Injective hulls are not natural.
type: Item
instance of: scholarly article
title: Injective hulls are not natural.
zbMATH Open document ID: 1061.18010
author: Jiří Adámek
author: Horst Herrlich
author: Jiří Rosický
author: Walter Tholen
published in: Algebra Universalis
publication date: 2005-04-19T00:00:00Z
review text: The question whether a given category has enough injectives (so that every object may be embedded into an injective one) or even injective hulls (so that such embeddings may be chosen to be essential), has been investigated for many categories, particularly in commutative and homological algebra, algebraic geometry, topology and in functional analysis. Existence of enough injectives is often facilitated by the existence of an injective cogenerator in the given category. The authors show that surprisingly in a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless the situation is trivial, in the sense that all objects are injective, in which case the injective-hull functor is given by the identity functor. In particular, assigning to a field its algebraic closure, to a poset its MacNeille completion, and to an \(R\)-module its injective envelope is not functorial, if one wants the injective embeddings to form a natural transformation.
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 2158815
zbMATH Keywords: injective object
zbMATH Keywords: projective object
zbMATH Keywords: injective hull
zbMATH Keywords: projective cover
Wikidata QID: Q114232179
MaRDI profile type: Publication
full work available at URL: s000120200006
OpenAlex ID: W2016415780
DOI: 10.1007/S000120200006
description: scientific article; zbMATH DE number 2158815