Categorical Glueing and Logical Predicates for Models of Linear Logic

Transcription

1 Categorical Glueing and Logical Predicate for Model of Linear Logic Maahito Haegawa Reearch Intitute for Mathematical Science, Kyoto Univerity January 1999 Abtract We give a erie of glueing contruction for categorical model of fragment of linear logic. Specifically, we conider the glueing of (i) ymmetric monoidal cloed categorie (model of Multiplicative Intuitionitic Linear Logic), (ii) ymmetric monoidal adjunction (for interpreting the modality!) and (iii) -autonomou categorie (model of Multiplicative Linear Logic); the glueing contruction for -autonomou categorie i a mild generalization of the double glueing contruction due to Hyland and Tan. Each of the glueing technique can be ued for creating intereting model of linear logic. In particular, we ue them, together with the free ymmetric monoidal cocompletion, for deriving Kripke-like parameterized logical predicate (logical relation) for the fragment of linear logic. A an application, we how full completene reult for tranlation between linear type theorie. 1

2 Content 1 Introduction 3 2 Preliminarie Symmetric Monoidal Structure Categorical Glueing Glueing Symmetric Monoidal Structure Glueing Symmetric Monoidal Cloed Categorie Glueing Symmetric Monoidal Adjunction Glueing -Autonomou Categorie Logical Predicate for Linear Logic Parameterized Predicate Logical 0-Predicate Binary Logical 0-Relation ual Intuitionitic Linear Logic Multiplicative Linear Logic Fully Complete Tranlation A Cae Study: From MILL to MLL Full Completene, Semantically Example Reference 28 A Syntax and Semantic of MILL 31 A.1 Syntax of MILL A.2 Semantic of MILL B Syntax and Semantic of ILL 32 B.1 Syntax of ILL B.2 Semantic of ILL C Sharing Theorie (Action Calculi) 34 C.1 Syntax C.2 Semantic C.3 Tranlation into ILL C.4 Control Operator and Proof of Fullne Proof of Propoition E Proof of Propoition

3 1 Introduction Logical predicate (logical relation, reducibility method) have been a powerful tool for proving both yntactic and emantic reult on intuitionitic type theorie. In particular, ince Plotkin work [38], a ubtantial tudy of characterizing the definability on model of the imply typed lambda calculu (and related typed language uch a PCF) uing logical predicate ha been carried out, ee for intance [26, 37, 3, 17]. From the category-theoretic point of view, it i known that a etting for logical predicate for the imply typed lambda calculu can be derived from the categorical glueing contruction (alo known a coning and Freyd covering) on carteian cloed categorie [29, 36]. In term of categorical logic, a glueing i to contruct a category of predicate from a (codomain or ubobject) fibration by a change-of-bae [24]. For carteian cloed categorie, it uffice to aume that the change-of-bae functor preerve finite product for making the glued category carteian cloed. In uch cae, the category of predicate on a model of the lambda calculu again give a model, and moreover the projection to the original model preerve the tructure. Thi obervation allow u to derive the Baic Lemma for logical predicate on categorical model of the imply typed lambda calculu [36]. Thi paper develop an analogou tory for fragment of linear logic [20]. We firt invetigate the glueing technique for ymmetric monoidal tructure which erve a category-theoretic model of linear logic. Specifically, we conider the glueing of 1. ymmetric monoidal cloed categorie (model of Multiplicative Intuitionitic Linear Logic), 2. ymmetric monoidal adjunction (for interpreting the modality!), and 3. -autonomou categorie (model of Multiplicative Linear Logic). The glueing contruction for -autonomou categorie i a mild generalization of the double glueing contruction due to Hyland and Tan [42]. Each of them can be ued for creating intereting model of linear logic. For intance, though not central for our development, we demontrate how phae emantic [21] and it variant can be derived ytematically from the glueing technique (Example 3.6, 3.9, 3.11, 3.18 and 3.23). Then we are ready to introduce a notion of logical predicate for model of linear logic. The predicate we introduce are parameterized, in the ame way a the Kripke-logical relation [3]; the role of parameterization i eential in dealing with connective of linear logic, epecially the multiplicative and modalitie, roughly by the following reaon. Suppoe that we have a predicate P b A b for each bae type b, where A σ i a et in which the cloed term of type σ are interpreted. A the tandard logical predicate, we hope to define a predicate P σ A σ for every type σ in an inductive way. However, we oon face a difficulty in contructing P σ τ from P σ and P τ. The naive contruction P σ τ a b a P σ b P τ make ene but can mi ome intereting undecompoable element of A σ τ; in particular aume a contant of type σ τ, then it interpretation may not belong to P σ τ for any P σ and P τ. The ame trouble appear when we contruct P!σ from P σ. We olve thi problem by parameterizing the predicate on a ymmetric monoidal category which pecifie a property cloed under the linear tructural contruction, o that the parameter indicate the linearly ued reource (or the linear context). Such parameterized predicate give rie to a model of the fragment of linear logic, and erve a a bai for contructing logical predicate. The problem oberved above diappear if each intereting element atifie the property. Thee parameterized logical predicate are derived from the glueing contruction together with the free ymmetric monoidal cocompletion [25] on ymmetric monoidal categorie. A mentioned above, it i known that a etting for tandard logical predicate can be obtained by glueing a carteian 3

4 cloed category to Set; our i derived by glueing a ymmetric monoidal cloed category to the op preheaf category Set 0 (free ymmetric monoidal cocompletion) of a mall ymmetric monoidal category 0 which play the role of world in the Kripke emantic [35]. A a conequence of thee obervation, we how full completene reult for tranlation between linear type theorie equivalently the fullne of the embedding into relatively free ymmetric monoidal tructure. Thi i carried out by contructing a logical predicate which pecifie the element of (the term model of) the target type theory definable in the ource theory, and then by appealing to the Baic Lemma. From a more application-oriented point of view, it might be fruitful to adapt our method to reaoning about the propertie of programming language. For example, the complexity-parameterized logical relation ued in [19] for howing the afety of a type-directed compilation with repect to the time complexity eem to have ome common idea with our parameterized logical predicate. Another intereting direction i to combine our approach to other technique of pecifying propertie of emantic categorie, for intance that of pecification tructure [1]. We alo note that an application of parameterized logical predicate for model of full propoitional claical linear logic i found in Streicher work [41] (ee Example 3.9). Some of the reult in thi paper have appeared in a preliminary verion [23], where only the cae of intuitionitic linear type theorie are dicued. Contruction of Thi Paper In Section 2 we review ome baic concept of ymmetric monoidal categorie and related tructure which will be ued throughout thi paper, and alo recall the definition of the categorical glueing. Section 3 decribe a erie of glueing contruction for ymmetric monoidal tructure, with everal example. In Section 4 we apply the glueing contruction for deriving a notion of parameterized logical predicate for three fragment of linear logic. A a direct application, in Section 5 we how the full completene of tranlation between linear type theorie. Appendice give ome yntactic detail of the linear type theorie a well a the proof of two reult in Section 3. Acknowledgement I thank Gordon Plotkin for dicuion at the initial tage of thi work. Thank are alo due to Martin Hyland and Audrey Tan, for introducing me to their work on double glueing. 2 Preliminarie 2.1 Symmetric Monoidal Structure While we will heavily make ue of concept related to (ymmetric) monoidal categorie, ome of them are given everal name in the literature; for avoiding poible confuion, here we ummarie the notion and terminology to be ued in thi paper. Many of them are found in the claical reference (e.g. [16]), in everal article on model of linear logic (e.g. [6, 11]), and alo in the econd edition of Mac Lane book [33]. A monoidal category I a l r conit of a category, a functor : (called the monoidal or tenor product), an object I (the unit object) and natural iomorphim a A B C : A B C A B C, l A : I A, the following two diagram commute: A and r A : A I 4 A uch that, for object A B C

5 A B C A A a B C a B C a A B A a A B a C C a A I B A r B A l A B I B A ymmetry for a monoidal category i a natural tranformation c A B : A B B A ubject to the following two commutative diagram: a c A B C A B C B C A A B c C B A C B a A C B B c a C A c A B B A A B A monoidal category equipped with a ymmetry i called a ymmetric monoidal category. A ymmetric monoidal cloed category i a ymmetric monoidal category uch that the functor A : ha a right adjoint for each object A; we often write A for a pecified right adjoint functor, and call it the exponent. Alo we write A B : A B A B for the counit of the adjunction, and Λ : B A C B A C for the bijection. For monoidal categorie I a l r and I a l r, a monoidal functor from to i a tuple F m m I where F i a functor from to, m i a natural tranformation from F F to F and m I : I FI i an arrow in, atifying the coherence condition below. FA FB m FC F A B FC m FC F A B C c FA a FB I FC FA FA m l FA FA F B C F A FA m I r FA Fa B C m I FA FI Fl FA m I FA m F I A FA FI m F A I Let I a l r c and I a l r c be ymmetric monoidal categorie. A ymmetric monoidal functor from to i a monoidal functor F m m I which additionally atifie the following condition: FA m FB c FB m FA F A B F B A Fc 5 Fr

6 Compoition of (ymmetric) monoidal functor i guaranteed by the following obervation [16]. Lemma 2.1 Given (ymmetric) monoidal functor F m m I : and G n n I :, G n n I F m m I G F G m n F F G m I n I i alo a (ymmetric) monoidal functor from to. Thi compoition i aociative, and atifiethe identity law for the identity (ymmetric) monoidal functor. A monoidal functor F m m I i trong, if m i a natural iomorphim and m I i an iomorphim; trict, if all component of m and m I are identitie. A trict ymmetric monoidal functor between ymmetric monoidal cloed categorie (with pecified exponent) i cloed if it preerve the exponent a well a the unit and counit. Given monoidal functor F m m I, G n n I with the ame ource and target monoidal categorie, a monoidal natural tranformation from F m m I to G n n I i a natural tranformation ϕ : F G uch that the following diagram commute: FA ϕ ϕ GA m FB F A B ϕ GB G A B n FI I m I A (ymmetric) monoidal adjunction between (ymmetric) monoidal categorie i an adjunction in which both of the functor are (ymmetric) monoidal and the unit and counit are monoidal natural tranformation. The following reult i tandard (Kelly [27]): Propoition 2.2 The left adjoint part of a monoidal adjunction i trong. Converely, if a trong (ymmetric) monoidal functor ha a right adjoint, then the adjunction i (ymmetric) monoidal. A -autonomou category [7, 8] i a ymmetric monoidal category equipped with a fully faithful functor : op uch that there exit a natural iomorphim A B C A B C. A -autonomou category i cloed becaue we have A B C A B C. Alo it i eay to verify A A and A B B A. If i -autonomou o i op, for unit ( fale ) I and tenor ( par ) A B A B. Alternatively, we can pecify a -autonomou category a a ymmetric monoidal cloed category with a dualiing object, i.e. an object uch that the canonical morphim A A i an iomorphim for any A. 2.2 Categorical Glueing Here we recall the notion of the categorical glueing contruction which will be ued throughout thi paper. See Section 7.7 of Taylor forthcoming book [43] for a comprehenive urvey on propertie, uage and hitorical remark on the glueing contruction. Given categorie, and with functor F : and G :, we write F G for the comma category [33] whoe object i a triple A B f : FA GB and an arrow from ϕ n I GI 6

7 A B f to A B f i a pair a : A A b : B B uch that the following diagram commute. Fa FA FA f f In the equel, we will be intereted in the comma categorie of the form G for a functor G :. An object of G may be written a C f : GC. An arrow from C f to C f i then a pair d : c : C C atifying Gc f f d. We note that there i an obviou projection functor p : G given by p C f C and p d c c. We may call G the glueing of to along G. We alo conider the full ubcategory G of G whoe object are ubobject in. We may write C X for an object of G, where X i a ubobject of GC. An arrow f : C X C X i then an arrow f : C C in o that (in et-theoretic notation) x X implie G f x X. So we regard X a a predicate (or a pecification) on GC, and a map from C X to C X i a map from C to C which repect the predicate. The projection p : G end f : C X C X to f : C C. The category G will be called the ubglueing of to along G. Alo it i ueful to notice that G and G together with the projection are characterized by the following pullback: G G Sub p G cod where cod i a forgetful functor which take the codomain, Sub i the full ubcategory of whoe object are ubobject in, and ι i the retriction of cod to Sub. It i often the cae that ha pullback, thu cod and ι are fibration (the codomain fibration and the ubobject fibration). In uch etting, p i a fibration obtained by change-of-bae along G. In particular, the ubglueing i the place where we talk about via the internal logic of. Thi point of view baed on fibration and categorical logic i exploited in Hermida thei [24], in the context of model of typed lambda calculi. It turn out that being a (bi)fibration ha imilar (but lightly indirect) impact in giving the ymmetric monoidal tructure on the glued categorie; later we will briefly addre thi iue (Propoition 3.2, 3.14 and 3.20). GB GB Gb p G ι 3 Glueing Symmetric Monoidal Structure In thi ection we give a erie of the glueing contruction for ymmetric monoidal tructure. We tart with a imple (perhap folklore) reult for glueing ymmetric monoidal cloed categorie. Baed on thi, we then conider two more involved etting. The firt i the glueing of ymmetric monoidal adjunction, that i, to contruct a ymmetric monoidal adjunction between the glued categorie from thoe between the component categorie. The econd i the double glueing of -autonomou categorie introduced by Hyland and Tan. To realize the duality in -autonomou categorie, we combine the glueing with it dual, thu ue the glueing contruction twice. We provide thee reult together with everal example. 7

8 3.1 Glueing Symmetric Monoidal Cloed Categorie Lemma 3.1 Suppoe that and are ymmetric monoidal cloed categorie and that Γ : a ymmetric monoidal functor. Moreover uppoe that ha pullback. Then the category Γ can be given a ymmetric monoidal cloed tructure, o that the projection p : Γ ymmetric monoidal cloed. Proof: We define the ymmetric monoidal tructure on Γ by I I I m I C f C f C C m C C f f d c d c d d c c i i trict where m I and m C C are the coherent morphim of the ymmetric monoidal functor Γ. Exponent are defined a C f which i given by the pullback in X C f π 2 X C C π 2 Γ C C π 1 θ C C f ΓC ΓC f ΓC ΓC where we write C C : C C C C for the counit of the adjunction, and θ C C : Γ C C ΓC ΓC i the adjunct of C C Γ C C m C C C : Γ C C ΓC ΓC. It i routine to ee the bijective correpondence between Γ C C m f f f Γ ΓC and f ΓC Γ C C ΓC f Γ f ΓC θ and f ΓC X Γ C C π 2 Γ Thi eem to be a folklore Lawvere ha tated thi reult in hi lecture in 1990, c.f. [31]. Caley et al. [13] decribe thi too. Alo ee Ambler thei [4] for a related obervation. In fact, a more abtract point of view i available, in term of fibration. Hermida [24] ha hown that, if we have a fibred ccc p :, with finite product and p with Con -product, then o i the fibration obtained by a change-of-bae of p along a functor preerving finite product. The following obervation i in pirit a parallel reult for ymmetric monoidal cloed categorie: 8

9 Propoition 3.2 Suppoe that, and are ymmetric monoidal cloed categorie, and that Γ : i a ymmetric monoidal functor, and moreover that p : i a trict ymmetric monoidal cloed functor which i alo a cloven bifibration(i.e. both p and p op are cloven fibration).conider the following pullback: q Γ Then can be given a ymmetric monoidal cloed tructure, o that the bifibrationq : i trict ymmetric monoidal cloed. Proof: See Appendix. The aumption that p i a cofibration i ued for making ymmetric monoidal, while the exponent are given by uing the fact that p i alo a fibration (ee Propoition 3.14 for a general reult for glueing uch adjunction). Lemma 3.1 can be regarded a an intance of Propoition 3.2, where p : i the codomain fibration cod :. We can then derive Lemma 3.1 from Propoition 3.2 jut by checking that i ymmetric monoidal cloed and cod preerve the tructure trictly. Lemma 3.5 below give another example, where the ubobject fibration ι : Sub take the place. Remark 3.3 One important reult in Hermida thei [24] tate that, if p : p i a fibred-ccc with Con -product and i a carteian cloed category, then o i and p trictly preerve the carteian cloed tructure. However, we do not know any anologou reult for ymmetric monoidal cloed categorie (we have no adequate notion of fibred mcc ). A a variation of Lemma 3.1, we have the tandard reult on glueing carteian cloed categorie (ee for intance [29, 15, 36]): Corollary 3.4 Suppoe that and are carteian cloed categorie and that Γ : i a functor which preerve finiteproduct. Moreover uppoe that ha pullback. Then the category Γ i a carteian cloed functor. i carteian cloed; and the projection p : Γ Similarly to Lemma 3.1, we can give a ymmetric monoidal cloed tructure on the ubglueing Γ, provided that the bae category admit epi-mono factorization: Lemma 3.5 In addition to the aumption in Lemma 3.1, uppoe that admit epi-mono factorization. Then Γ can be given a ymmetric monoidal cloed tructure, o that the projection p : Γ i trict ymmetric monoidal cloed. Proof Sketch: The decription of the ymmetric monoidal cloed tructure of Γ i eaier than that of Γ, uing et-theoretic notation: I I m I x x I C X C X C C m C C x x x X x X C X C X C C f Γ C C x X implie f x X Alternatively we can apply Propoition 3.2; ince the bae category admit epi-mono factorization, the ubobject fibration ι : Sub i a bifibration. So we only need to check that Sub i ymmetric monoidal cloed o that ι i trict ymmetric monoidal cloed. C C 9

10 Example 3.6 (Phae emantic) A ymmetric monoidal functor from the one-object one-arrow category 1 (with the trivial ymmetric monoidal cloed tructure) to Set (with carteian cloed tructure a the ymmetric monoidal cloed tructure) i no other than a commutative monoid M M e the underlying et M i the image of the unique object of 1, while the unit e and the multiplication correpond to m I and m repectively. It ubglueing i the poet P M of ubet of M with the incluion ordering. By Lemma 3.5, we can give a ymmetric monoidal cloed tructure on P M a follow. I e X Y x y x X y Y X Y u x X implie u x Y In fact P M i a free (commutative) quantale on the monoid M. By the way, thi ymmetric monoidal tructure determine the multiplicative tructure of the phae emantic [21]; it follow that, for a fixed X M, the ubpoet P M X of P M whoe object take the form A X ha a -autonomou tructure given by I X X, A B A B X X and A A X. (In fact, for any ymmetric monoidal cloed preordered et and X, the Kleili category of the monad X X become a -autonomou preordered et in thi way, c.f. Example 3.9.) Example 3.7 (Subconing) Let be a locally mall ymmetric monoidal cloed category. The functor I : Set i ymmetric monoidal, with m I id I : 1 I I and m A B : I A I B I A B which end x : I A y : I B to I I x I y A B. The ubglueing Set I ha the following ymmetric monoidal cloed tructure: I I id I A X B Y A B x y x X y Y A X B Y A B f 1 I A B x X implie f x Y where X I A, Y I B, and indicate the canonical iomorphim I category the ubconing of and may write for it. I I. We call thi Example 3.8 (Parameterized predicate) A more ophiticated (and ueful) example i obtained by combining the (ub)glueing contruction with the free ymmetric monoidal cocompletion. Let : 0 1 be a trict ymmetric monoidal functor from a mall ymmetric monoidal category 0 to a locally mall ymmetric monoidal cloed op category 1. We firt note that the preheaf category Set 0 ha a ymmetric monoidal cloed tructure given by I 0 I F G X Y FX GY 0 X Y F G op Set 0 F G op Set 0 given by Γ X 1 X, equipped with m I : 0 I I and m A B : 1 A 1 B 1 A B where m I X end f 0 X I to f 1 X I and m A B X end the equivalence cla of f 1 U A g 1 V B h 0 X U V to f g h 1 which in fact i a free ymmetric monoidal cocompletion of monoidal functor Γ : 1 1 X A B 10 0 [25]. Now we have a ymmetric

11 op 0 Γ ha object of the form A P, where A 1 and P i a ub- A. An arrow from A P to B Q i an arrow f : A B in 1 uch that, for The ubglued category Set functor of 1 any X a I I I, A P B Q 0, x P X implie f x Q X. It ymmetric monoidal cloed tructure i decribed A B P Q and A P B Q A B P Q where I X h h 0 X I P Q X a b h Y Z 0 h 0 X Y Z a P Y b Q Z P Q X f 1 X A B Y 0 a P Y f a Q X Y Note that the ubcone in Example 3.7 i the pecial cae of thi contruction in which 0 i equivalent to 1. The ubglueing Set 0 Γ will appear in Section 4.1 a the category 0 of the op parameterized predicate. Example 3.9 (Proof-relavant phae emantic) The following contruction appear in Streicher work on denotationally complete model of claical linear logic [41]. In example 3.8, uppoe that the category 1 i -autonomou. The reulting ubglueing need not be -autonomou; however, it i eaily een that, for any ubfunctor P of 1, the canonical natural tranformation from P to P P P i aniomorphim. It then follow that the Kleili category of the monad P P i a -autonomou category, where the unit, tenor and duality are given a the phae emantic (Example 3.6). In [41] Streicher ha hown that, with uitable condition on 0, one can cover not only the -autonomou tructure (multiplicative) but alo additive and exponential. Remark 3.10 We can drop all ymmetric from the reult above and talk about monoidal (bi)cloed categorie, which are model of non-commutative linear logic and alo the yntactic calculu of Lambek [30]. The example below how that a non-commutative verion of the phae emantic can be given a an intance of the ubglueing contruction. We alo note that Shirau [40] tudied the glueing of monoidal (bi)cloed lattice (called FL-algebra ) along a monoidal meet-emilattice morphim ( fringe morphim ) for howing dijunction and exitence propertie of ubtructural logic; hi contruction can be derived from the non-ymmetric verion of Lemma 3.1. Example 3.11 (Non-commutative phae emantic) A monoidal functor from the one-object one-arrow category 1 to Set i no other than a monoid M M e. It ubglueing i the poet P M, a decribed in Example 3.6. The non-ymmetric variant of Lemma 3.5 implie that P M ha the following monoidal bicloed tructure. I e X Y x y x X y Y Y X u x X implie u x Y X Y u x X implie x u Y It i eay to verify the adjunction X X and X X. 3.2 Glueing Symmetric Monoidal Adjunction F F Lemma 3.12 Suppoe that 1 2 and 1 2 are (ymmetric monoidal) adjunction, U U with (ymmetric monoidal) functor Γ 1 : 1 1 and Γ 2 : 2 2 together with a (monoidal) 11

12 natural iomorphim τ : Γ 1 U U Γ 2. For G 1 monoidal) functor F : G 1 G 2 and U : G 2 F C f U C g 1 Γ 1 and G 2 G 1 given by F FC σ C F f F d c U UC τ 1 C U g U d c 2 Γ 2, there are (ymmetric F d Fc U d Uc where σ C ε Γ 2 FC F τ FC F Γ 1 η C : F Γ 1 C Γ 2 FC (η i the unit of F U and ε i the counit of F U ). F i (trong and) left adjoint to U. Moreover the projection p 1 : G 1 1 and F F p 2 : G 2 2 give a map of adjunction [33] from G 1 G 2 to 1 2. Proof Sketch: See the bijective correpondence between U U F F f F σ Γ 1 C Γ 2 FC g Γ 2 Γ 2 C and f Γ 1 C U U Γ 2 C Γ 1 UC U g τ 1 C Γ 1 While for mot of our development Lemma 3.12 i ufficiently general, we can drop the aumption that τ i an iomorphim, provided 1 ha pullback: F F Lemma 3.13 Conider 1 2, 1 2, Γ 1 : 1 1 and Γ 2 : 2 2 a in Lemma 3.12, U U with a (monoidal) natural tranformation τ : Γ 1 U U Γ 2. Moreover uppoe that 1 ha pullback. For G 1 1 Γ 1 and G 2 2 Γ 2, there are (ymmetric monoidal) functor F : G 1 G 2 and U : G 2 G 1 given by F C f F FC σ C F f and U C g X UC π 2, where σ i given a in Lemma 3.12, and π 2 : X Γ 1 UC i given by the following pullback in 1. X π 2 Γ 1 UC π 1 τ C U U Γ 2 C U g F i (trong and) left adjoint to U, and the projection p 1 : G 1 of adjunction. 1 and p 2 : G 2 2 give a map 12

13 Proof Sketch: It i eay to ee the natural bijection between F σ F f g Γ 2 FC Γ 2 Γ 2 C and U f U g Γ 1 C Γ 1 Γ 1 UC τ U Γ 2 C and X f π 2 Γ 1 C Γ 1 Γ 1 UC Almot all example below will atify the aumption of Lemma 3.12; we ue Lemma 3.13 only in Example In term of fibration Lemma 3.13 can be generalized a follow. Propoition 3.14 Let p 1 : 1 1 be a cloven fibrationand p 2 : 2 2 be a cloven cofibration F o that they give a map of adjunction from 1 2 to F 1 2. Alo uppoe that there U U F are an adjunction 1 2, functor Γ 1 : 1 1, Γ 2 : 2 2, and a natural tranformation U τ : Γ 1 U U Γ 2. Conider the following pullback: Then there i an adjunction 1 C E q 1 1 F p 1 U U E, where σ : F Γ 1 q 2 Γ Γ 2 2 with functor given by F 2 p 2 C E F C σ C! F E and U U C τ C Γ 2 F i given a in Lemma 3.12 (the notation for carteian and cocarteian lifting are thoe in Appendix ). Moreover, q 1 and q 2 give a map of adjunction from F F 1 2 to 1 2. U U Thi not only generalie Lemma 3.13 but alo cover the contruction of exponent in Propoition 3.2 where we have τ θ : Γ C ΓC Γ and σ m : Γ ΓC Γ C. The ubglueing verion of Lemma 3.12 and 3.13 are alo available, provided the categorie 1 and 2 admit epi-mono factorization; for Lemma 3.12, we have F C X FC σ C x x F X U C Y UC τ 1 C y y U Y 13

14 and for Lemma 3.13 F C X FC σ C x x F X U C Y UC z Γ 1 UC τ C z U Y Note that, if τ i an iomorphim, then thee two reult agree. Remark 3.15 Again we can drop all ymmetric from the reult above. Example 3.16 (Adjunction between ubconing) F Let 1 2 be a ymmetric monoidal adjunction between mall ymmetric monoidal categorie U F Id and. By ubglueing 1 2 to the trivial adjunction Set Set along the glueing functor 1 I U Id : 1 Set and 2 I : 2 Set together with a monoidal natural iomorphim τ : 1 I U 2 FI 2 I, we obtain a ymmetric monoidal adjunction between ubconing (Example 3.7) 1 2 where F U F A X FA Fx m I x X U B Y UB U y m 1 I η I y Y Example 3.17 (Adjunction between categorie of parameterized predicate) In Example 3.8 we have contructed a category of parameterized predicate a a ubglueing. Conider a commutative diagram of functor F F 1 in which 0, 1, 0 and 1 are ymmetric monoidal categorie, F 0, F 1 are trong ymmetric monoidal, while, are trict ymmetric monoidal. Moreover aume that F 1 ha a right adjoint U 1 : 1 1. For thi etting, we hall give a ymmetric monoidal adjunction between the categorie of parameterized predicate. We firt note that there i a ymmetric monoidal adjunction between the preheaf categorie Set 0 Lan op F op 0 op Set 0, where LanF op G X 0 F 0 X 0 F 0 op op GX : 0 Set i a left Kan extenion [33] of G : 0 Set along F op 0. By ubglueing F U 1 Lan op F op 0 op to Set 0 0 via the glueing functor Γ 1 : X 1 X op : 1 Set 0 and Γ 2 : X 1 X : 1 F 0 Set τ X : Γ 1 U 1 X 1 Set op 0 together with a monoidal natural iomorphim U 1 X 1 F 1 X 1 op we obtain a ymmetric monoidal adjunction Set 0 Γ 1 U are given by F A P F 1 A F P and U B Q F P Y F 1 f U Q X U 1 g η X g Q F 0 X F U F 0 X F 0 Γ 2 X Set op 0 Γ 2. Explicitly, F and U 1 B U Q where h X 0 f P X h 0 Y F 0 X 14

15 Thi example will be ued in Section 4.4 for modelling the modality! of intuitionitic linear logic in the category of parameterized predicate. Example 3.18 (Adjunction between phae emantic and the interpretation of!) A noted in Example 3.6, to give a ymmetric monoidal functor from 1 to Set i to give a commutative monoid. It i eaily een that to give a monoidal natural tranformation between uch ymmetric monoidal functor i to give a monoid homomorphim between the correponding monoid. Now let u conider commutative monoid M, N and a monoid homomorphim τ : M N. We then obtain F P N by ubglueing 1 Id Id 1 to Set Set via a ymmetric monoidal adjunction P M U the glueing functor and natural tranformation correponding to M, N and τ, uing the ubglueing verion of Lemma Explicitly, for X M and Y N, F X τ X τ x x X U Y τ 1 Y x τ x Y The induced comonad on P N end Y to Y τ M. By compoing thi comonad with the Kleili adjunction of the monad X X decribed in Example 3.6, we obtain a comonad T on P N X which end Y to Y τ M X X. The (co)kleili adjunction of T i ymmetric monoidal (with repect to the finite product of the (co)kleili category and the -autonomou tructure of P N X) if and only if τ M I and alo Y Z τ M X Y τ M Z τ M X for Y Z P N X; in uch cae T give a ound interpretation of the modality!. In [21], M i choen to be the ubmonoid u I u u u of N, with τ : M N the incluion. 3.3 Glueing -Autonomou Categorie We give a mild generalization of the double glueing contruction of Hyland and Tan [42] (ee Example 3.24 below). The eential idea i that we double the object of the glued category o that the duality of the underlying -autonomou category cale up to the glued category. While an object in the glued category conidered o far i eentially a predicate on an object of the underlying category, in the double-glued category an object i a pair of predicate, one on an object of the underlying category and the other on it dual. Though uch a doubled category obviouly ha a elf-duality, it i perhap urpriing to ee that we can alo give a ymmetric monoidal tructure on it o that together with the duality it form a -autonomou category. Propoition 3.19 Suppoe that monoidal cloed category and that Γ : I Id i a -autonomou category, i a ymmetric i a ymmetric monoidal functor. Moreover uppoe that ha pullback. Then the category G Γ determined by the following pullback can be given a -autonomou tructure, o that the projection p : G Γ preerve the -autonomou tructure trictly. G Γ Γ op Id (p 1 : Γ Γ and p 2 : Γ p p 1 p op 2 op are the projection from the glued categorie.) 15

16 Proof Sketch: Explicitly, object of G Γ are tuple t C f f t where t, C, ΓC in. An arrow from t C f f t to t C f f t i a tuple d : d t : t t c : C C uch that the following diagram commute. f : ΓC and f t : t f ΓC d ΓC f Γc f t t ΓC d t t ΓC The -autonomou tructure on G Γ i given a follow. t C f f t X and x : X Γ C C t C f f t t C f f t I ft Γc I ΓI I m I id ΓI X C C m C C f f x t C f t f ΓC ΓC are given by the following three pullback in. X t x Γ C C θ Γ C C f ΓC f t Γ C C ΓC ΓC ΓC Γ Γ C C θ C C ΓC ΓC f ΓC t f ΓC t θ C C : Γ C C ΓC ΓC i given a in the proof of Lemma 3.1. It i not hard to ee that the unit and tenor product given above determine a ymmetric monoidal tructure on G Γ ; alo it i eay to ee that determine a contravariant fully faithful functor on G Γ. The mot nontrivial part i to check the natural bijection G Γ G Γ it require ome calculation, a found in Appendix E. p : G Γ end t C f f t to C and d d t c to c, and obviouly preerve the -autonomou tructure trictly. We call G Γ the double glueing of to along Γ. In fact it i poible to give a more general tatement in term of fibration, in the imilar way 16

17 to Propoition 3.2, a follow. It ubume Propoition 3.19 and alo 3.22 below. The proof i eentially the reworking of that of Propoition 3.19 (Appendix E) uing the idiom of fibration. Propoition 3.20 Suppoe that i a -autonomou category,, are ymmetric monoidal cloed categorie, and that Γ : i a ymmetric monoidal functor while p : i a trict ymmetric monoidal cloed functor which i alo a cloven bifibrationwith fibredfiniteproduct. Conider the category G Γ p determined by the following pullback: G G t G Γ p G op t p Γ p p t op Γ Then G Γ p can be given a -autonomou tructure, o that the projection to the -autonomou tructure. p G p p op t trictly preerve Proof Sketch: An object of G Γ p i a triple A A A A t uch that Γ A p A and Γ A p A t hold. An arrow from A A A A t to B B B B t i a triple f f : A B f : A B f t : B t A t which atifie Γ f p f and Γ f p f t. Let u define (uing the notation in Appendix ) I I m I! I 1 ΓI A B A B m A B! A B θ 1 A B t Γ A B θ 2 B A t A A A t σ A where θ 1 : Γ A B θ Γ A B Γ A Γ B θ 2 : Γ A B θ Γ B A Γ B Γ A σ : Γ A Γ A and we write 1 X and E X E for the terminal object and binary product in the fibre over X. Then one can check that thee data give a -autonomou tructure on G Γ p, imilarly to the proof of Propoition Remark 3.21 The functor Γ i ymmetric monoidal with repect to the ymmetric monoidal tructure on op. ouble glueing make ue of thi duality between I and op. Propoition 3.22 In addition to the aumption in Propoition 3.19, uppoe that admit epi- mono factorization. Then the category G Γ determined by the following pullback can be given a -autonomou tructure, o that the projection p : G Γ preerve the -autonomou tructure trictly. G Γ Γ op Γ p p 1 p op 2 17

18 Proof Sketch: Let u write A A A A t for an object of G Γ, where A and A, A t are ubobject of Γ A and Γ A repectively. Then a map f from A to B i an arrow f : A B in uch that, x A implie Γ f x B and y B t implie Γ f y A t. The -autonomou tructure i given a I A B A I m I x x I ΓI A B m A B x y x A y B u Γ A B Γ A B Γ B A x A implie u x B t y B implie u y A t A A t x x A We call G Γ the double ubglueing of to along Γ. Example 3.23 ( ouble phae emantic) A in Example 3.6, let u conider the cae that functor Γ : i 1 and i Set, thu a ymmetric monoidal M e. By applying the double i determined by a commutative monoid M ubglueing contruction we have a poet P M P M op (with the ordering A A t iff A B and alo B t A t ) with the following -autonomou tructure. I I M A B t A A t B B t A A t A B A t A B A t B B t where I, and in the right hand ide are thoe of Example 3.6. Other connective are derived from them. Explicitly: A A t B B t A A t B B t M I A t B A B Example 3.24 (ouble glueing of Hyland and Tan) B t A A t B t B t A t A B t If i a locally mall compact cloed category [28], Set and Γ I, then G Γ i exactly the double glueing G of Hyland and Tan. Explicitly, G object i a triple A A A I A A t I A and an arrow f : A B in G i an arrow f : A B in atifying f x B for x A and alo f y A t for y B t. A a leading example, from the compact cloed category Rel of et and binary relation, we obtain it double glueing GRel which i the category of linear logical predicate of Loader [32]. In her thei [42], Tan ha hown that the full completene of G (a a -autonomou category) i reduced to that of (a a compact cloed category), and derived full completene reult of variou model of MLL. Example 3.25 (Parameterized predicate) We can conider the double verion of the parameterized predicate in Example 3.8. Let : 0 1 be a trict ymmetric monoidal functor from a mall ymmetric monoidal category 0 to a locally mall -autonomou category 1. By applying the double ubglueing contruction to op the ymmetric monoidal functor Γ : X 1 X : 1 Set 0, we obtain the -autonomou category G Γ. We will concretely decribe it in Section 4.5 a the category 0 of the double parameterized predicate. 18

19 4 Logical Predicate for Linear Logic We conider the notion of parameterized logical predicate for three fragment of linear logic: MILL (Multiplicative Intuitionitic Linear Logic), ILL (ual Intuitionitic Linear Logic = MILL + modality!), and MLL (Multiplicative Linear Logic). In fact, the category-theoretic etting for thee logical predicate have already appeared a example of the glueing contruction in lat ection (Example 3.8, 3.17 and 3.25); in thi ection we turn them in a more yntax-oriented form. 4.1 Parameterized Predicate Let 0 be a (mall) ymmetric monoidal category, 1 a (locally mall) ymmetric monoidal cloed category and be a trict ymmetric monoidal functor from 0 to 1. efinition 4.1 An Obj 0 -indexed et P P X X i a 0 0-predicate on A 1 when P X 1 X A for X 0, and for f 0 X Y, g P Y implie g f P X. X A repreent (a denotation of) the et of proof of a equent We may intuitively think that 1 X A, and 0 (imported into 1 via ) determine a property on proof which i cloed under tenor, compoition and tructural contruction. Unlike the traditional non-linear calculi and logical predicate over them, we explicitly tate the reource X, which will play ome ignificant role in our work. Then, for a 0-predicate P on A, P X i a predicate on the proof of X A. The econd condition tell u that P i table under the change of reource along ome proof of X that it atifie the property 0. efinition 4.2 efine the category of 0-predicate 0 a follow: Y, provided an object of 0 i a pair A P where P i a 0-predicate on A 1; an arrow from A P to B Q i an arrow h 1 A B uch that g P X implie h g Q X. efinition 4.3 For 0-predicate P on A and Q on B, define 0-predicate P Q on A B and P Q on A B a follow. P Q P Q X g h f Y Z 0 f 0 X Y Z g P Y h Q Z X f 1 X A B Y 0 g P Y f g Q X Y op X. A explained in Example 3.8, the definition of P Q above i The reader hould notice that 0 i no other than the ubglueing of 1 to Set 0 along the glueing functor X 1 derived from the free ymmetric monoidal cocompletion together with Lemma 3.5. However, here we alo give a proof-theoretic explanation: a equent X A B can be derived a X Π f. Y Z X Y Π g. A Y Z A B Z Π h. A B B -Intro -Elim 19

20 where X Y Z plit a reource X to Y and Z which are ued to prove A and B repectively. In general, uch a plitting of reource i not unique, o we conider all poible cae uch that (i) the proof Π f of the plitting atifie the tenor-cloed property 0 and (ii) the proof Π g of Y A and Π h of Z B atify the predicate P Y and Q Z repectively in uch cae we ay that the derivation atifie the property P Q X. The definition of P Q i in pirit the ame a the uual definition of logical predicate; M : A B atifie P Q if and only if MN : B belong to Q for any N : A atifying P. However, ince our type theory i linear, we have to deal with the reource of term linearly, and we explicitly tate them in the definition: intuitively, M : A B atifie P Q if and only if MN : B atifie Q for any N : A atifying P. Lemma 4.4 For each X A A i a 0-predicate on A. 0 define A X f f 0 X A. Then: f : A A B B in 0 if and only if f g for ome g 0 A B. A B A B. Propoition become a ymmetric monoidal cloed category by the following data: the unit object i I I, tenor i given by A P B Q A B P Q, and we have exponent A P B Q A B P Q. Moreover extend to a trict ymmetric monoidal functor from 0 to 0 which i full. Remark 4.6 If 0 i cloed and preerve exponent trictly, then o i in particular we have A B A B. 4.2 Logical 0-Predicate Suppoe that we have 0, 1 and : 0 MILL in 1 (ee Appendix A). 1 a before. Alo we fix an interpretation 1 of efinition 4.7 A type-indexed family P σ of 0-predicate i a logical 0-predicate if P σ i a 0-predicate on σ 1, P I I, P σ τ P σ P τ, P σ τ P σ c 1 : σ 1 P σ Note that a logical P τ, and τ 1 P τ for each contant c : σ τ. 0-predicate i completely determined by it intance at bae type. Given a logical 0-predicate P σ, we can interpret MILL in 0 by b b 1 P b for each bae type b and c c 1 : σ 1 P σ τ 1 P τ for each contant c : σ τ. Thu we have Lemma 4.8 (Baic Lemma for MILL) Let P σ be a logical M : τ, M : τ 1 : 1 P τ 1 P τ hold. Corollary 4.9 If no occur in, M :τ 1 P τ 0 hold for any M :τ predicate. Then, for any term

21 0 itelf determine a logical 0-predicate in a canonical way, provided that for each bae type b there i an object b 0 0, and for each contant c : σ τ there i an arrow c 0 0 σ 0 τ 0 where σ 0 i defined inductively by I 0 I and σ τ 0 σ 0 τ 0. Then we automatically have an interpretation 1 in 1 determined by b 1 b 0 and c 1 c 0. Now define the canonical logical 0-predicate σ by b b (a noted above, a logical 0 0-predicate i determined by it intance at bae type). Baic Lemma for the canonical logical 0-predicate implie that, at -free type (at any type if 0 and are cloed) a definable element mut be in the image of. Example 4.10 If 0 i equivalent to the one object one arrow category (thu we have no bae type nor contant), then the canonical logical 0-predicate pecifie the canonical iomorphim between object generated from I. If 0 1 and i the identity functor, then every morphim between MILL-definable object atifie the canonical logical 0-predicate. 4.3 Binary Logical 0-Relation It i traightforward to generalize (or pecialize) our logical predicate to multiple argument, i.e. logical relation, in the ame way a demontrated in [36]. Here we pell out the cae of binary logical relation. Suppoe that 0 i a (mall) ymmetric monoidal category, 1 and 2 are (locally mall) ymmetric monoidal cloed categorie and that 1 : 0 1 and 2 : 0 2 are trict ymmetric monoidal functor. A binary 0-relation i no other than a 0-predicate obtained by replacing 1 by 1 2 and by 2 : Explicitly: 1 efinition 4.11 An Obj 0 -indexed et R R X X i a 0 0-relation on A B 1 when R X 1 1X A 2 2X B for X 0, and for f 0 X Y, g h P Y implie g 1 f h 2 f P X. efinition 4.12 efine the category of 0-relation 0 a follow: 2 an object of 0 i a triple A B R where R i a 0-relation on A B ; an arrow from A B R to A B R i a pair of arrow h 1 A A k 2 B B uch that f g R X implie h f k g R X. 0 i a ymmetric monoidal cloed category. Again explic- From Propoition 4.5 we know that itly: efinition 4.13 For 0-relation R on A B and R on A B, define 0-relation R R on A A B B and R R on A A B B a follow. 21

22 R R R R X X g g 1 f f g Y 0 h h 2 f Y Z 0 f 0 X Y Z g h R Y g h R Z a b R Y implie f a g b R X Y Note that, in the relational notation, R f R R R can be given a X g iff a R Y b implie f a R X Y g b Therefore the reult of the application of related function to related element are again related thi i the key property of the logical relation. The only novelty in our definition i that we take care about the linearly ued reource X and Y. Now fix interpretation 1 and 2 of MILL in 1 and 2 repectively. efinition 4.14 A type-indexed family R σ of 0-relation i a logical 0-relation if R σ i a 0-relation on σ 1 σ 2, R I X 1 f 2 f f 0 X I, R σ τ R σ R τ, R σ τ R σ R τ, and c 1 c 2 : σ 1 σ 2 R σ τ 1 τ 2 R τ for each contant c : σ τ. Lemma 4.15 (Baic Lemma, binary verion) Let R σ be a logical M : τ, M : τ 1 M : τ 2 : 1 2 R τ 1 τ 2 R τ hold. 4.4 ual Intuitionitic Linear Logic 0-relation. Then, for any term Now we enrich our logic and calculu with the modality!. There are many poible choice for thi, ee for intance [10]. Here we chooe the formulation due to Barber and Plotkin, called ual Intuitionitic Linear Logic (ILL) [6] for it imple yntax and equational theory, a well a for the well-etablihed category-theoretic model of ILL in term of ymmetric monoidal adjunction 1. Alternatively we could ue Benton Linear Non-Linear Logic (LNL Logic) [9] which ha eentially the ame cla of category-theoretic model a ILL. The yntax and emantic of ILL are recalled in Appendix B. Conider a commutative diagram of functor F F The term dual refer to the double context of the type ytem, and ha nothing to do with the duality in claical linear logic 22

23 in which 0 and 1 are carteian categorie, 0 ymmetric monoidal and 1 ymmetric monoidal cloed; and F 0, F 1 are trong ymmetric monoidal while, are trict ymmetric monoidal. Moreover aume that F 1 ha a right adjoint U 1 : 1 1. A before, we define the categorie of 0- and 0-predicate let u call them 0 and 0 repectively. Note that 0 i a carteian category (actually carteian cloed if 1 i cloed) with product given by A P B Q A B P Q where P Q X f g f P X g Q X for 0-predicate P and Q (which coincide with P Q in efinition 4.3). Now we give a ymmetric monoidal adjunction between 0 and 0, by applying Example For a 0-predicate P on A 1, define a 0-predicate L P on F 1 A 1 by L P Y F 1 g f X 0 f 0 Y F 0 X g P X and, for a 0-predicate Q on B 1, define a 0-predicateF 0 Q on U 1 B 1 by F 0 Q X f 1 X U 1 B f Q F 0 X 1 F 0 X B 1 F 1 X B where f : X U 1 B i the adjunct of f : F 1 X B. Propoition 4.16 L andf 0 extend to functor between 0 and 0. Moreover L i trong ymmetric monoidal, and left adjoint tof 0. Therefore we have a ymmetric monoidal adjunction 0 L F 0 0 between a carteian category 0 and a ymmetric monoidal cloed category 0. Let! be the induced comonad on 0, that i, we define a 0-predicate!P on F 1 U 1 A by!p Y F 1 g f X 0 f 0 Y F 0 X g P F 0 X for a 0-predicate P on A. A explained in Example 3.17,!P i derived from a left Kan extenion together with the ubglueing verion of Lemma 3.12, but it can be explained more or le intuitively (proof-theoretically) a follow. A equent /0 ; Y!A can be proved a Π f Π g.. X ; /0 A /0 ; Y!X X ; /0!A /0 ; Y!A!-Intro!-Elim where /0 ; Y!X convert a linear reource Y to!x which i ued non-linearly in X ; /0!A to produce!a. Taking all uch poible cae into account, we ay that the proof atifie!p Y when Π f belong to 0 and Π g atifie P X. F 1 Now let u fix an interpretation 1 of ILL in 1 1 (ee Appendix B). efinition 4.17 A type-indexed family P σ of 0-predicate i a logical 0 P σ i a 0-predicate on σ 1, U 1 F 0 0 -predicate if 23

24 P I I, P σ τ P σ P τ, P σ τ P σ P τ and P!σ!P σ hold, and c 1 : σ 1 P σ τ 1 P τ for each contant c : σ τ. Lemma 4.18 (Baic Lemma for ILL) Let P σ F be a logical predicate. Then, for any term Γ ; M : τ, Γ ; M : τ 1 : Γ ; 1 P Γ ; τ 1 P τ hold. F F itelf determine the canonical logical predicate when for each bae type b there i an object b 0 0, and for each contant c : σ τ there i an arrow c 0 0 σ 0 τ 0 where σ 0 i defined inductively by I 0 I and σ τ 0 σ 0 τ 0. In uch cae we automatically have an interpretation 1 in 1 determined by b 1 b 0 and c 1 c 0, and F the canonical logical predicate σ i determined by b b Multiplicative Linear Logic Let 0 be a (mall) ymmetric monoidal category, 1 a (locally mall) -autonomou category and be a trict ymmetric monoidal functor from 0 to 1. efinition 4.19 A double 0 -predicate on A 1 i a pair P P P t uch that P i a on A and P t i a 0-predicate on A. efinition 4.20 efine the category of double 0-predicate 0 a follow: 0 -predicate an object of 0 i a pair A P where P P P t i a double 0-predicate on A 1; an arrow from A P to B Q i an arrow h 1 A B uch that g P X implie h g Q X and g Q t X implie h g P t X. P P t on A and Q Q Q t on B, define a double 0-predicate P Q P Q P Q t on A B by P Q X a b h h 0 X Y Z a P Y b Q Z f 1 X A B P Q t X 1 X A a P Y B 1 X B implie f id a Q t X Y b A Q Y implie f id b P t X Y P P t on A, define a double 0-predicate P P P t on X A f P X. 0, define A X f f 0 X A and At X 1 X A efinition 4.21 For double 0-predicate P Alo, for a double 0-predicate P X f A A by P P t and P t Lemma 4.22 For each X A Then A f : A A A At i a double 0-predicate on A. B B in 0 if and only if f g for ome g A B.. 24

25 A B A B. Propoition become a -autonomou category by the following data: the unit object i I I, tenor i given by A P B Q A B P Q while the duality i A P A P. Moreover extend to a trict ymmetric monoidal functor from 0 to 0 which i full. Let u fix an interpretation 1 of MLL in 1. efinition 4.24 A type-indexed family P σ of double 0 -predicate i a double logical 0-predicate if P σ i a double 0-predicate on σ 1, P I I, P σ τ P σ P τ and P σ P σ, c 1 : σ 1 P σ τ 1 P τ for each contant c : σ τ. Lemma 4.25 (Baic Lemma for MLL) Let P σ be a double logical term M : σ of MLL, P σ I hold. M : σ 1 5 Fully Complete Tranlation 0-predicate. Then, for any A an application of our logical predicate (hence of our glueing contruction), we can how that everal tranlation between fragment of linear logic are fully complete, i.e. it i not jut conervative but alo full. 5.1 A Cae Study: From MILL to MLL Let u pell out the cae of the embedding from MILL to MLL, under the aumption that they have the ame bae type and contant. The tranlation at the type level i given by b b, I I, σ τ σ τ and σ τ σ τ. We omit the tranlation at the proof level, ince it hould be obviou for thoe familar with linear logic, and alo it require u to preent the proof theory (proof net) of MLL which need ome pace; ee e.g. [12] for a complete decription. While MLL i a richer theory than MILL (MLL contain lot of type which are not definable in MILL), it i not very obviou how proof of thee two theorie can be related; we how that they are in fact in a tronget relation. Let 0 be a term model of MILL, and 1 be that of MLL. Thu 0 i a mall ymmetric monoidal cloed category which i freely generated from a fixed et of bae object and contant arrow, and 1 i a mall -autonomou category freely generated from the ame et of bae object and contant arrow. The trict ymmetric monoidal cloed embedding : 0 1 then correpond to the yntactic tranlation from MILL to MLL. The tranlation i ound, becaue preerve the ymmetric monoidal cloed tructure trictly. It i alo conervative, becaue i faithful; thi can be hown by the following obervation: given a mall ymmetric monoidal cloed category, we can alway contruct a -autonomou category to which (fully and) faithfully embed uch op i obtained, for example, by applying the Chu contruction [8] to Set. In general thi kind of model contruction technique i ueful for howing the conervativity of yntactic tranlation, ee for example [18, 5]. A much harder property to how i that the tranlation i full, i.e. 25