On cyclic star-autonomous categories
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 the relationships between these axioms.NEWLINENEWLINENEWLINENEWLINEIn Section 3 the authors consider the links between the notion of cyclicity and profunctors. They ``show that such cyclic structures are the natural setting in which to consider enriched profunctors. Specifically, if \(V\) is a cyclic star-autonomous category, then the collection of \(V\)-enriched profunctors carries a canonical cyclic structure.''NEWLINENEWLINENEWLINENEWLINEIn Section 4 they consider ``the case of braided star-autonomous categories... the correspondences between cyclic structures and balances or tortile structures.'' Section 5 is devoted to strictification.NEWLINENEWLINENEWLINENEWLINEThe graphical calculus proposed by \textit{R. F. Blute} et al. [J. Pure Appl. Algebra 113, No. 3, 229--296 (1996; Zbl 0858.03064)] is used extensively. Some parts of the paper are sketchy: for example, the authors refer to \textit{D. N. Yetter} [J. Symb. Log. 55, No. 1, 41--64 (1990; Zbl 0701.03026)] for defintion of \textit{cyclic star-autonomous poset} but this term is not explicitly used by Yetter in his paper; there are cases when notation is not explained.
scientific article; zbMATH DE number 6094134
scientific article