J. Lambek, Multicategories revisited, Contemporary Mathematics, 92 (1989) 217239.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....b2Act Y ) Gamma Thus we can define T 2 (p) T 2 (p) a;b;c ) a;b;c2Act where T 2 (p) a;b;c : Y ) Gamma flZ. We set T 2 (p) a;b;c = flp m(a; b) c 0 otherwise. This definition forms part of the extension of T to a multifunctor on SProc viewed as a multicategory [38]. Since both T and T 2 are guarded, there is a unique invariant ff : A = TA, and a unique fixpoint m : A Gamma A such that TA Omega TA ff Omega T 2 ( m) TA 36 JP Theta QK = I JP K ThetaJQK Note that any generalized static operation [40] could be treated in the ....

J. Lambek. Multicategories revisited. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 217--240, 1989.

....Proposition 5.8 (i) Suppose 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 ....

J. Lambek. Multicategories revisited. In J. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 217-239. American Mathematical Society, 1989.

....permutations from holding, their proposal disallows such succedents. Thus, goal formulas are weaker than those presented here, but contexts are richer. The loss of ed goals, however, means that the examples in Section 5 cannot be coded directly. In the area of natural language parsing, Lambek [16, 17] used a logic that can be identified with a noncommutative variant of linear logic for inferring the syntactic categories of phrases. Recently, Pereira handled gaps using a (commutative) linear logic like context mechanism [26] Neither of these approaches use or the of course operator and, ....

J. Lambek. Multicategories revisited. In Categories in Computer Science, volume 92 of Contemporary Mathematics, pages 217 -- 239. AMS, June 1987.

.... and intuitionistic logic that is more subtle than the standard negative interpretation, see Girard [49, 52] Detailed descriptions of linear logic rules may be found in, e.g. 45, 72, 89] Let us mention that the nonmodal fragment of linear logic was anticipated by a calculus proposed by Lambek [66, 67], motivated by linguistic considerations of syntax of natural languages. A remarkable result of Lincoln and Winkler [75] shows that there is no simple minded truth table characterization of provability even for the multiplicative fragment of linear logic, unless p = np (see below) In this sense ....

J. Lambek. Multicategories revisited. In Categories in Computer Science and Logic, pages 217--239. Contemporary Math., vol. 92, American Math. Soc., Providence, RI, 1989.

....G, multi maps j : F 1 ; Fn ; F Gamma G are in bijective correspondence with multi maps j : F 1 ; Fn Gamma G F . Proposition 10 could be restated in more abstract terms by saying that functors in Predom Sigma , together with multi maps, comprise a closed multi category [Lambek 1989]. We regard this as a kind of categorical justification for, or commentary on, the form of the translation of SCI. At this point the reader may be wondering why we arranged the ( Delta) ffi translation in this multi map form: It would be more standard to follow the structure of a monoidal closed ....

Lambek, J. 1989. Multicategories revisited. In J. W. Gray and A. Scedrov Eds., Categories in Computer Science and Logic, Volume 92 of Contemporary Mathematics, pp. 217--240. American Mathematical Society.

....Research Council of Canada. We thank Narciso Mart i Oliet and Robert Seely for helpful comments on a preliminary version of this work. 1 ffl Natural Transformations from category theory. ffl Parametricity from the foundations of polymorphism. Familiar work of Curry, Howard, Lambek and others [12, 15, 17] has shown how we may consider constructive proofs as programs. For example, Gentzen s intuitionistic sequents A 1 ; A k B may be interpreted as functional programs mapping k inputs of types A i , 1 i k, to outputs of type B. More precisely, proofs are interpreted as certain terms of ....

.... Lambda Terms Although the Curry Howard procedure of associating typed lambda terms to natural deduction proofs (cf [12] is by now quite familiar, a similar process applied to Gentzen sequent calculus appears less so, despite the related work of Lambek connected to categorical coherence theorems [17, 18]. One motivation of this term assignment is to think of an intuitionist sequent 0 B (where 0 = fA 1 ; A k g) as an input output device, accepting k inputs of types A 1 ; A k and returning an output of type B. To this end, recall that our language of typed lambda calculus has ....

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J. Lambek. Multicategories Revisited, Contemp. Math.92, pp. 217-239.

....or with history. Whereas of certain interest in physics, it is not clear what impact this might have in computer science. The general considerations of this paper make sense not just for categories and functors, but also in other contexts; for example, for multicategories and multifunctors [Lam89] and for computads (see [Str96] 23 ....

J. Lambek. Multicategories revisited. In Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 217--239. AMS, 1989.

....understanding of Music as Language . More specifically, we employ techniques from Categorial Grammar to represent a rather specific and simple problem of music theory, which we believe nevertheless to be of widespread interest: functional harmonic analysis [Bri79] The aim of Categorial Grammar [Ben87, Ben90, Ben91, Lam58, Lam89] is the analysis of syntactic well formedness of sentences. The fundamental concept underlying Categorial Grammar is that of syntactic categories, which are classes to which words in a sentence must belong. Syntactic categories can be organised as formulae of some substructural logic e.g. the ....

J. Lambek. Multicategories Revisited. Contemporary Mathematics, 92:217--239, 1989.

.... quite easy, and in true categorical spirit, one finds that it was answered long before being put, namely by Barr [1979] Our intent here is mainly to supply a few details to make the matter more precise (though we leave many more details to the reader) to point out some similarities with work of Lambek [1987] (see these proceedings) and to appeal for a change in some of the notation of Girard [1987] Second, what is the meaning of Girard s exponential operator Since Girard has in fact offered several variants of in [1987] and another in Girard and Lafont [1987] one cannot be too dogmatic ....

.... of sequents needed to get the structure of a polycategory (Szabo [1975] these equations essentially make (cut) into a polycomposition of polyarrows which is associative, partially commutative , and has units (id A ) Analagous equations for multicategories may be found in this volume in Lambek [1987]; for this reason I will not go into detail here for these or the remaining equations. Next, we must account for the monoidal structure of I , Omega (and their duals OE; fi) by adding equations which make sequents A 1 ; An B 1 ; Bm equivalent to sequents A 1 Omega Delta ....

J. Lambek, "Multicategories revisited," Proc. A.M.S. Conf. on Categories in Computer Science and Logic, 1987.

....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, "Multicategories revisited", pp. 217 -- 239 in Categories in Computer Science and Logic, J.W. Gray and A. Scedrov, eds. (Contemporary Mathematics 92 (1989) American Mathematical Society).

....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 ....

....the graded sequent calculus described above will have much more structure, e.g. it will be graded closed since it has a graded internal hom given by the graded implication. Of course, all this is directly analagous to the ungraded case, and follows the paradigm case of a Gentzen multicategory [10]. Also, in the intended example the grading is filtered if a morphism has grade k then it also has grade k 0 , for any k 0 k, in a suitable ordering. However, it is not clear this should be part of the general definition. One final point: for simplicity, in this note we shall not consider ....

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J. Lambek, "Multicategories revisited", Proceedings of the A.M.S. Summer Conference on Categories in Logic and Computer Science, Boulder 1987.

.... of Cambridge 1 Introduction This paper contributes to the area of categorical combinatory logic or categorical combinators , following the steps of Curien [Cur93] and Ritter [Rit92] We provide a precise syntactic formulation of the notion of a multicategory (a recent reference is Lambek [Lam89]) and based on that we give categorical combinators for (multiplicative) Intuitionistic Linear Logic, following the general approach of Ritter for the Calculus of Constructions [Rit92] Multicategories are usually thought of as just like categories, except that instead of arrows A B one has ....

J. Lambek. Multicategories revisited. In Categories in Computer Science and Logic, volume 92 of AMS Contemporary Mathematics, pages 217--239, June 1989.

....A Delta; Sigma Gamma A (restrictedweakening) Delta Gamma A A; Sigma Gamma B Delta; Sigma Gamma B (cut) permutation at the left side of the arrow no contraction rule Here Sigma , Delta are lists of formulas, A,B,C formulas. This calculus was introduced by Lambek (see, e.g. [5,6]) Its description may also be found in the book by Szabo [7] a natural deduction system, which is equivalent to it with respect to categories) was described by Mints [8] and investigated by Babaev [9,10] Babaev s works have no English translation) The structure of a free closed category on our ....

J.Lambek. Multicategories Revisited.In J.W.Gray and A.Scedrov, editors, Categories in Computer Science and Logic. AMS, Providence, 217-239.

....Research Council of Canada. We thank Narciso Mart i Oliet and Robert Seely for helpful comments on a preliminary version of this work. ffl Natural Transformations from category theory. ffl Parametricity from the foundations of polymorphism. Familiar work of Curry, Howard, Lambek and others [12, 15, 17] has shown how we may consider constructive proofs as programs. For example, Gentzen s intuitionistic sequents A 1 ; A k B may be interpreted as functional programs mapping k inputs of types A i , 1 i k, to outputs of type B. More precisely, proofs are interpreted as certain terms of ....

.... Lambda Terms Although the Curry Howard procedure of associating typed lambda terms to natural deduction proofs (cf [12] is by now quite familiar, a similar process applied to Gentzen sequent calculus appears less so, despite the related work of Lambek connected to categorical coherence theorems [17, 18]. One motivation of this term assignment is to think of an intuitionist sequent Gamma B (where Gamma = fA 1 ; A k g) as an input output device, accepting k inputs of types A 1 ; A k and returning an output of type B. To this end, recall that our language of typed lambda ....

[Article contains additional citation context not shown here]

J. Lambek. Multicategories Revisited, Contemp. Math.92, pp. 217-239.

.... the connection between this categorical view of proofs and the two main styles of proof introduced by Gentzen (Natural Deduction and Sequent Calculus) introducing the notion of multicategory to model sequents in which more than one proposition occurs on either side of the turnstile, see [ Lambek, 1989 ] This has resulted in applications of proof theory to category theory, for example in the use of cut elimination theorems to prove coherence results: see [ Minc, 1977 ] Mac Lane, 1982 ] In the reverse direction of applications of category theory to proof theory, general categorical ....

J. Lambek. Multicategories revisited. In Gray and Scedrov

....as is needed. The relationship between L and Bollen s system is discussed in greater depth in the first author s dissertation, where an extension of L allowing direct representation of relevant and affine implication is also presented (Hodas, 1993) In the area of natural language parsing, Lambek (Lambek, 1958, 1987) has used a logic that can be identified with a non commutative variant of linear logic to infer the syntactic categories of phrases. Recently, Pereira described how the semantics of gaps could be computed using a linear logic like context mechanism (Pereira, 1990) his approach can be formalized ....

Lambek, J. (1987). Multicategories Revisited. In Categories in Computer Science, Vol. 92 of Contemporary Mathematics, pp. 217 -- 239. AMS.

.... numbers object in ccc s [20] In this paper we shall use recursors R since (i) we do not necessarily assume product types, ii) recursors are necessary for more general type theories (e.g. Girard s system F) as well as for more general categorical frameworks (e.g. Lambek s multicategories, [19]) b) We may consider extensions of typed lambda calculus by adding first order logic to the equational theory [23, 22] This will be discussed in the next section. Finally, we may extend by additional data types. For example, in the last section we consider adding the data type of Brouwer ....

....y = 0. Similarly, y . x = 0. Hence, using equation Proposition 3.4(ix) above, we obtain: x = x (y . x) y (x . y) y. 3.6. Definable Mal cev Operations. The rule M f;h is equational, provided the specification of the Mal cev operations mA are. In the pure typed lambda calculus, Lambek [19] showed that Mal cev operations are actually definable from cut off subtraction . provided we adjoin some simple equations (see below) Consider the inductively defined family of Theory and Applications of Categories, Vol. 6, No. 4 53 closed terms mA : A 3 ) A, uniquely determined (by ....

J. Lambek, Multicategories Revisited. Contemp. Math.92, 1989, pp. 217-239.

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J. Lambek, Multicategories revisited, Contemporary Mathematics, 92 (1989) 217239.

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J. Lambek, Multicategories revisited, in: Contemporary Mathematics 92 (Amer. Math. Soc., Providence 1989), pp. 217-239.

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