The change-base issue for \(\Omega\)-categories
type: Item
instance of: scholarly article
title: The change-base issue for \(\Omega\)-categories
zbMATH Open document ID: 1147.18006
author: Chengling Fang
author: Dexue Zhang
published in: Chinese Annals of Mathematics. Series B
publication date: 2008-09-08T00:00:00Z
review text: If \((\Omega ,*,I)\) is a commutative and unital quantale (that is, the unit element \(I\) of \(\Omega \) coincides with the top element in \(\Omega \)), an \( \Omega\)-category is a set \(A\) together with an assignment of an element \( A(a,b)\in \Omega \) to every ordered pair \((a,b)\in A\times A\) such that \( I\leq A(a,a)\) for every \(a\in A\) and \(A(a,b)*A(b,c)\leq A(a,c)\) for all \( a,b,c\in A.\) An \(\Omega\)-functor between \(\Omega\)-categories \(A\) and \(B\) is a function \(f:A\rightarrow B\) such that \(A(a,b)\leq B(f(a),f(b)),\) for all \(a,b\in A.\) If \(G:(\Omega ,*,I)\rightarrow (\Omega ^{\prime },*^{\prime },I^{\prime })\) is a closed unital map between commutative, unital quantales, then \(G\) induces a canonical functor \(\overline{G}:\Omega \)-\({\mathcal C}at\rightarrow \Omega ^{\prime }\)-\({\mathcal C}at\). This paper is concerned with some basic questions about \( \overline{G}:\) when does \(\overline{G}\) have a left adjoint which is also of this form; when \(\overline{G}\) is an equivalence of categories and when does \(\overline{G}\) preserve completeness? The main results are: (1) when \(\Omega\), \(\Omega ^{\prime }\) are integral, \( G:\Omega \rightarrow \Omega ^{\prime }\) and \(F:\Omega ^{\prime }\rightarrow \Omega \) are closed unital maps, then \(\overline{F}\) is a left adjoint of \( \overline{G}\) if and only if \(F\) is a left adjoint of \(G;\) (2) \(\overline{G}\) is an equivalence of categories if and only if \(G\) is an isomorphism in the category of commutative unital quantales and closed unital maps; and (3) a sufficient condition is obtained for \(\overline{G}\) to preserve completeness in the sense that \(\overline{G}A\) is a complete \(\Omega ^{\prime }\)-category whenever \(A\) is a complete \(\Omega \)-category.
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 06F07
Mathematics Subject Classification ID: 18B35
zbMATH DE Number: 5322654
zbMATH Keywords: commutative unital quantale
zbMATH Keywords: closed unital map
zbMATH Keywords: enriched category
zbMATH Keywords: change-base
reviewed by: Dimitru Buşneag
Wikidata QID: Q58331932
MaRDI profile type: Publication
full work available at URL: s11401-007-0401-y
OpenAlex ID: W1981565079
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DOI: 10.1007/S11401-007-0401-Y
description: scientific article; zbMATH DE number 5322654