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Zbl 1061.18010
Adámek, Jiř\'i; Herrlich, Horst; Rosický, Jiř\'i; Tholen, Walter
Injective hulls are not natural. (English)
[J] Algebra Univers. 48, No. 4, 379-388 (2002). ISSN 0002-5240; ISSN 1420-8911/e

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. \par 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. \par 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.
[Radom\'{i}r Halaš (Olomouc)]

MSC 2000:: *18G05 Projectives and injectives (homological algebra)
16D50 Injective modules, self-injective rings (assoc. rings and algebras)

Keywords: injective object; projective object; injective hull; projective cover

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