M. Barr. #-autonomous categories, volume 752 of LNM. Springer-Verlag, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....another object (W; W 0 ) just when (X; X 00 ) W; W 0 ) that is when and X W and W 0 X 0 in X . We now show that this H Theta H op is a autonomous category. We show this via the Chu construction. The Chu construction was introduced by Chu, a student of Barr s, in the appendix of (Barr 1979), as a method of producing autonomous categories. Definition 17 Chu (Barr 1979) Let V be a finitely complete symmetric monoidal closed category, and let be an object of V. We define a category Chu(V; such that ffl the objects are Chu spaces, that is, triples (V; V 0 ; v) where V; V 0 ....

....X W and W 0 X 0 in X . We now show that this H Theta H op is a autonomous category. We show this via the Chu construction. The Chu construction was introduced by Chu, a student of Barr s, in the appendix of (Barr 1979) as a method of producing autonomous categories. Definition 17 Chu (Barr 1979) Let V be a finitely complete symmetric monoidal closed category, and let be an object of V. We define a category Chu(V; such that ffl the objects are Chu spaces, that is, triples (V; V 0 ; v) where V; V 0 are objects of V and v: V Omega V 0 is a morphism in V, ffl a morphism from ....

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Barr, M. 1979. -Autonomous categories, LNM 752. Springer-Verlag.

....the order; it implies that the commutative tensor is stronger than the non commutative one . For a more substantial survey on NL, see [2, 18] In the present paper, we investigate the de nition of categorical models of multiplicative NL (section 2) which have two autonomous structures [3, 4] together with a monoidal natural transformation, called entropy, from one tensor product to the other one. The entropy maps turn out to be the most interesting part of the de nition, and for this reason, we call such categories entropic. Section 2 comes with what is assumed from a denotational ....

M. Barr. -autonomous categories. Springer LNM, 752, 1979.

....to 2 functors and the adjunction is a 2 adjunction. The same is true for similar results to be discussed below, but since we are only interested in the one dimensional aspects we will not mention this explicitly. The category V lt: SLat is self dual (in fact, autonomous in the sense of [1]) and the induced adjunction between SLat and V lt: SLat op is induced by the schizophrenic object 2. Next we restrict the functor F: SLat V lt: SLat to interesting sub categories. We start with DLat, the category of distributive lattices. Lemma 1.3 The functor F restricts to a ....

M. Barr. -autonomous categories, revisited. J. Pure Appl. Algebra, 111:1-20, 1996.

....pairs of morphisms from X such that (X; X 0 ) 2 X Theta X op has a morphism to another object (W; W 0 ) just when f : X W and g: W 0 X 0 in X that is whenever X W and W 0 X 0 . 11.1 Chu The Chu construction was introduced by Chu, a student of Barr s, in the appendix of (Barr 1979), as a method of producing autonomous categories. Definition 33 Chu(Barr 1979) Let V be a finitely complete symmetric monoidal closed category, and let be an object of V. We define a category Chu(V ; such that ffl the objects are Chu spaces, that is, triples (V; V 0 ; v) where V; V 0 ....

....to another object (W; W 0 ) just when f : X W and g: W 0 X 0 in X that is whenever X W and W 0 X 0 . 11.1 Chu The Chu construction was introduced by Chu, a student of Barr s, in the appendix of (Barr 1979) as a method of producing autonomous categories. Definition 33 Chu(Barr 1979) Let V be a finitely complete symmetric monoidal closed category, and let be an object of V. We define a category Chu(V ; such that ffl the objects are Chu spaces, that is, triples (V; V 0 ; v) where V; V 0 are objects of V and v: V Omega V 0 is a morphism in V, 8 ffl a ....

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Barr, M. 1979. -Autonomous categories, LNM 752. Springer-Verlag.

.... autonomous categories are exactly the Chu coalgebras. 1. Introduction The Chu construction was devised by Michael Barr and his student Po Hsiang Chu as means to show that there are plenty of autonomous categories. It first appeared in Chu s master s thesis and in the appendix of Barr s book (Barr 1979). It differed in spirit from the methods pursued in the rest of the book, and looked a bit mysterious and ad hoc. Strangely enough, starting from a completely different background, Hyland and his student de Paiva (de Paiva 1989) arrived at a very similar structure, which they called Dialectica ....

Barr, M. (1979) -Autonomous Categories, Lecture Notes in Mathematics 752. Springer-Verlag.

....to 2 functors and the adjunction is a 2 adjunction. The same is true for similar results to be discussed below, but since we are only interested in the one dimensional aspects we will not mention this explicitly. The category V filt: SLat is self dual (in fact, autonomous in the sense of [1]) and the induced adjunction between SLat and V filt: SLat op is induced by the schizophrenic object 2. Next we restrict the functor F : SLat V filt: SLat to interesting sub categories. We start with DLat, the category of distributive lattices. Lemma 1.3 The functor F restricts to a ....

M. Barr. -autonomous categories, revisited. J. Pure Appl. Algebra, 111:1--20, 1996.

....with constructivity calling for the substitution of categories for algebras. A constructive model of linear logic consists of a autonomous category D with all finite products and a comonad with natural isomorphisms (A Theta B) A Omega B and 1 = A autonomous category [Bar79] is a representably self dual closed symmetric monoidal category D. Monoidal means that there exists a binary operation (functor) Omega : D 2 D and an object of D having A Omega (B Omega C) A Omega B) Omega C and Omega A = A = A Omega , satisfying certain coherence ....

M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....algebra to combine eks s in various ways, showing how these operations correspond to the usual operations for concurrent processes. Finally we connect them with event structures, showing how any event structure can be represented as an eks. These structures have previously been studied by Barr[Bar79], as instances of Chu s construction on sets, and by Lafont and Streicher[LS91] as games over 2. Brown and Gurr[BG90] have used similar structures to study Petri nets. 1.1 Theory of an eks An eks (E; Q) is a set Q of subsets of 2 E . We can therefore write it out as a formula in Disjunctive ....

M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....is highly dependent on the ability of the observer to understand the observed object. 6 Notes and Acknowledgements Historical notes. Chu spaces are the case V = Set of the construction described by Po Hsiang Chu in the appendix of Barr s book on autonomous (i.e. self dual closed) categories [Bar79]. Chu s construction takes a closed monoidal category V with pullbacks and completes it to a self dual category Chu(V; k) De Paiva [dP89a, dP89b] and Brown and Gurr [BG90, BGdP91] apply the Chu construction to respectively a version of Godel s Dialectica and Petri nets. Lafont and Streicher study ....

M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....these as gate based schedules that mention each event only once. This eliminates the usual dependence on labeling needed to distinguish these two notions from respectively late branching a(b c) and true concurrency akb. In addition, by tying this framework into the algebra of Chu spaces [Bar79, LS91, Bar91, Pra93] we implicitly bring to bear on gates as acceptors the by now considerable machinery of both Chuology and linear logic, with the expectation moreover of seeing many further developments in this very interesting and (we conjecture) extremely rich new field. 2 Processes as Gates 2.1 Dynamic ....

....as its open sets) and homomorphisms of PDLat s. The category Chu(V; k) of Chu spaces and their associated transforms, with dualizing object k, was first defined in Po Hsiang Chu s master s thesis, which appeared as an appendix to his advisor Michael Barr s monograph on autonomous categories [Bar79]. Here and in [Pra93] we follow Lafont and Streicher [LS91] in focusing on Chu s construction for V = Set. In the present paper we further take k = f0; 1g. Numeric quantities are not the only mathematical objects on which one can perform arithmetic. One may also add and multiply vector spaces, ....

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M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....and so on. Chu spaces were first described in enriched generality by M. Barr to his student P. H. Chu, whose master s thesis on Chu(V; k) the V enriched category produced by what since came to be called the Chu construction, became the appendix to Barr s monograph on autonomous categories [Bar79]. The latter subject generated no interest at the time but a decade later was recognized by Seely [See89] as furnishing Girard s linear logic [Gir87] with a natural constructive semantics. Barr then proposed the Chu construction as a means of producing constructive models of linear logic [Bar91] ....

M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....a comprehensive summary of the state of the art in 1990. This paper is based on a model that is, we feel, a particularly clean example of the state of that art. It has two main sources for its basic structure, the event spaces of Winskel [NPW81, Win88] and the autonomous categories of Barr [Bar79], originally done entirely independently of any possibility of its application to computer science. More specifically it makes use of those autonomous categories arising from a construction studied by Barr s student Chu and reported in an appendix to Barr s book. Chu spaces specialize this ....

M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....composition respectively, which for some time now in our own writing about models of behavior [Pra86] we have been calling respectively concatenation, or sequence, and orthocurrence, or flow. This juxtaposition of the calculi is achieved by translating Chu s construction of autonomous categories [Bar79], ordinarily given in the rarefied atmosphere of commuting diagrams, into the same elementary set theoretic terms in which the Peirce calculus is customarily described. Although the Chu interpretation of linear logic is conventionally understood via adjunctions in terms of (co)products and tensor ....

....value or a set of proofs of Delta from Gamma. But when Girard presented his logic at a category theory conference in Boulder in 1988, M. Barr recognized the suitability for modeling linear logic of his autonomous categories in general and his student P. Chu s construction of such in particular [Bar79]. Chu spaces, as the objects of Chu s construction for the category of sets, seem to be a particularly attractive constructive model of linear logic. We have been using A B and A Omega B in our concurrency work, starting with [Pra86] where they are notated respectively AkB, called concurrence ....

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M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....then M(A) 1. Also, one may generalize these conditions somewhat, replacing all instances of 1 with any arbitrary constant c, and allowing propositions to have different (although fixed) values, where p has value v p , and p has value c Gamma v p [3] Other related work is given in [17] and [4]. Since the above is only a necessary condition, there has been a question as to whether some form of simple truth table or numerical evaluation function like the above could yield a necessary and sufficient condition for provability of constant multiplicative (comll) expressions. The main ....

M. Barr. *-autonomous categories. In: Lecture Notes in Mathematics 752, Springer, 1979.

....of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the Heisenberg Schrodinger quantum mechanical solution to analogous problems of causal interaction in physics. 1 Cartesian Dualism The Chu construction [Bar79] strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93] metamathematics [Pra93, Pra94a] and physics [Pra94b] Here we make a first attempt at applying them to philosophy. It might seem that ....

....the physical object A. States are possible, making a Chu space a kind of a Kripke structure [Gup93] only one state at a time may be chosen from the menu X of alternatives. Lafont and Streicher [LS91] were the first to single out Chu spaces as a case of the more general Chu construction Chu(V; k) [Bar79, Bar91], namely V = Set, worthy of separate attention as a natural model of linear logic [Gir87] embedding topological spaces, vector spaces, and coherent spaces. They referred to these objects as games, understanding j= as the payoff matrix of a vonNeumann Morgenstern two person game. There is a ....

M. Barr. -Autonomous categories, LNM 752. Springer-Verlag, 1979.

....Applications of Categories, Vol. 6, No. 1, pp. 5 24. AUTONOMOUS CATEGORIES: ONCE MORE AROUND THE TRACK To Jim Lambek on the occasion of his 75th birthday MICHAEL BARR ABSTRACT. This represents a new and more comprehensive approach to the autonomous categories constructed in the monograph [Barr, 1979]. The main tool in the new approach is the Chu construction. The main conclusion is that the category of separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those categories. ....

....is the Chu construction. The main conclusion is that the category of separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those categories. 1. Introduction The monograph [Barr, 1979] was devoted to the investigation of autonomous categories. Most of the book was devoted to the discovery of autonomous categories as full subcategories of seven different categories of uniform or topological algebras over concrete categories that were either equational or reflective ....

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M. Barr (1979), -Autonomous Categories. Lecture Notes in Mathematics 752.

.... more recent reference is [1] It associates to an autonomous (symmetric monoidal closed) category V with pullbacks and a fixed object K in V a autonomous category called Chu(V; K) This construction gives us many examples of autonomous categories, which are difficult to obtain directly (see [2]) For a more detailed discussion of the advantages of using the Chu construction to define and study the autonomous categories of [2] see [3] 2.2. The category Chu(V; K) Let V be an autonomous category that has pullbacks and let K be an object in V. Then define a category Chu(V; K) as ....

.... K in V a autonomous category called Chu(V; K) This construction gives us many examples of autonomous categories, which are difficult to obtain directly (see [2] For a more detailed discussion of the advantages of using the Chu construction to define and study the autonomous categories of [2] see [3] 2.2. The category Chu(V; K) Let V be an autonomous category that has pullbacks and let K be an object in V. Then define a category Chu(V; K) as follows. The objects are of the form (V; V 0 ; h Gamma; Gammai) where V and V 0 are objects of V and h Gamma; Gammai : V Omega V 0 ....

M. Barr, -Autonomous Categories, Lect. Notes in Math. 752, Springer-Verlag, Berlin 1979

....also autonomous under weaker conditions than had been given previously ( Barr, 1991) In the process we find conditions under which the intersection of a full reflective subcategory and its coreflective dual in a Chu category is autonomous. 1. Introduction 1.1. Chu categories. An appendix to [Barr, 1979] was an extract from the master s thesis of P. H. Chu that described what seemed at the time a too simple to be interesting construction of autonomous categories [Chu, 1979] In fact, this construction, now called the Chu construction has turned out to be surprisingly interesting, both as a way ....

M. Barr (1979), -Autonomous Categories. Lecture Notes in Mathematics 752, Springer-Verlag, Berlin, Heidelberg, New York.

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M. Barr. #-autonomous categories, volume 752 of LNM. Springer-Verlag, 1979.

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M. Barr. *-autonomous categories. In: Lecture Notes in Mathematics 752, Springer, 1979.

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M. Barr. -autonomous categories. Springer LNM, 752, 1979.

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Barr, M. (1979) -Autonomous Categories. Springer Lecture Notes in Math. 752.

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M. Barr (1979) -Autonomous Categories. Lecture Notes in Mathematics 752, SpringerVerlag, Berlin, Heidelberg, New York.

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M. Barr. *-Autonomous Categories . Springer LNM 752, 1979.

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M. Barr. *-Autonomous Categories . Springer LNM 752, 1979.

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