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Injective Hulls are not Natural Ji r Ad amek Horst Herrlich Ji r Rosick y y

Summary: Injective Hulls are not Natural
Jir Adamek  Horst Herrlich Jir Rosicky y
Walter Tholen z
In a category with injective hulls and a cogenerator, the embeddings
into injective hulls can never form a natural transformation, unless all
objects are injective. In particular, assigning to a eld its algebraic closure,
to a poset or Boolean algebra its MacNeille completion, and to an R-
module its injective envelope is not functorial, if one wants the respective
embeddings to form a natural transformation.
Mathematics subject classi cation: 18G05, 16D50, 12F99, 06A23
Keywords: injective object, projective object, injective hull, projective cover.
1 Introduction
Projectivity and injectivity are fundamental concepts of modern mathematics.
The question whether a given category has enough injectives (so that every ob-
ject may be embedded into an injective one) or even injective hulls (so that such
embeddings may be chosen to be essential ), as well as the dual questions (enough
projectives, projective covers), have been investigated for many categories, par-
ticularly in commutative and homological algebra, algebraic geometry, topology,
and in functional analysis.


Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig
Rosický, Jirí - Department of Mathematics and Statistics, Masaryk University
Tholen, Walter - Department of Mathematics and Statistics, York University (Toronto)


Collections: Computer Technologies and Information Sciences; Mathematics