nLab injective envelope

Idea

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

Examples

• In Vect every object $A$ has an injective hull, $A \stackrel{id_A}{\longrightarrow} A$. In other words, every vector space is 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 ring $R$, the category $R-Mod$ of left $R$-modules has injective envelopes. Moreover, every essential monomorphism whose domain is an injective $R$-module is an isomorphism. The injective envelope of a module $M$ is a terminal object in the subcategory of the slice category $M/R-Mod$ consisting of the essential morphisms. (These terminal objects are called maximal essential extensions.)

Generalization

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

1. Any morphism $A \longrightarrow E'$ to an object in $\mathcal{E}$ factors through $h$ via some morphism $E \longrightarrow E'$.

2. Whenever a morphism $f: E \longrightarrow E$ satisfies $f\circ h = h$, 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 $A$ is a morphism $h:A\to E$ in $\mathcal{H}$ such that:

1. $E$ is an $\mathcal{H}$-injective object and

2. $h$ is $\mathcal{H}$-essential, i.e., if $k\circ h \in \mathcal{H}$, then $k\in\mathcal{H}$.

Related concepts

• projective object, projective presentation, projective cover, projective resolution

  • projective module

• injective object, injective presentation, injective envelope, injective resolution

  • injective module

• free object, free resolution

  • free module

• flat object, flat resolution

  • flat module

References

Discussion in homological algebra:

• Peter Hilton, Urs Stammbach, Section I.9 in: A course in homological algebra, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 (doi:10.1007/978-1-4419-8566-8, pdf)

Discussion in general concrete categories:

• Jiří Adámek, Horst Herrlich, George Strecker, Definition 9.16 (page 156) in: The Joy of Cats (2004)

See also:

• Jiří Adámek, Horst Herrlich, Jiří Rosický, Walter Tholen, Injective hulls are not natural, Algebra univers. 48 (2002) 379–388 doi pdf ps
• 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:

• Joachim Lambek, Michael Barr, John Kennison, Robert Raphael, Injective hulls of partially ordered monoids, Theory Appl. Categories 26 13 (2012) 338–348 [tac:16-13]