injective hull



In a concrete category, an injective hull of an object AA is an extension AmBA \stackrel{m}{\longrightarrow} B of AA such that BB is injective and mm 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 AA has an injective hull, Aid AAA \stackrel{id_A}{\longrightarrow} 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).


Given a class \mathcal{E} of objects in a category, an \mathcal{E}-hull (or \mathcal{E}-envelope) of an object AA is a map h:AEh\colon A\longrightarrow E such that the following two conditions hold:

  1. Any map k:AEk\colon A\longrightarrow E' to an object in \mathcal{E} factors through hh via some map f:EEf: E\longrightarrow E'.

  2. Whenever a map f:EEf\colon E\longrightarrow E satisfies fh=hf\circ h = h then it must be an automorphism.

On the other hand, given a class \mathcal{H} of morphisms in a category, an \mathcal{H}-injective hull of an object AA is a map h:AEh:A\to E in \mathcal{H} such that:

  1. EE is a \mathcal{H}-injective object and

  2. hh is \mathcal{H}-essential, i.e. if khk\circ h \in \mathcal{H} then kk\in\mathcal{H}.


Last revised on April 8, 2015 at 02:18:11. See the history of this page for a list of all contributions to it.