrects an error in an earlier draft BPS This is discussed in section Linear Logic and Monoidal Categories We shall b egin with a few preliminaries concerning linear logic We shall not repro duce the formal syntax of linear logic nor the usual discussion of its intuitive interpretation or utilityfor this the reader is referred to the standard references such as G We do recall Se that a categorical semantics for linear logic may b e based on Barrs notion of autonomous categories B If only to establish notation here is the denition Denition A category C is autonomous if it satises the fol lowing C is symmetric monoidal closed that is C has a tensor product A B and an internal hom A B which is adjoint to the tensor in the second variable Hom A B C Hom B A C op C has a dualizing object that is the functor C C dened by A A is an involution viz the canonical morphism A A is an isomorphism In addition various coherence conditions must holda go o d account of these may b e found in MOM Coherence theorems may b e found in BCST Bl An equivalent characterization of autonomous categories is given in CS based on the notion of weakly distributive categories That characterization is useful in contexts where it is easier to see how to mo del the tensor ... .. .. .. .. ...... .. and linear negation and the coherence conditions may b e expressed in terms of those .