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Category theory for linear logicians
 Linear Logic in Computer Science
, 2004
"... This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categori ..."
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Cited by 13 (2 self)
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This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus. 0
The Shuffle Hopf Algebra and Noncommutative Full Completeness
, 1999
"... We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformati ..."
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Cited by 9 (3 self)
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We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cutfree proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *autonomous category, canonically enriched over vector spaces. This paper
A Noncommutative Full Completeness Theorem (Extended Abstract)
 Elsevier Science B.V
, 1996
"... ) R.F. Blute 1 P.J. Scott 1 Dept. of Mathematics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA E. N. T. C. S. Elsevier Science B. V. Abstract We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (Cy ..."
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Cited by 3 (3 self)
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) R.F. Blute 1 P.J. Scott 1 Dept. of Mathematics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA E. N. T. C. S. Elsevier Science B. V. Abstract We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (CyLL), first defined by Yetter. The semantics is obtained by considering dinatural transformations on a category of topological vector spaces which are invariant under certain actions of a noncocommutative Hopf algebra, called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that the space has the denotations of cutfree proofs in CyLL+MIX as a basis. This work is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic. In that paper, we consider dinaturals which are invariant under certain actions of the additive group of integers. The passage from group...
Edinburgh, Scotland
, 1998
"... Most tensored \Lambdacategories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact ..."
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Most tensored \Lambdacategories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored \Lambdacategories, all morphisms are nuclear, and in the tensored \Lambdacategory of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored \Lambdacategories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations.