Constant morphisms and constant subcategories
type: Item
instance of: scholarly article
title: Constant morphisms and constant subcategories
zbMATH Open document ID: 0821.18002
DOI: 10.1007/BF00877632
author: Maria Manuel Clementino
published in: Applied Categorical Structures
publication date: 1995-09-25T00:00:00Z
review text: Let \({\mathcal K}\) be a category with an \(({\mathcal E}, {\mathcal M})\)-factorization system and let \({\mathcal C}\) be a subcategory of \({\mathcal K}\) closed under \({\mathcal M}\)-subobjects and \({\mathcal E} \)-quotients. A morphism \(f : A \to B\) of \({\mathcal K}\) is called constant if \(f = g \circ h\) where \(h : A \to C\) for a \(C \in{\mathcal C}\). We say that a full subcategory \({\mathcal A}\) of \({\mathcal K}\) is right-constant if there exists a class \({\mathcal B}\) of \({\mathcal K}\)-objects such that any \({\mathcal K}\)-morphism \(f : B \to A\) is constant whenever \(B \in {\mathcal B}\) and \(A \in {\mathcal A}\). The aim of this paper is to give characterizations of right-constant subcategories of \({\mathcal K}\). For example, a right-constant category is \({\mathcal E}\)-reflective if and only if \({\mathcal C}\) is an \({\mathcal E}\)-reflective subcategory of \({\mathcal K}\), or if \({\mathcal K}\) is \({\mathcal M}\)-complete and \({\mathcal M}\)-wellpowered then a reflective subcategory \({\mathcal A}\) is right-constant just when it is strongly upwards-closed (i.e., if any terminal \({\mathcal A}\)-fan belongs to \({\mathcal A})\). Hence for an \({\mathcal M}\)-complete, \({\mathcal M}\)-wellpowered, \({\mathcal E}\)-cocomplete category \({\mathcal K}\) and its reflective subcategory \({\mathcal C}\), a subcategory \({\mathcal A}\) is right-constant if and only if \({\mathcal A}\) is reflective and strongly upwards-closed. Many illustrating examples are presented.
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 18E40
zbMATH DE Number: 758377
zbMATH Keywords: constant morphisms
zbMATH Keywords: constant subcategories
zbMATH Keywords: \(({\mathcal E}, {\mathcal M})\)- factorization
zbMATH Keywords: reflective subcategory
reviewed by: Václav Koubek
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00877632
OpenAlex ID: W2015267920
description: scientific article; zbMATH DE number 758377