J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for singleton strings, and in the compact closed case, when both tensors agree, but not in general. b) Multiplicative linear logic can be modeled by autonomous categories. Dropping the explicit negation led J. R. B. Cockett and R. A. G. Seely to introduce weakly distributive categories [7] and [8] later renamed to linearly distributive categories , that carry two monoidal structures ( tensors ) and linked by so called linear distributions (B C) A B) C and (A B) A (B C) 4) subject to certain coherence conditions. In view of (a) these are just ....

Cockett, J. R. B. and R. A. G. Seely, Weakly distributive categories, in: M. P. Fourman, P. T. Johnstone and A. M. Pitts, editors, Applications of Categories to Computer Science (Durham,

....commas on the right are treated as disjunction. Thus for a proper categorical interpretation of polycategories, one needs categories with two monoidal structures which interact in an appropriate fashion. Such categories are called linearly or weakly distributive, a notion due to Cockett and Seely [CS97, BCST96]. Linearly distributive categories are the appropriate framework for considering a speci c logical system known as linear logic, introduced by Girard [Gir87, Gir89] For a brief exposition of linear logic, see Appendix B. As we will see, the re ned logical connectives of linear logic will be ....

....This corresponds to having a symmetric polycategory. This is very much related to having a symmetric tensor or tensors, i.e. ones with the property that = A. We will always assume our polycategories are symmetric. We now give a more formal de nition of polycategory. We refer the reader to [CS97, Sza75] for further details. De nition 4.2 A polycategory C consists of the following data: A set of objects, denoted jCj. If A 1 ; A 2 ; A n and B 1 ; B 2 ; Bm are nite sequences of objects, then we have a set of morphisms of the form f : A 1 ; A 2 ; A n B 1 ; B 2 ; ....

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J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133-172, 1997. 25

....which is not allowed. b) Planar polycategories have (at least) two advantages over multicategories. Certain aspects of linear logic can be modeled by autonomous categories. Dropping the explicit negation led J. R. B. Cockett and R. A. G. Seely to introduce weakly distributive categories [7], later renamed to linearly distributive categories , that carry two monoidal structures ( tensors ) linked by so called linear distributions (B C) A B) C and (A B) A (B C) 3) subject to certain coherence conditions. Tensors that need not be symmetric are most ....

Cockett, J. R. B. and R. A. G. Seely, Weakly distributive categories, in: M. P. Fourman, P. T. Johnstone and A. M. Pitts, editors, Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992), pp. 45-65.

....and homomorphisms to set . Finally, consider an Eilenberg Moore algebra hB; i for this monad. Associative binary operations and on B are given by a b : h(a) b)i and a b : h(ab)i and clearly obey axioms (3 03) 3.02 Remark. Cockett and Seely introduced weakly distributive categories [5], later renamed to linear distributivity categories , that carry two monoidal structures linked by socalled linear distributions A (B C) A B) C and (A B) C A (B C) subject to certain coherence conditions. While these structural morphisms resemble the axioms (3 03) in general ....

Cockett, J. R. B., and Seely, R. A. G. Weakly distributive categories. In Applications of Categories to Computer Science (1992), M. P. Fourman, P. T. Johnstone, and A. M. Pitts, Eds., vol. 177 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 45-65.

....is a precise orthogonality on a symmetric monoidal closed category C. Then T J is a multicategory. ii) Suppose is a precise symmetric orthogonality on a autonomous category C. Then T is a polycategory. 38 For more information about multicategories see [41, 42] and for polycategories see [50, 20]. polycategories are explained in [32] A multicategory in which the multimaps are fully representable can be regarded as a symmetric monoidal closed category; and a polycategory in which the polymaps are fully representable can be regarded as a autonomous category. Implicitly there is a ....

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133-173, 1997.

....commas on the right are treated as disjunction. Thus for a proper categorical interpretation of polycategories, one needs categories with two monoidal structures which interact in an appropriate fashion. Such categories are called linearly or weakly distributive, a notion due to Cockett and Seely [CS97, BCST96]. Linearly distributive categories are the appropriate framework for considering a speci c logical system known as linear logic, introduced by Girard [Gir87, Gir89] For a brief exposition of linear logic, see Appendix B. As we will see, the re ned logical connectives of linear logic will be used ....

....corresponds to having a symmetric polycategory. This is very much related to having a symmetric tensor or tensors, i.e. ones with the property that A B = B A. We will always assume our polycategories are symmetric. We now give a more formal de nition of polycategory. We refer the reader to [CS97, Sza75] for further details. De nition 4.2 A polycategory C consists of the following data: A set of objects, denoted jCj. If A 1 ; A 2 ; A n and B 1 ; B 2 ; Bm are nite sequences of objects, then we have a set of morphisms of the form f : A 1 ; A 2 ; A n B 1 ; B 2 ; ....

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J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133-172, 1997.

.... the exponentials ) This semantics is equivalent to the phase semantics of [Gi87] see [Av88] for more details) 3) Another general semantical framework which seems to have close relationships to that of [Av88] but was developed independently is that of Weakly Distributive Categories (see [CS91] and [Ba90] 5 III. Semantics in the integers We start with the following obvious observation: if we take the two truth values of classical logic to be 0 and 1 then the operations which correspond to the connectives are defined by :a = 1 Gamma a, a b = max(a; b) a b = min(a; b) Moreover, ....

....less general, idea has been used by Girard with connection to the variable free fragment of LL m . Structures which are practically identical to the integral models which are defined below and their various generalizations (like the c models of IV.2. 1) were independently introduced also in [CS91] and in [Ba91] In [CS91] they are called interpretations of the initial model in shift monoids on Z . 7 III.3 Theorem. 1) If T LL a A and v validates all the sentences of T then v validates A. 2) If T LL m A and v strongly validates all the sentences of T then v strongly ....

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Cockett, J.R.B. and Seely R.A.G. Weakly distributive categories, in Fourman M.P., Johnstone P.T., Pitts A.M., eds: Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992), pp. 45-65

....sequential composition, and angelic or demonic parallel composition. Some recent contributions to this research, resulting from the study of semantics for linear logic and process calculi, include Chu spaces (Pratt 1995) interaction categories (Abramsky 1994) and weakly distributive categories (Cockett Seely 1997). Residuated lattices and related mathematical structures are well known and have been used to model substructural logics (Dosen Schroeder Heister 1993) Ono (1993) gives an algebraic semantics for a class of substructural logics, which is based on a variant of quantales called complete full ....

Cockett, J. R. B. & Seely, R. A. G. (1997), `Weakly distributive categories', Journal of Pure and Applied Algebra 114(2), 133--173.

.... Just as in [14] below we will frequently use Joyal and Street s very useful string diagrams, cf. 13] 22] and [21] It should be noted that these are to be read from the bottom up, in contrast to the circuit diagrams developed by Cockett, Seely and their collaborators, cf. e.g. [10], which are inspired by proof nets. In the absence of symmetry the familiar notion of closedness for a symmetric monoidal category splits into left closedness and rightclosedness. These notions make perfect sense in any bicategory X . We recall the formalization by Street and Walters [23] where ....

....same hom categories as X . 2. The equivalences [A; B] B; A] extend to a biequivalence X ( X ) coop that xes the objects. Bicategories with two compatible 1 cell compositions naturally generalize Cockett and Seely s linearly (formerly weakly) distributive categories [10]. A forthcoming paper [9] will study this in depth. kchu.tex; 4 06 1999; 13:24; p.7 8 2. Modules and their reversal Consider a cyclic autonomous bicategory X that locally has coequalizers. Since X is closed, these coequalizers are stable, i.e. are preserved by 1 cell composition. Hence we ....

Cockett, J. R. B., and Seely, R. A. G. Weakly distributive categories. In Applications of Categories to Computer Science (1992), M. P. Fourman, P. T. Johnstone, and A. M. Pitts, Eds., vol. 177 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 45-65.

....lattice theoretic models as a borrowing of models from conditional equational logic. We explain below the details of this borrowing process and its good properties. Following [21] we call the lattice models Girard algebras. Exploiting results of Cockett and Seely on weakly distributive categories [4] we can define a Girard algebra G as a set, together with binary operations Phi, Omega , unary operations ....

.... : Bm ) and in general Phi(S; Delta) Sigma G [ S; EG [ ff( Delta) It can be shown using ideas in [21,4] that the above map ( Phi; ff) is indeed conservative and therefore, by Proposition 31, that the borrowing of the conditional logic institution via ( Phi; ff) makes LL into a complete logic. This borrowing process amounts to a systematic reduction of propositional linear logic to algebra in a way ....

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. In M. Fourman, P. Johnstone, and A. Pitts, editors, Proceedings Symposium on Applications of Categories in Computer Science, pages 45--65, Cambridge, 1992. Cambridge University Press. 40

.... logic connections between mll and autonomous categories (see e.g. 3, 26] so that proof nets correspond to maps of the category, and the fact that autonomous categories coincide with symmetric weakly distributive categories 9 (swdc) with negation in the sense of Cockett and Seely (see [7]) Take the proof net Gamma; A 0 A Omega B B; Delta which, under the categorical logic correspondence, describes the composite map Gamma Omega Delta Omega 0 A Omega B l Gamma1 O O (A Omega B) where OA l O A is the left unit natural isomorphism ....

J. R. B. Cockett and R. A. G. Seely. Weakly distributive categories. In M. P. Fourman, P. T. Johnstone, and A. M. Pitts, editors, Applications of Categories to Computer Science, volume 177, pages 45--65. Cambridge University Press, 1991.

....structure required for interpreting linear logic, an important question is how to interpret the connective P without using negation, and how to axiomatize its relationship with the tensor Omega . Cockett and Seely have answered this question with the notion of a weakly distributive category [20]. A weakly distributive category consists of a category C with two symmetric tensor products Omega ; P: C Theta C C, and a natural transformation A Omega (B P C) Gamma (A Omega B) P B (weak distributivity) satisfying some coherence equations 11 . Negation is added to a weakly ....

....category C with two symmetric tensor products Omega ; P: C Theta C C, and a natural transformation A Omega (B P C) Gamma (A Omega B) P B (weak distributivity) satisfying some coherence equations 11 . Negation is added to a weakly distributive category 11 Cockett and Seely develop in [20] the more general case in which the tensor products are not assumed to be symmetric. by means of a function ( jCj jCj on the objects of C, and natural transformations 1 Gamma A P A and A Omega A Gamma satisfying some coherence equations. Cockett and Seely then prove that ....

J. R. B. Cockett and R. A. G. Seely, Weakly distributive categories, in: M. P. Fourman, P. T. Johnstone, and A. M. Pitts (eds.), Applications of Categories in Computer Science, Cambridge University Press, 1992, pages 45--65.

.... morphism A Gamma ( A Gammaffi ) Gammaffi ) is an isomorphism) In addition various coherence conditions must hold a good account of these may be found in [M OM89] Coherence theorems may be found in [BCST, Bl91, Bl92] An equivalent characterization of autonomous categories is given in [CS91], based on the notion of weakly distributive categories. That characterization is useful in contexts where it is easier to see how to model the tensor Omega , the par . ....

....depending on whether or not the additives are wanted) There is an intermediate notion, full intuitionistic linear logic due to de Paiva [dP89] in which the morphism A Gamma A need not be an isomorphism. And as mentioned above, there is the notion of weakly distributive category [CS91, BCST], where negation and internal hom are not required. One classically important class of autonomous categories are the compact categories [KL80] where the tensor is self dual: A Omega B) A Omega B . Linear logicians often regard with derision those models in which tensor and ....

Cockett, J.R.B. and R.A.G. Seely "Weakly distributive categories", in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45--65.

....categorical structure required for interpreting linear logic, an important question is how to interpret the connective P without using negation, and how to axiomatize its relationship with the tensor Omega . Cockett and Seely answer this question with the notion of a weakly distributive category [13]. A weakly distributive category consists of a category C with two symmetric tensor products Omega ; P: C Theta C C, and a natural transformation A Omega (B P C) Gamma (A Omega B) P C (weak distributivity) satisfying some coherence equations 6 . Negation is added to a weakly ....

....is parameterized by a theory ATOM providing atomic formulas. Propositions are of the form [A] for A an expression in Prop0. The purpose of the operation [ is to transform a formula A into its equivalent negation normal form [A] using the equations in the theory 6 Cockett and Seely develop in [13] the more general case in which the tensor products are not assumed to be symmetric. PROP0[X] mentioned before. The rewrite rules for Omega ; P; and negation correspond to the natural transformations in the definition of a weakly distributive category, as explained above. The rules for ( Phi, ....

J. R. B. Cockett and R. A. G. Seely, Weakly Distributive Categories, in: M. P. Fourman, P. T. Johnstone, and A. M. Pitts (eds.), Applications of Categories in Computer Science, Cambridge University Press, 1992, pages 45--65.

.... Gamma Gammaffi Gamma i is derivable. ffl Gamma is derivable if and only if Gamma i is derivable for i = 1; 2. The set of sequents mentioned in the above theorem is then obtained by using three canonical morphisms which exist in any model of MLL MIX. These are the weak distributivity [13] and the MIX morphism [15] ffi: A Omega (B . C) A Omega B) ....

J.R.B. Cockett, R.A.G. Seely, Weakly Distributive Categories, in: Applications Of Categories in Computer Science, London Mathematical Society Lecture Notes Series 177, (1992)

....is from The Logic of Relatives [Klo86, p. 456] Peirce says Two formulae so constantly used that hardly anything can be done without them are a; b c) a; b c (a b) c a b; c These correspond to half of the weak distributivity laws for linear logic studied by Cockett and Seely [CS91]. 6 Tarski s School In 1939 Tarski was visiting the US when Germany invaded his native Poland, and he accordingly took up permanent residence in the US, taking a position at UC Berkeley in 1942 where he remained until his death in 1983. In 1941 Tarski revived the long dormant calculus with a ....

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. In P.T. Johnstone, editor, Proc. LMS Symp. on the Applications of Category Theory in Computer Science, Durham, 1991.

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Cockett, J.R.B. and Seely, R.A.G. (1991) Weakly distributive categories. In Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177, pp. 45--65.

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Cockett, J.R.B. and Seely, R.A.G. (1991) Weakly distributive categories. In Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177, pp. 45--65.

....linear (or lax) natural transformations, which can only be defined for representable poly bicategories. These in fact correspond to modules having special properties. Introduction Linear bicategories [6] were introduced as a natural 2 dimensional extension of linearly distributive categories [7]. When compared to ordinary bicategories, the most striking feature is the presence of two global (as opposed to local , inside the hom categories) compositions B#A, B# B#B, C# ## B#A, C# or tensors ( tensor ) and ( par , to remind us of the connection with Girard s linear logic, even ....

....categories and linear bicategories. Initially Cockett and Seely had considered a sequent calculus (for the tensor par fragment of linear logic) with an input output symmetry: the desire to model Gentzen s cut rule categorically then motivated the introduction of linearly distributive categories [7]. The definition proceeded via the poly categories mentioned above. Although not explicitly stressed in the definition [6] linear bicategories have a similar underlying (albeit 2dimensional) structure. We introduce this in Section 1 under the name poly bicategory . Gentzen s cut then provides ....

[Article contains additional citation context not shown here]

Cockett, J. R. B., and Seely, R. A. G. Weakly distributive categories. In Applications of Categories to Computer Science (1992), M. P. Fourman, P. T. Johnstone, and A. M. Pitts, Eds., vol. 177 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 45--65.

....the categorical aspects of this work, some discussion is necessary so as to make the philosophical basis for the logics intelligible. The setting we propose to use corresponds to the tensor par fragment of linear logic, whose categorical semantics lies in linearly distributive categories [7, 10, 13, 14, 15]. Linearly distributive categories were originally introduced by the latter two authors [13] with the idea that the tensor par fragment of linear logic [17] was crucial categorically, and that the remaining structure associated with linear logic could then be added to this basic fragment in a ....

....basis for the logics intelligible. The setting we propose to use corresponds to the tensor par fragment of linear logic, whose categorical semantics lies in linearly distributive categories [7, 10, 13, 14, 15] Linearly distributive categories were originally introduced by the latter two authors [13] with the idea that the tensor par fragment of linear logic [17] was crucial categorically, and that the remaining structure associated with linear logic could then be added to this basic fragment in a modular fashion. A linearly distributive category is a category equipped with two monoidal ....

[Article contains additional citation context not shown here]

J.R.B. Cockett and R.A.G. Seely (1997) "Weakly distributive categories." Journal of Pure and Applied Algebra 114 133--173. (Updated version available on http://www.math.mcgill.ca/rags.)

....at the poly level. However, in the representable case it can be subsumed by the notion of poly module. To Saunders Mac Lane, on the occasion of his ninetieth birthday 0 Introduction We introduced linear bicategories [5] as a natural 2 dimensional extension of linearly distributive categories [6]. When compared to ordinary bicategories, the most striking feature is the presence of two global (as opposed to local , inside the hom categories) compositions BhA;Bi BhB;Ci BhA;Ci or tensors ( tensor ) and ( par , to remind us of the 1 connection with Girard s linear logic, even ....

....categories and linear bicategories. Initially Cockett and Seely had considered a sequent calculus (for the tensor par fragment of linear logic) with an input output symmetry: the desire to model Gentzen s cut rule categorically then motivated the introduction of linearly distributive categories [6]. The de nition proceeded via Szabo s poly categories mentioned above. Although not explicitly stressed in the de nition [5] linear bicategories have a similar underlying (albeit 2 dimensional) structure. We introduce this in Section 1 under the name poly bicategory . Gentzen s cut then provides ....

[Article contains additional citation context not shown here]

Cockett, J. R. B., and Seely, R. A. G. Weakly distributive categories. In Applications of Categories to Computer Science (1992), M. P. Fourman, P. T. Johnstone, and A. M. Pitts, Eds., vol. 177 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 45-65.

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J.R.B. Cockett and R.A.G. Seely (1997) Weakly distributive categories. Journal of Pure and Applied Algebra 114 133-173. (Updated version available on http://www.math.mcgill.ca/rags.)

....the notions necessary for the development of category theory enriched in a linear bicategory. To Saunders Mac Lane, on the occasion of his ninetieth birthday 1 Introduction In [Cockett et al. 1999] we introduced linear bicategories as a natural extension of the linearly distributive categories of [Cockett Seely 1992] to the 2 dimensional world. When compared to ordinary bicategories, the most striking feature is the presence of two 1 cell compositions or tensors. These we denote by ( tensor ) and ( par , to remind us of the connections with Girard s linear logic, even though we do not use his notation) ....

....) B(X;Z) B(Z;W ) B(X;W ) L B(X;Y ) B(Y;Z) B(Z;W ) 1 1 B(X;Y ) B(Y;W ) B(X;Z) B(Z;W ) B(X;W ) CK R that must satisfy several coherences expressing the linear distributivity of the tensor over the par. These may be found in [Cockett Seely 1992, Cockett et al. 1999] Central in analyzing the structure of linear bicategories is a linearized notion of adjunction. We recall the basic idea; for the full story the reader is referred to [Cockett et al. 1999] A linear adjunction v = h ; i: A a B: X Y ( A is a left linear adjoint to B ) ....

J.R.B. Cockett and R.A.G. Seely (1992) Weakly distributive categories, in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177, 45-65.

....We develop a calculus of proof nets for linear functors, and show how linearity accounts for the essential coherence structure of the exponentials and the additives. Introduction What is the appropriate notion of a functor between linearly (formerly weakly ) distributive categories In [CS92] we were content to think of the functors between linearly distributive categories as being those which preserved all the structure on the nose. However, this very restrictive notion does not allow the expression of common linear structure such as the exponentials 1 Research partially supported ....

....More controversial perhaps is our insistence upon the use of for par and for the coproduct sum . As category theorists we are unrepentant upon this point, and there the matter must rest. 1 Linear functors For the full de nition of a linearly distributive category, we refer the reader to [CS92,CS92j,BCST] (where the term weakly distributive category is used) Brie y, a linearly distributive category is a category with two tensors ; and two strength natural transformations, making each strong (respectively costrong) with respect to the other. These two strength transformations shall be ....

[Article contains additional citation context not shown here]

J.R.B. Cockett and R.A.G. Seely \Weakly distributive categories", in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45-65.

....logic, it is quite productive to ignore the closed structure entirely and instead focus on the interaction between the tensor product and its dual cotensor, par. This was one of the motivations of the latter two authors in introducing linearly distributive categories. In a sequence of papers [CS92,BCST96,BCS96,CS97a,CS97b], it has been amply demonstrated that once one understands the linearly distributive structure, the extension of crucial structural results to autonomy is straightforward. These results are achieved by introducing a graph theoretic language for specifying morphisms which is inspired by proof ....

....preferring for the coproduct sum . We wish to thank the anonymous referee for several helpful remarks and suggestions. 1 The core of a MIX category 1.1 Preliminaries 1.1. 1 Linearly distributive categories For the full de nition of a linearly distributive category, we refer the reader to [CS92,CS92j,BCST96] (where the term weakly distributive category is used) Brie y, a linearly distributive category is a category with two tensors ; and two strength natural transformations, making each tensor strong (respectively costrong) with respect to the other. These strength transformations will be ....

[Article contains additional citation context not shown here]

J.R.B. Cockett and R.A.G. Seely \Weakly distributive categories", in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45-65.

....We show cut elimination for this fragment, and we introduce a simple notion of proof circuit, which provides a description of free contextual categories. 1 Introduction T HIS document has its roots in the attempt to elucidate the structure of linear logic using weakly distributive categories [CS91]. In that programme we started by investigating the categorical proof theory of the (two sided) linear cut rules, giving rise to the notion of weakly distributive categories. Our rationale was that by so doing we could better 1 Department of Mathematics, University of Ottawa, 585 King Edward ....

Cockett, J.R.B. and R.A.G. Seely "Weakly distributive categories", in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45--65. (Expanded version to appear in Journal of Pure and Applied Algebra.)

....other than that which occurs naturally as part of the reduction process; these must be the same apart from the wiring of thinning links if the nets are to be equivalent. The Rewiring Theorem then decides the equivalence of the nets at this point. 2 5. Adding negation As in the earlier papers (Cockett and Seely 1991; Blute et al. 1992) we can extend the current context to include negation, by adding two new links and two new rewrites, namely, one reduction and one expansion: fl) A A i : i : A A i : A A A i : A i : A A A i : A What is of interest in this case is that by adding negation to ....

Cockett, J.R.B., and Seely, R.A.G. (1992) Weakly distributive categories. In M.P. Fourman, P.T.

....by context categories; cut elimination holds for this fragment. Categorical cut elimination also is valid, but a proof of this fact is deferred to a sequel. Introduction This document has its roots in the attempt to elucidate the structure of linear logic using weakly distributive categories [CS91]. In that programme we started by investigating the categorical proof theory of the (two sided) linear cut rules, giving rise to the notion of weakly distributive categories. Our rationale was that by so doing we could better modularize the structure of linear logic which would facilitate the ....

.... on the left of the turnstile (i.e. in the hypothesis) Our next step is therefore to address the remaining half: we dualize the material above, giving a notion of cocontext. For context and cocontext to fit together properly, there are three weak distributivities (very much in the spirit of [CS91]) which allow the construction of a fibrational fork from the fibration and cofibration induced by context and cocontext. These distributivities we will discover correspond precisely to the cut rules of the sequent calculus for unified logic. To be able to handle at least the tensor par fragment ....

[Article contains additional citation context not shown here]

Cockett, J.R.B. and R.A.G. Seely "Weakly distributive categories", in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45--65. (Expanded version to appear in Journal of Pure and Applied Algebra.)

....paper were produced with the help of the T E Xcad drawing program of G. Horn and the diagram macros of F. Borceux. 4 Collaboration on this work partially supported by funds from NSERC, Canada. 0 Introduction Weakly distributive categories were defined by J.R.B. Cockett and R.A.G. Seely in [CS91]. The basic structure is that of a category with two tensors, but the usual distributive law is modified to be resource sensitive . The usual distributive law, as stated for example in [Lp72] has an implicit asymmetry in that the number of occurrences of variables is not the same on the left ....

....distributivity is a natural transformation which acts simultaneously as a linear strength and costrength linking the two monoidal structures, and a weakly distributive category is a category equipped with two monoidal structures so linked. For more details as well as the complete definition, see [CS91]. This resource sensitive character is something that weakly distributive categories share with linear logic [G87] In fact, weakly distributive categories correspond precisely to multiplicative linear logic without negation. This is reflected by the fact that adding negation is precisely what is ....

[Article contains additional citation context not shown here]

Cockett, J.R.B. and R.A.G. Seely "Weakly distributive categories", in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45--65. (Expanded version to appear in Journal of Pure and Applied Algebra.)

No context found.

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J. Cockett and R. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J. Cockett and R. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J. Cockett and R. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J. Cockett and R. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

Cockett, J.R.B., Seely, R.A.G.: Weakly distributive categories. In Applications of Categories in Computer Science: Proceedings LMS Symp., Durham, UK, 20--30 July 1991. Volume 177. Cambridge University Press, Cambridge (1992) 45--65

No context found.

J.R.B. Cockett and R.A.G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114:133--173, 1997.

No context found.

Cockett, J. R. B., and Seely, R. A. G. Weakly distributive categories (corrected version at http://www.math.mcgill.ca/rags/linear/wdc.ps.gz). J. Pure Appl. Algebra 114 (1997), 133173.

No context found.

Cockett, J. R. B., and Seely, R. A. G. Weakly distributive categories. In Applications of Categories to Computer Science (Durham,

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J. R. B. Cockett and R. A. G. Seely, Weakly distributive categories, in: M. P. Fourman, P. T. Johnstone, and A. M. Pitts, eds., Applications of Categories in Computer Science (Cambridge University Press, 1992) 45--65.

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