A remark on fixed points of functors in topological categories
type: Item
instance of: scholarly article
title: A remark on fixed points of functors in topological categories
zbMATH Open document ID: 0879.18003
DOI: 10.1007/BF00124120
author: Jiří Adámek
published in: Applied Categorical Structures
publication date: 1997-11-19T00:00:00Z
review text: Let \(F\) be an endofunctor of a concrete category \(\mathcal K\). A cardinal \(\alpha\) is a fixed cardinal of \(F\) if for every \(\mathcal K\)-object \(U\) with an underlying set of cardinality \(\alpha\) the cardinality of the underlying set of \(FU\) is at most \(\alpha\). We say that a \(\mathcal K\)-object \(U\) is a fixed point of \(F\) if there exists an isomorphism \(f:FU@>>> U\). A fixed point \(U\) of \(F\) with an isomorphism \(f:FU@>>>U \) is the least (greatest) fixed point of \(F\) if for every fixed point \(V\) of \(F\) with an isomorphism \(g:FV@>>> V\) there exists exactly one \(\mathcal K\)-morphism \(h:U@>>> V\) (or \(h:V@>>> U\)) with \(g\circ Fh=h\circ f\) (or \(f^{-1}\circ h=Fh\circ g^{-1}\), respectively). If \(\mathcal K\) is a fibre-small monotopological category and if \(F\) preserves bimorphisms then \(F\) has the least fixed point whenever there exists a fixed cardinal of \(F\). If \(\mathcal K\) is a fibre-small topological category and if \(F\) preserves monomorphisms then \(F\) has the greatest fixed point whenever there exists a cardinal \(\alpha\) such that every cardinal \(\beta\) with \(\alpha\leq\beta\leq 2^{\alpha}\) is a fixed cardinal of \(F\).
Mathematics Subject Classification ID: 18A22
Mathematics Subject Classification ID: 68Q65
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 68Q55
zbMATH DE Number: 1053695
zbMATH Keywords: fixed point of functor
zbMATH Keywords: topological category
zbMATH Keywords: least (greatest) fixed point of functor
zbMATH Keywords: endofunctor
zbMATH Keywords: concrete category
reviewed by: Václav Koubek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00124120
OpenAlex ID: W1589210210
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description: scientific article; zbMATH DE number 1053695