There is a very general notion of injective objects in a category , and then there is a sequence of more concrete notions as is equipped with more structure and property, in particular for an abelian category or a relative thereof.
The concept of resolutions by injective objects – injective resolutions – is crucial notably in the discussion of derived functors (in the context of abelian categories: derived functors in homological algebra).
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.
Also one can define various notions of internally injective objects. These turn out to be equivalent:
The implications “1. 2.”, “3 4.”, “3. 1.”, and “4. 2.” are trivial.
The equivalence “3. 5.” follows directly from the interpretation rules of the stack semantics.
The implication “2. 4.” employs the extra left-adjoint to , as in the usual proof that injective sheaves remain injective when restricted to smaller open subsets: We have that , the functor preserves monomorphisms, and one can check that reflects the property that global elements locally possess preimages. Details are in (Harting, Theorem 1.1).
The implication “4. 3.” follows by performing an extra change of base, since any non-global element becomes a global element after a suitable change of base.
Somewhat surprisingly, and in stark contrast with the situation for internally projective objects, internal injectivity coincides with external injecticity.
Let be an externally injective object. Then satisfies condition 2. of Proposition 5, even without having to pass to a cover.
Conversely, let be an internally injective object. Let be a monomorphism and let be an arbitrary morphism. We want to show that there exists an extension of along . To this end, consider the sheaf
In the category of spaces (see separation axiom), the injective objects are the terminal spaces.
In the above two cases, this refers to injectivity with respect to monomorphisms.
In the category of spaces, the injective objects with respect to homeomorphic embeddings are precisely those given by Scott topologies on continuous lattices; as locales these are locally compact and spatial. (Such spaces are usually called, perhaps confusingly, injective spaces.)
In the category of all spaces, the injectives with respect to homeomorphic embeddings (i.e., regular monomorphisms) are the spaces whose reflections are continuous lattices under the Scott topology.
Injective objects in the category of Boolean algebras are precisely complete Boolean algebras. This is the dual form of a theorem of Gleason, saying that the projective objects in the category of Stone spaces are the extremally disconnected ones (the closure of every open set is again open).
Observe that an object is injective precisely if the hom-functor into it sends monomorphisms to epimorphisms, and that preserves monomorphisms by assumption of (left-)exactness. With this the statement follows via adjunction isomorphism
Additivity of the left adjoint follows from the remaining assumptions, since exact functors preserve biproducts.
We discuss a list of classes of categories that have enough injectives 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.
Now by the discussion at projective module every abelian group receives an epimorphism from a free abelian group, hence is the quotient group of a direct sum of copies of . Accordingly it embeds into a quotient of a direct sum of copies of .
Here is divisible because the direct sum of divisible groups is again divisible, and also the quotient group of a divisible groups is again divisble. So this exhibits an embedding of any into a divisible abelian group, hence into an injective -module.
Then if has enough injectives, also has enough injectives.
Consider . By the assumption that has enough injectives, there is an injective object and a monomorphism . The adjunct of this is a morphism
and so it is sufficient to show that
is injective in ;
is a monomorphism.
The first point is the statement of lemma 1.
By the assumption that is an exact functor, the image of this sequence under is still exact
Now observe that is a monomorphism: this is because its composite with the adjunction unit is (by the formula for adjuncts) the original morphism , which by construction is a monomorphism. Therefore the exactness of the above sequence means that is the zero morphism; and by the assumption that is a faithful functor this means that already is zero, hence that , hence that is a monomorphism.
In particular if the axiom of choice holds, then has enough injectives.
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. Thus the statement follows via lemma 2 from prop. 8.
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
For injective objects in a topos see
Francis Borceux, Handbook of Categorical Algebra vol. 3 , Cambridge UP 1994. (section 5.6, pp.314-315)
D. Higgs, Injectivity in the topos of complete Heyting algebra valued sets , Can. J. Math. 36 (1984) pp.550-568. (pdf)
Fred Linton, Injective Objects in Topoi III: Stability under coproducts , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 29 (1981) pp.341-347.
For a detailed discussion of internal notions of injectivity see
For injective toposes in the 2-category of bounded toposes see
Peter Johnstone, Injective Toposes , pp.284-297 in LNM 871 Springer Heidelberg 1981.
Peter Johnstone, Sketches of an Elephant vol. 2 , Cambridge UP 2002. (section C4.3, pp.738-745)