*-autonomous categories, revisited
type: Item
instance of: scholarly article
title: *-autonomous categories, revisited
zbMATH Open document ID: 0857.18010
DOI: 10.1016/0022-4049(95)00040-2
author: Michael Barr
published in: Journal of Pure and Applied Algebra
publication date: 1996-11-25T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 18D15
zbMATH DE Number: 920019
zbMATH Keywords: linearly topologized vector spaces
zbMATH Keywords: abelian topological groups
zbMATH Keywords: coalgebras
zbMATH Keywords: symmetric monoidal closed category
zbMATH Keywords: Chu construction
zbMATH Keywords: *-autonomous categories
zbMATH Keywords: Chu spaces
reviewed by: Heinrich Kleisli
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00040-2
OpenAlex ID: W2083853821
cites work: Q5620723
cites work: *-Autonomous categories and linear logic
cites work: Q4773818
cites work: Uniform Boundedness for Groups
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description: scientific article; zbMATH DE number 920019