*-autonomous categories, revisited
During the 70ties, the author studied in a series of papers the notion of a symmetric monoidal closed category endowed with a duality. He collected the results and a number of new examples in the book ``*-autonomous categories'', Lect. Notes Math. 752 (1979; Zbl 0415.18008), where he adopted the term autonomous for symmetric, monoidal, closed and introduced the name *-autonomous for the autonomous categories with a duality. In an appendix of that volume, there is a paper by \textit{P.-H. Chu} which describes a rather formal construction of a class of *-autonomous categories. At that time it was not anticipated that the construction would give interesting categories. Now, there are at least two reasons for revisiting *-autonomous categories. In the meantime, \textit{J.-Y. Girard} had created linear logic, and *-autonomous categories, especially those with an exponential, had become interesting as models of that logic. Secondly, it was of considerable interest to learn that the Chu construction could replace the complications of topology and completeness that characterized the authors original examples of *-autonomous categories. The paper under review starts with a finitely complete autonomous category \({\mathcal V}\) and an object \(K\) of \({\mathcal V}\). A category \(\text{Chu}({\mathcal V},K)\) and full subcategories \(\text{Chu}_{\text{se}}({\mathcal V},K)\) are introduced which all yield *-autonomous categories. Then the latter are employed in order to clarify the *-autonomous structure of the category \(SLC\) of sublinearly compact spaces and of the category \(SC\) of subcompact abelian groups. Finally, an example is given which shows how systematic use of the duality can simplify some aspects of the theory of coalgebras. The reviewer believes that the Chu construction will play a similar role in other situations, so that it is worthwile to describe here the objects of the category \(\text{Chu}_{\text{se}}({\mathcal V},K)\), called extensional, separated Chu spaces, by means of the following paradigmatic example. Let \({\mathcal V}\) be the category of vector spaces over a field \(K\). Then an extensional, separated Chu space is given by a couple \((V,V')\) of \(K\)-vector spaces together with separated duality \(\langle\;,\;\rangle:{\mathcal V}\times{\mathcal V}'\to K\). In order to generalize to arbitrary autonomous categories \({\mathcal V}\), just observe that a separated duality can be considered as a morphism \(V\otimes V'\to K\) in \({\mathcal V}\) such that the adjoint morphisms \(V\to[V',{\mathcal V}]\) and \(V'\to[V,{\mathcal V}]\) are monic. A morphism \((f,g):(V,V')\to (W,W')\) in \(\text{Chu}({\mathcal V},K)\) is given by a couple of morphisms \(f:V\to V'\) and \(g:W'\to W\) in \({\mathcal V}\) such that \[ \langle f(v),w'\rangle=\langle w,g(w')\rangle \quad\text{for all \(v\) in \(V\) and \(w'\) in }W'. \] The dual \((V,V')^*\) of a Chu space \((V,V')\) is then \((V,V')^*= (V',V)\). What is non trivial is the construction of the tensor product and of the internal hom in \(\text{Chu}_{\text{se}}({\mathcal V},K)\).
scientific article; zbMATH DE number 920019
scientific article