The Chu Construction: History of an Idea

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 self-dual categories such as complete semi-lattices, 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 semi-lattices, 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 desired properties. The desired properties were what I subsequently called ∗-autonomy. I do not want to go into the technical details here (see [Barr, 1999], for example). Basically, a ∗-autonomous category has an internal hom, denoted −◦, a symmetric monoidal structure, ⊗, along with the usual coherent isomorphisms Hom(A ⊗ B,C) ' Hom(A,B−◦C). In addition, there should be a “dualizing object”, ⊥, such that if we define A∗ = A−◦⊥, the subsequent canonical map (A−◦B) // (B∗−◦A∗) should be an isomorphism. There is a certain redundancy in this definition. For instance, the most efficient (and perhaps the most natural) way is to assume just −◦ and ⊥ and define A∗ = A−◦⊥ and then A⊗B = (A−◦B∗)∗, since the latter two are provably naturally isomorphic. By the end of the year, I had in fact produced a moderate number of examples of ∗autonomous categories. One of them was a full subcategory of topological abelian groups that included all the locally compact abelian (LCA) groups in such a way that the duality restricted to them was the well-known duality of LCA groups. As one would expect, the the circle group was the dualizing object. In addition, the category was complete and cocomplete. In fact, it was built, essentially, by completing the LCA groups. This work was supported by a grant from the NSERC of Canada Received by the editors 2003-06-06 and, in revised form, 2006-11-09. Published on 2006-12-16 in the volume Chu spaces: theory and applications. 2000 Mathematics Subject Classification: 18-03, 01A65, 18D15.