J. Lambek. Deductive systems and categories II. In Lecture Notes in Mathematics 87. Springer-Verlag, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be such spaces, and a morphism of the above form will be a linear function: Bm Thus polycategories have proven to be quite useful in the analysis of (ordinary) categories in which one can form tensor products of objects. Indeed this was the original motivation for their de nition. See [Lam69, Sza75]. Categories in which one has a reasonable notion of tensor product 15 are called monoidal, and have recently gured prominently in several areas of mathematical physics, most notably topological quantum eld theory [Ati90, BD95] The second well known application of polycategories is to logic. ....

J. Lambek. Deductive systems and categories II. In Lecture Notes in Mathematics 87. Springer-Verlag, 1969.

....existential requirement for bicategories could be elegantly avoided by dispensing with the global composition and considering multi 2 cells with a finite sequence of inputs, but just one output. This yields a 2 dimensional generalization of multi categories , as introduced by Lambek [19], that (together with double categories) is subsumed by Tom Leinster s fc multi categories [20] Parallel to Szabo s [26] generalization of multi categories to poly categories (with finite strings of objects as inputs and outputs) it was natural to consider generalizing these ....

.... to three axioms: ID) cut has identities: # 1 #) i = y implies # 1 y = # and (# 0 #) j = y implies 1 y #; AS) cut is associative: if # # are defined, then (# i k l j (IC) cut satisfies the interchange property (referred to as commutativity by Lambek [19]) # are defined and j l, then # #) # 0 # l # are defined and i k, then (# k # 0 # # = # #. The poly bicategories P arise by reversing the 1 cells and the poly 2 cells of P, respectively. 1.2. Remark. 1. For any object A of ....

Lambek, J. Deductive systems and categories II. In Category Theory, Homology Theory and their Applications I (1968.

....This is a preliminary version. The nal version will be published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl locate entcs 1 Introduction In 1969 Jim Lambek introduced multicategories as a framework for his logicinspired syntactic calculus [10]. In terms of pasting diagrams or circuit diagrams [5] which we prefer here) the theory of multicategories concerns the compositional properties of multi 2 cells of the form 33 or g 0 g 1 : gn 1 h The input or source is a nite, possibly empty, string of 1 cells , ....

Lambek, J., Deductive systems and categories II, in: Category Theory, Homology Theory and their Applications I, LNM 86 (1968), pp. 76-122.

....to be double semigroups where each multiplication associates with the other, re ecting the de ning property of linearly distributive categories. This provides a further link between the latter and planar poly categories. 0 Introduction Multi categories were introduced in 1969 by Jim Lambek [7] as a framework for his logicinspired syntactic calculus. In more geometric terms, speci cally, in the language of circuit diagrams [3] the theory of multi categories may be viewed as describing the compositional properties of multi 2 cells of the form g0 g1 : gn 1 h (0 00) Read from top ....

Lambek, J. Deductive systems and categories II. In Category Theory, Homology Theory and their Applications I (

....# B = df A#B, as the primary connective [191, 192] 2.5 Lambek Calculus Lambek worked on his calculus to model the behaviour of syntactic and semantic types in natural languages. He used technique from proof theory [149, 150] as well as techniques from category theory which we will see later [151]) His techniques built on work of Bar Hillel [15] and Ajdukiewicz [4] who in turn formalised some insights of Husserl. The logical systems Lambek studied contain implication connectives and a fusion connective. Fusion in this language is not commutative, so it naturally motivates two implication ....

JOACHIM LAMBEK. "Deductive Systems and Categories II". In PETER HILTON, editor, Category Theory, Homology Theory and their Applications II, volume 86 of Lecture Notes in Mathematics. Springer-Verlag, 1969.

....the above form will be a linear function: f : A 1 A 2 : A n B 1 B 2 : Bm Thus polycategories have proven to be quite useful in the analysis of (ordinary) categories in which one can form tensor products of objects. Indeed this was the original motivation for their de nition. See [Lam69, Sza75]. Categories in which one has a reasonable notion of tensor product are called monoidal, and have recently gured prominently in several areas of mathematical physics, most notably topological quantum eld theory [Ati90, BD95] The second well known application of polycategories is to logic. ....

J. Lambek. Deductive systems and categories II. In Lecture Notes in Mathematics 87. Springer-Verlag, 1969.

.... treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5] Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the internal language supports a compositional dinatural interpretation (between definable functors) In ....

J. Lambek. Deductive Systems and Categories II, Springer LNM86 (1969), pp. 76-122.

....and so by coding strings of variables as tuples, we recover the complete class of p time functions as part of the class of polynomially curried functions. We expect to present more details in a separate paper. An aside for the category theorist: our structure is essentially a closed multicategory [9, 10] with finite products and coproducts: oe gives the internal hom but this is not a hom for the product structure given by . We do not have a tensor Omega the comma in the sequent notation takes that role but if we did, it would not be symmetric. We can find no operational significance for any ....

J. Lambek, "Deductive systems and categories II", Springer LNM 86 (1969), 76 -- 122.

....A . Second, among the structure rules we keep thinning, but drop exchange and contraction, roughly the opposite of Girard s linear logic [7] Again, it was shown in [4] that one could have k Gamma A oe (B oe C) without having k Gamma B oe (A oe C) Our structure is then a closed multicategory [9, 10] with finite products and coproducts: oe gives the internal hom but this is not a hom for the product structure given by . We do not have a tensor Omega the comma in the sequent notation takes that role but if we did, it would not be symmetric. Furthermore, it would not satisfy the expected ....

....that we have something more complicated there than a straightforward notion of weak adjoint , especially with the universal quantifiers k 8 . The existential quantifier is a little more straightforward, and will be remarked upon briefly at the end of the paper. 2. 1 Definitions Recall from [9, 10] that Definition 10 A multicategory C consists of a set Ob(C) of objects and a set M (C) of morphisms, also called arrows, multimorphisms, just like a category, except that the source of a morphism is a finite sequence of objects, rather than a single object. The target of a morphism is ....

J. Lambek, "Deductive systems and categories II", Springer LNM 86 (1969), 76 -- 122.

....and morphisms in monoidal categories was exploited. For, when categorical morphisms can be represented as proof nets, the strongly normalizing reduction systems of these nets translate into categorical coherence theorems. Lambek s technique of representing morphisms as proofs of a sequent calculus (Lambek 1969) was intended to accomplish the same goal. While proof nets are a representation of the same proofs, they use a natural deduction style. For monoidal categories, natural deduction systems have a significant advantage: they capture the basic tensor coherences very succinctly and naturally. This, in ....

Lambek, J. (1969) Deductive systems and categories II. Lecture Notes in Mathematics 87, Springer-Verlag, Berlin, Heidelberg, New York.

.... treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5] Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the internal language supports a compositional dinatural interpretation (between definable functors) In ....

J. Lambek. Deductive Systems and Categories II, Springer LNM86 (1969), pp. 76-122.

....been a problem in the coherence theory for these categories. In the past most coherence results have had to make restrictive assumptions on the units; such restrictions are avoided with our approach. It is natural to expect that the logical approach to coherence, originally introduced by Lambek in [L69], could be exploited in this setting using linear logic. In [B92, B91] R. Blute proposed an approach to the coherence question for various theories of monoidal categories based on a natural deduction system for linear logic, as presented graphically by proof nets [G87] The advantage of this ....

....identity net. j Omega j Phi j Omega j Phi j Phi j Phi ( B Phi I) Omega I ) Phi I I I I B ( B Phi I) Omega I ) Phi I In the nets above, if I were not a unit, these would be the expanded normal forms, and clearly these nets are not the same. An old idea of Lambek s [L69] may be seen here: the generality of the first net is clearly a derivation of the sequent ( B Phi C) Omega C ) Phi D Gamma ( B Phi D) Omega E ) Phi E, whereas the generality of the second is ( B Phi C) Omega D) Phi E Gamma ( B Phi C) Omega D) Phi E. This is no surprise; ....

Lambek, J. "Deductive systems and categories II", Lecture Notes in Mathematics 87 (Springer-Verlag, Berlin, Heidelberg, New York, 1969).

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J. Lambek, "Deductive systems and categories II," Springer Lecture Notes in Mathematics 86, pp. 76 - 122, Berlin, 1969.

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