In a concrete category, an injective hull of an object is an extension of such that is injective and 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.
- In Vect every object has an injective hull, . 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 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).
Given a class of objects in a category, an -hull (or -envelope) of an object is a map such that the following two conditions hold:
Any map to an object in factors through via some map .
Whenever a map satisfies then it must be an automorphism.
On the other hand, given a class of morphisms in a category, an -injective hull of an object is a map in such that:
is a -injective object and
is -essential, i.e. if then .
Revised on April 8, 2015 02:18:11
by Mike Shulman