We take another look at the Chu construction and show how to simplify it by looking at

. 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 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 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...

. 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...

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 V-arrows, 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 =?...

. Starting from symmetric monoidal closed (= autonomous) categories, Po-Hsiang Chu showed how to construct new - autonomous categories, i.e., autonomous categories that are self-dual by virtue of having a dualizing object. Recently, Michael Barr extended this to the non-symmetric, but closed, case, utilizing monads and modules between them. Since these notions are well-understood 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 Chu-construction 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...