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Summary: Injective Hulls are not Natural
Jir Adamek Horst Herrlich Jir Rosicky y
Walter Tholen z
Abstract
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.
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