There is a very general notion of injective objects in a category , and a sequence of refinements as is equipped with more structure and property, in particular for an abelian category or a relative thereof.
Frequently is the class of all monomorphisms or a related class.
An object in is -injective if all diagrams of the form
admit an extension
If is the class of all monomorphisms, we speak merely of an injective object.
Ones says that a category has enough injectives if every object admits a monomorphism into an injective object.
The dual notion is a projective object.
Assuming the axiom of choice, we have the following easy result.
An arbitrary small product of injective objects is injective.
If has a terminal object then these extensions are equivalently lifts
and hence the -injective objects are precisely those that have the right lifting property against the class .
The term injective object is used most frequently in the context that is an abelian category.
is a pullback, hence is exact.
An object of an abelian category is then injective if it satisfies the following equivalent conditions:
See homotopically injective object for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.
The following criterion says that for identifying injective modules it is sufficient to test the right lifting property which characterizes injective objects by def. 1, only on those monomorphisms which include an ideal into the base ring .
This is due to (Baer).
Let be a monomorphism in , and let be a map. We must extend to a map . Consider the poset whose elements are pairs where is an intermediate submodule between and and is an extension of , ordered by if contains and extends . By an application of Zorn's lemma, this poset has a maximal element, say . Suppose is not all of , and let be an element not in ; we show that extends to a map , a contradiction.
is well-defined and extends , as desired.
By Baer’s criterion, theorem 1, it suffices to show that for any ideal of , a module homomorphism extends to a map . Since is Noetherian, is finitely generated as an -module, say by elements . Let be the projection, and put . Then for each , is nonzero for only finitely many summands. Taking all of these summands together over all , we see that factors through
for some finite . But a product of injectives is injective, hence extends to a map , which completes the proof.
This is due to Bass and Papp. See (Lam, Theorem 3.46).
An explicit proof is spelled out at Planet math – abelian group is divisible if and only if it is an injective object
The group of rational numbers is injective in Ab, as is the additive group of real numbers and generally that underlying any field of characteristic zero. The additive group underlying any vector space is injective. The quotient of any injective group by any other group is injective.
We discuss a list of classes of categories that have enough injective according to def. 2.
Every topos has enough injectives.
Every power object can be shown to be injective, and every object embeds into its power object by the “singletons” map.
Full AC is much more than required, however; small violations of choice suffices.
Observe that the forgetful functor has both a left adjoint (extension of scalars from to ) and a right adjoint (coextension of scalars). Since it has a left adjoint, it is exact, and so its right adjoint preserves injective objects. Thus given any -module , we can embed in an injective abelian group , and then embeds in .
A proof of can be found in Peter Johnstone’s book Topos Theory, p261.
This is in stark contrast to the situation for projective objects, which generally do not exist in categories of sheaves.
This slick proof of this important fact was pointed out by Colin McLarty in an email to the categories list dated 10 Oct 2010.
flat object, flat resolution
The notion of injective modules was introduced in
(The dual notion of projective modules was considered explicitly only much later.)
A general discussion can be found in
The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.
Using tools from the theory of accessible categories, injective objects are discussed in
Baer’s criterion is discussed in many texts, for example