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THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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Cited by 12 (1 self)
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We take another look at the Chu construction and show how to simplify it by looking at
The Separated Extensional Chu Category
 Theory and Applications of Categories
, 1998
"... . This paper shows that, given a factorization system, E=M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also autonomous under weaker conditions than had been given previously ([Barr, 1991)]. In the process we find conditions u ..."
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Cited by 9 (0 self)
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. This paper shows that, given a factorization system, E=M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is 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 toosimpletobeinteresting 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 of providing models of Girard's linear logic [Seely, 1988], in theoretical computer science [Pratt, 1993a, 1993b, 1995] and as a general approach to duality [Barr and Kleisli, to apear] and [Schlapfer, 1998].. Given a...
*Autonomous Categories: Once More Around The Track
 AND CHU CONSTRUCTIONS: COUSINS? 149
, 1999
"... . 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 equationa ..."
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Cited by 6 (1 self)
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. 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. 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 subcategories of equational categories. The base categories were: 1. vector spaces over a discrete field; 2. vector spaces over the real or complex numbers; 3. modules over a ring with a dualizing module; 4. abelian groups; 5. modules ove...
Chu I: cofree equivalences, dualities and *autonomous categories
, 1993
"... ing from the technique of dual pairs in functional analysis (Kelley, Nanmioka et al. 1963, ch. 5), they defined the objects of their category to be the triples hA; B; A\Omega B OE !?i, where A and B are arbitrary objects of an autonomous category V, and ? is a fixed object, chosen to become duali ..."
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Cited by 5 (1 self)
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ing from the technique of dual pairs in functional analysis (Kelley, Nanmioka et al. 1963, ch. 5), they defined the objects of their category to be the triples hA; B; A\Omega B OE !?i, where A and B are arbitrary objects of an autonomous category V, and ? is a fixed object, chosen to become dualizing. A morphism from hA; B; OEi to hC; D; fli was defined as a pair hu : A ! C; B / D : vi of Varrows, making the square A\Omega D A\Omega B C\Omega D ? u\Omega D<Fnan><Fnan> fflffl A\Omega v<Fnan><Fnan> // OE<Fnan><Fnan> fflffl fl<Fnan><Fnan> (1) Cofree equivalences, dualities and autonomous categories 3 commute. This is the setting in which the autonomous structure of a Chu category was originally discovered. The starting point of the present paper is the fact that the category described by Chu is isomorphic to the comma category V=? ? , induced by the homming functor ? ? : V op \Gamma! V : A 7\Gamma! A ? = A \Gammaffi? : (2) By definition, the objects of V=? ? (i.e. Id V =?...
Beyond the Chuconstruction
, 1999
"... . Starting from symmetric monoidal closed (= autonomous) categories, PoHsiang Chu showed how to construct new  autonomous categories, i.e., autonomous categories that are selfdual by virtue of having a dualizing object. Recently, Michael Barr extended this to the nonsymmetric, but closed, case, ..."
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. Starting from symmetric monoidal closed (= autonomous) categories, PoHsiang Chu showed how to construct new  autonomous categories, i.e., autonomous categories that are selfdual by virtue of having a dualizing object. Recently, Michael Barr extended this to the nonsymmetric, but closed, case, utilizing monads and modules between them. Since these notions are wellunderstood for bicategories, we introduce a notion of cyclic  autonomy for these that implies closedness and, moreover, is inherited when forming bicategories of monads and of interpolads. Since the initial step of Barr's construction also carries over to the bicategorical setting, we recover his main result as an easy corollary. Furthermore, the Chuconstruction at this level may be viewed as a procedure for turning the endo1 cells of a closed bicategory into the objects of a new closed bicategory, and hence conceptually is similar to constructing bicategories of monads and of interpolads. Keywords: closed bicate...
THE CHU CONSTRUCTION: HISTORY OF AN IDEA
"... Abstract. This paper describes the historical background and motivation involved in the discovery (or invention) of Chu categories. In 1975, I began a sabbatical leave at the ETH in Zürich, with the idea of studying duality in categories in some depth. By this, I meant not such things as the duality ..."
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Abstract. This paper describes the historical background and motivation involved in the discovery (or invention) of Chu categories. In 1975, I began a sabbatical leave at the ETH in Zürich, with the idea of studying duality in categories in some depth. By this, I meant not such things as the duality between Boolean algebras and Stone spaces, nor between compact and discrete abelian groups, but rather selfdual categories such as complete semilattices, finite abelian groups, and locally compact abelian groups. Moreover, I was interested in the possibilities of having a category that was not only self dual but one that had an internal hom and for which the duality was implemented as the internal hom into a “dualizing object”. This was already true for the complete semilattices, but not for finite abelian groups or locally compact abelian groups. The category of finite abelian groups has an internal hom, but lacks a dualizing object, while locally compact groups have a dualizing object, but not an internal hom that is defined everywhere. Although you could define an abelian group of continuous homomorphisms between locally compact abelian groups, there was no way of systematically putting a locally compact topology on the hom set that would lead to the
The Chu Construction
, 1996
"... . We take another look at the Chu construction and show how to simplify it by looking at it as a module category in a trivial Chu category. This simplifies the construction substantially, especially in the case of a nonsymmetric biclosed monoidal category. We also show that if the original category ..."
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. We take another look at the Chu construction and show how to simplify it by looking at it as a module category in a trivial Chu category. This simplifies the construction substantially, especially in the case of a nonsymmetric biclosed monoidal category. We also show that if the original category is accessible, then for any of a large class of "polynomiallike" functors, the category of coalgebras has cofree objects. 1. Introduction In a recent paper, I showed how the Chu construction, given originally in [Chu, 1979] for symmetric monoidal closed categories, could be adapted to monoidal biclosed (but not necessarily symmetric) categories. The construction, although well motivated by the necessity of providing a doubly infinite family of duals, was rather complicated with many computations involving indices. Recently I have discovered that the autonomous structure of Chu categories can be put into the familiar context of bimodules over a not necessarily commutative "algebra" objec...