Barr M. [1979] *-Autonomous Categories, Springer Lecture Notes in Mathematics 752, Berlin.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....measurement. As a result, it is not possible to uniquely reconstruct an object from measurement results. In other words, each measurement is a function r(x; y) of two variables: an object x and a (not completely known) measuring device y. Such a function describes a so called Chu space (see, e.g. [1, 2, 7, 8, 20, 21, 22, 23, 24, 25, 26]) 1.3 Precise Definition of a Chu Space To be more precise, to define a Chu space, we must fix a set K (of possible values) Then, a K Chu space is defined as a triple (X; r; Y ) where X and Y are sets, and r : X Theta Y K is a function which maps every 1 pair (x; y) of elements x 2 X and ....

M. Barr, *-Autonomous Categories, Springer Lecture Notes in Mathematics, Vol. 752, Springer-Verlag, Berlin, 1979.

....are Banach spaces, i.e. complete normed vector spaces, and whose morphisms are linear maps of norm less than or equal to 1. This is a symmetric monoidal closed category, when the tensor product is taken to be the completed projective tensor [36, 6] One can then apply the Chu construction to BAN 1 [7]. In so doing, we obtain a autonomous category of topological vector spaces in which products and coproducts no longer coincide. Explicitly, if V; W 2 BAN 1 , then we have the following formulas: Products jj(v; w)jj = maxfjjvjj; jjwjjg Coproducts jj(v; w)jj = jjvjj jjwjj These correspond ....

M. Barr, -Autonomous Categories, Springer Lecture Notes in Mathematics 752, (1980).

.... We first got acquainted with the notion of Chu spaces through the paper [6] presented at the First International Workshop on Current Trends and Developments in Fuzzy Logic (Thessaloniki, Greece, October 1998) This notion was originally introduced as a purely mathematical notion, by Po Hsiang Chu [2]. Since then, Chu spaces were used to model concurrency in computer science (see, e.g. 7] and more recently, to model information flow in distributed systems [3] 1.4 It Is Desirable to Use Chu Spaces to Describe Uncertainty In the above applications, Chu spaces have a clear advantage over ....

M. Barr, *-Autonomous Categories, Springer Lecture Notes in Mathematics, Vol. 752, SpringerVerlag, Berlin, 1979.

....As a result, it is not possible to uniquely reconstruct an object from measurement results. In other words, each measurement is a function r(x; y) of two variables: an object x and a (not completely known) measuring device y. Such a function describes a so called Chu space (see, e.g. [1, 2, 7, 8, 16, 17, 18, 19, 20, 21, 22]) 1.3. Precise definition of a Chu space To be more precise, to define a Chu space, we must fix a set K (of possible values) Then, a K Chu space is defined as a triple (X; r; Y ) where X and Y are sets, and r : X Theta Y K is a function which maps every pair (x; y) of elements x 2 X and y ....

M. Barr, *-Autonomous Categories, Springer Lecture Notes in Mathematics, Vol. 752, Springer-Verlag, Berlin, 1979.

....A Omega (B Gammaffi A 0 ) f 0 (ev AA 0 (a Omega g o f) ev BB 0 (f(a) Omega f 0 o g) 5) A generalization of this example to nested evaluations is in Lemma 10.2 below. 7. 2 Interpreting MLL Sequents Functorial polymorphism can be extended to handle Barr s autonomous categories [7], i.e. smc categories C equipped with an involution functor ( C op C given by a dualising object. Such categories interpret the multiplicative fragment of classical linear logic [42, 9] We modify the functorial interpretation of formulas mentioned earlier to the Omega ; ....

M. Barr, -Autonomous Categories, Springer Lecture Notes in Mathematics 752, (1980).

.... category means the structure making valid the proportion linear logic : linear category = typed calculus : cartesian closed category then what is a linear category This question is 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, ....

.... : is a contravariant functor (in view of ( var) that it is strong (in view of ( AB ) and that it is an involution (in view of (d) which thus must be inverse to (d Gamma1 ) These yield further equations, including the following, if we are to have a autonomous category, as defined in Barr [1979]) for any A; B, these derivations of the sequent A GammaffiB : A Gammaffi: B are equal: d Gamma1 Gammaffi d) B:A ) AB ) Here (d Gamma1 Gammaffi d) is a case of the schema C (f) Gamma A B (g) Gamma D A GammaffiB (f Gammaffig) Gamma C GammaffiD given by A; A ....

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M. Barr, -autonomous categories, Springer Lecture Notes in Mathematics 752, Berlin, 1979.

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Barr M. [1979] *-Autonomous Categories, Springer Lecture Notes in Mathematics 752, Berlin.

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M. Barr, *-Autonomous Categories, Springer Lecture Notes in Mathematics, Vol. 752, Springer-Verlag, Berlin, 1979.

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