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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).
Being injective is a property of an object; the corresponding structure is called an algebraic injective.
Let be a category and a class of morphisms in .
Frequently is the class of all monomorphisms or a related class.
This is notably the case for is a category of chain complexes equipped with the injective model structure on chain complexes and is its class of cofibrations.
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.
If has a terminal object , then the -injective objects according to def. are those for which the unique morphism is a -injective morphism.
One says that a category has enough injectives if every object admits a monomorphism into an injective object.
(The dual notion is that of a projective object.)
Assuming the axiom of choice, we have the following easy result:
Any small Cartesian 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 .
If is a locally small category then is -injective precisely if the hom-functor
takes morphisms in to epimorphisms in Set.
The term injective object is used most frequently in the context that is an abelian category.
For an abelian category the class of monomorphisms is the same as the class of morphisms such that is exact.
By definition of abelian category every monomorphism is a kernel, hence a pullback of the form
for the (algebraic) zero object. By the pasting law for pullbacks we find that also the left square in
is a pullback, hence is exact.
An object of an abelian category is then injective if it satisfies the following equivalent conditions:
the hom-functor is exact where is the category of abelian groups;
for all morphisms such that is exact and for all , there exists such that .
By the formal dual of this prop..
See homotopically injective object for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.
Let be a commutative ring and the category of -modules. We discuss injective modules over (see there for more).
The following criterion says that for identifying injective modules it is sufficient to test the right lifting property which characterizes injective objects by def. , only on those monomorphisms which include an ideal into the base ring .
If the axiom of choice holds, then a module is an injective module precisely if for any left -ideal regarded as an -module, any homomorphism in can be extended to all of along the inclusion .
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.
The set is an ideal of , and we have a module homomorphism defined by . By hypothesis, we may extend to a module map . Writing a general element of as where , it may be shown that
is well-defined and extends , as desired.
(Assume that the axiom of choice holds.) Let be a Noetherian ring, and let be a collection of injective modules over . Then the direct sum is also injective.
By Baer’s criterion, theorem , 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.
Conversely, is a Noetherian ring if direct sums of injective -modules are injective.
This is due to Bass and Papp. See (Lam, Theorem 3.46).
Let Ab be the abelian category of abelian groups.
If the axiom of choice holds, then an abelian group is an injective object in Ab precisely if it is a divisible group, in that for all integers we have .
This follows for instance using Baer's criterion, prop. .
An explicit proof is spelled out at Planet math – abelian group is divisible if and only if it is an injective object
By prop. the following abelian groups are injective in Ab.
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.
Not injective in Ab is for instance the cyclic group for .
In any topos, the subobject classifier is an injective object, as is any power of (Mac Lane-Moerdijk, IV.10).
Also one can define various notions of internally injective objects. These turn out to be equivalent:
In any elementary topos with a natural numbers object, the following statements about an object are 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 the topos of sheaves over a locale. Then an object is internally injective (in any of the senses given by Proposition ) if and only if is injective as in Definition .
Let be an externally injective object. Then satisfies condition 2. of Proposition , 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
One can check that is flabby (this is particularly easy using the internal language, details will be added later) and therefore has a global section.
The analogs of Proposition and Proposition for abelian group objects instead of unstructured objects hold as well, with mostly the same proofs. Condition 1. then refers to the functor .
In the category Top of all topological spaces, the injective objects are precisely the inhabited indiscrete spaces.
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).
Given a pair of additive adjoint functors
between abelian categories such that the left adjoint is a left exact functor (thus automatically exact), then the right adjoint preserves injective objects.
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.
The preceding lemma has the following variant:
Let , be categories and be an adjunction. If maps monos to monos, then maps injectives to injectives.
Let be injective. Consider the following diagram in :
Let the natural bijection given by the adjunction. Consider now the following diagram in where the assumptions ensure that is mono:
Since is injective, there there exists a filler which by the adjunction must come from a uniquely determined . But the naturality of the bijection with respect to composition says that
correspond to each other under the bijection whence but from the commutativity of the second diagram we have . Since is a bijection it follows that which proves that is injective.
The proof transposes the proof of the dual statement 10.2. in (Hilton-Stammbach 1971, p. 82): In situation , if maps epis to epis then maps projectives to projectives.
Let such that exists and has a right adjoint . Since it is easy to check that preserves monos it follows that preserves injectives.
In particular, for toposes this implies that all power objects are injective since the injectivity of follows more or less straightforwardly from its classifying properties.
We discuss a list of classes of categories that have enough injectives according to def. .
Every topos has enough injectives.
Every power object can be shown to be injective (cf. the above remark), and every object embeds into its power object by the “singletons” map.
Assuming some form of the axiom of choice, the category of abelian groups has enough injectives.
Full AC is much more than required, however; small violations of choice suffices.
By prop. an abelian group is an injective -module precisely if it is a divisible group. So we need to show that every abelian group is a subgroup of a divisible group.
To start with, notice that the group of rational numbers is divisible and hence the canonical embedding shows that the additive group of integers embeds into an injective -module.
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.
Given a pair of additive adjoint functors
between abelian categories such that the left adjoint is
an exact functor,
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 .
For the second point, consider the kernel of as part of the exact sequence
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.
As soon as the category Ab of abelian groups has enough injectives, so does the abelian category Mod of modules over some ring .
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 from prop. .
For a field, hence Mod = Vect, every object is both injective as well as projective.
The category of abelian sheaves on any small site , hence the category of abelian groups in the sheaf topos over , has enough injectives.
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.
The category of sheaves of modules over any sheaf of rings? on any small site also enough injectives.
This slick proof of this important fact was pointed out by Colin McLarty in an email to the categories list dated 10 Oct 2010.
Let be an abelian category. Then for every object there is an injective resolution, hence a chain complex
equipped with a a quasi-isomorphism of cochain complexes
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution
The notion of injective modules was introduced in
The dual notion of projective modules was considered explicitly only much later in
Textbook accounts:
(used, e.g., for the proof of Lemma);
Using tools from the theory of accessible categories, injective objects are discussed in
Baer’s criterion is discussed in many texts, for example
See also
For injective objects in a topos see
Francis Borceux, Handbook of Categorical Algebra vol. 3 , Cambridge UP 1994. (section 5.6, pp.314-315)
Saunders Mac Lane, Ieke Moerdijk, Sheaves in geometry and Logic , Springer Heidelberg 1994. (section IV.10, pp.210-213)
D. Higgs, Injectivity in the topos of complete Heyting algebra valued sets , Can. J. Math. 36 (1984) pp.550-568. (pdf)
Peter Johnstone, Fred Linton, Robert Paré, Injective Objects in Topoi II: Connections with the axiom of choice , pp.207-216 in LNM 719 Springer Heidelberg 1979.
T. Kenney, Injective Power Objects and the Axiom of Choice , JPAA 215 (2011) pp.131–144.
Fred Linton, Injective Objects in Topoi III: Stability under coproducts , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 29 (1981) pp.341-347.
Fred Linton, Robert Paré, Injective Objects in Topoi I: Representing coalgebras as algebras , pp.196-206 in LNM 719 Springer Heidelberg 1979.
Discussion of injective objects (types) in homotopy type theory:
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)
Last revised on October 23, 2024 at 09:28:15. See the history of this page for a list of all contributions to it.