nLab injective envelope



Category theory

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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.

Beware that, 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 from the identity functor to 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 Field 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}.


Discussion in homological algebra:

Discussion in general concrete categories:

See also:

  • Adámek, Herrlich, Rosický, Tholen; 2002; Injective hulls are not natural
  • Jinzhog Xu; 1996; Flat Covers of Modules, SLNM 1634.
  • Walter Tholen, “Essential weak factorization systems”. Contributions to General Algebra 13 (2001) 321-333. ps.

On injective hulls of partially ordered monoids:

Last revised on February 17, 2024 at 12:06:21. See the history of this page for a list of all contributions to it.