On cyclic star-autonomous categories
type: Item
instance of: scholarly article
title: On cyclic star-autonomous categories
zbMATH Open document ID: 1327.18014
author: J. M. Egger
author: M. B. McCurdy
published in: Journal of Pure and Applied Algebra
publication date: 2012-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.5099
reviewed by: Serguei V. Solov'ev
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 03F52
zbMATH DE Number: 6094134
MaRDI profile type: Publication
OpenAlex ID: W2094816363
arXiv ID: 1007.5099
cites work: Nonsymmetric \(^{\ast}\)-autonomous categories
cites work: Natural deduction and coherence for weakly distributive categories
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cites work: Weakly distributive categories
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cites work: The geometry of tensor calculus. I
cites work: Q4851998
cites work: \(*\)-autonomous categories of bimodules
cites work: Q5715381
cites work: Quantales and (noncommutative) linear logic
cites work: Q4693954
DOI: 10.1016/J.JPAA.2012.02.011
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review text: The authors consider star-autonomous categories in which the left and right duals of every object \(p\) are linked by coherent natural isomorphism. They call these categories ``cyclic star-autonomous''. They emphasize that initially they do not assume existence of any braided structure. In fact the definition is discussed in the introduction of the paper: ``It is well understood, at least in principle, that the term \textit{cyclic star-autonomous category} should mean a star-autonomous category equipped with a coherent natural isomorphism \(p^*\rightarrow\, ^*p\). But this raises the question: what are the right coherence axioms? This is complicated by the fact that there is a second approach to the fenomenon of cyclicity which does not explicitly refer to dual objects.'' The authors remark also that there are several (equivalent) definitions of star-autonomous category itself. In their paper is used the definition from \textit{J. R. B. Cockett} and \textit{R. A. G. Seely} [J. Pure Appl. Algebra 114, No. 2, 133--173 (1997; Zbl 0867.18008)]: ``a linearly distributive category with chosen left and right duals for every object''.NEWLINENEWLINENEWLINENEWLINEThey explain that coherence may be defined either in terms of isomorphisms \(p*\rightarrow *p\) or isomorphisms between \(Hom\)-sets involving \(tensor\) and \(par\). the first was used, e.g., by \textit{K. I. Rosenthal} [J. Pure Appl. Algebra 97, No. 2, 189--202 (1994; Zbl 0819.18005)] and another by Blute, Lamarche and Ruet (see [\textit{R. F. Blute} et al., Theory Appl. Categ. 10, 424--460 (2002; Zbl 1017.03037)]). They show (in Section 2) that the first approach in fact is weaker and the second stronger. To settle the question they introduce the additional notions of \(tensor\)- and \(par\)-semicyclicity, and call \textit{cyclicity} the stronger notion (it turns out to be a conjunction of two semicyclicities) and \textit{quasicyclicity} the weaker one. Section 2 contains coherence axioms and lemmas that describe …
description: scientific article; zbMATH DE number 6094134