Closure operators and functorial topologies
type: Item
instance of: scholarly article
title: Closure operators and functorial topologies
zbMATH Open document ID: 0659.18006
DOI: 10.1016/0022-4049(88)90118-1
author: G. Castellini
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: The basic idea for a closure operator on a category \({\mathcal X}\) is to have, for each object X, an extensive, increasing and idempotent operation [ ]\({}^ X\) on the ordered class of its subobjects \(M\to X\) that belong to a fixed class \({\mathcal M}\) of monomorphisms of \({\mathcal X}\); M is said to be closed if \([M]^ X\) is isomorphic to M. The pairs of the form (X,[ ]\({}^ X)\) and the ``continuous'' morphisms f of \({\mathcal X}\) form a concrete category CS(\({\mathcal X})\) over \({\mathcal X}\). Now, consider a concrete category \({\mathcal A}\) over \({\mathcal X}\). The author sets up a Galois correspondence between the subcategories \({\mathcal C}\) of \({\mathcal A}\) and the faithful functors F: \({\mathcal A}\to CS({\mathcal X})\) (also called global closures over \({\mathcal A})\). The ensuing theory is exploited and applied to the cases where both \({\mathcal A}\) and \({\mathcal X}\) consist of commutative groups or of commutative topological groups.
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18B30
zbMATH DE Number: 4077513
zbMATH Keywords: closed objects
zbMATH Keywords: abelian groups
zbMATH Keywords: Galois correspondence between subcategories and faithful functors
zbMATH Keywords: closure operator on a category
zbMATH Keywords: global closures
zbMATH Keywords: commutative topological groups
reviewed by: J. Sonner
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90118-1
OpenAlex ID: W1979191841
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description: scientific article; zbMATH DE number 4077513