nLab
injective hull

In a concrete category, an injective hull of an object $A$ is an extension $A\stackrel{m}{⟶}B$ of $A$ such that $B$ is injective and $m$ is an essential embedding?. It is the dual concept to projective cover?.

In general, there is no way of making the assignment of the injective hull to an object into a functor such that there is a natural transformation between the identity functor and that functor.

Examples

• In Vect every object $A$ has an injective hull, $A\stackrel{{\mathrm{id}}_A}{⟶}A$. In other words, every vector space is already an injective object.
• In Pos every object has an injective hull, its MacNeille completion.
• In Ab every object has an injective hull. The embedding $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is an example.
• In the category of fields and algebraic field extensions, every object has an injective hull, its algebraic closure.
• In the category of metric spaces and short maps, the injective hull is a standard construction also known as the tight span? (see Wikipedia).

References

• Jiří Adámek?, Horst Herrlich?, George Strecker?; 2004; The Joy of Cats, Definition 9.16 (page 156)
• Adámek, Herrlich, Rosický, Tholen; 2002; Injective hulls are not natural