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injective object

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Definition

There is a very general notion of injective objects in a category CC, and then there is a sequence of more concrete notions as CC is equipped with more structure and property, in particular for CC 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).

General definition

Let CC be a category and JMor(C)J \subset Mor(C) a class of morphisms in CC.

Example

Frequently JJ is the class of all monomorphisms or a related class.

This is notably the case for CC is a category of chain complexes equipped with the injective model structure on chain complexes and JJ is its class of cofibrations.

Definition

An object II in CC is JJ-injective if all diagrams of the form

X I jJ Z \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow \\ Z }

admit an extension

X I jJ Z. \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} \\ Z } \,.

If JJ is the class of all monomorphisms, we speak merely of an injective object.

Remark

If CC has a terminal object *\ast, then the JJ-injective objects II according to def. 1 are those for which I!*I \stackrel{\exists!}{\to} \ast is called a JJ-injective morphisms.

Definition

Ones says that a category CC 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.

Proposition

An arbitrary small product of injective objects is injective.

Remark

If CC has a terminal object ** then these extensions are equivalently lifts

X I jJ Z * \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} & \downarrow \\ Z &\to& * }

and hence the JJ-injective objects are precisely those that have the right lifting property against the class JJ.

Remark

If CC is a locally small category then II is JJ-injective precisely if the hom-functor

Hom C(,I):C opSet Hom_C(-,I) : C^{op} \to Set

takes morphisms in JJ to epimorphisms in Set.

In abelian categories

The term injective object is used most frequently in the context that CC is an abelian category.

Proposition

For CC an abelian category the class JJ of monomorphisms is the same as the class of morphisms f:ABf : A \to B such that 0AfB0 \to A \stackrel{f}{\to} B is exact.

Proof

By definition of abelian category every monomorphism ABA \hookrightarrow B is a kernel, hence a pullback of the form

A 0 B C \array{ A &\to& 0 \\ \downarrow && \downarrow \\ B &\to& C }

for 00 the (algebraic) zero object. By the pasting law for pullbacks we find that also the left square in

0 A 0 0 B C \array{ 0 &\to& A &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& B &\to& C }

is a pullback, hence 0AB0 \to A \to B is exact.

Corollary

An object II of an abelian category CC is then injective if it satisfies the following equivalent conditions:

  • the hom-functor Hom C(,I):C opSetHom_C(-, I) : C^{op} \to Set is exact;

  • for all morphisms f:XYf : X \to Y such that 0XY0 \to X \to Y is exact and for all k:XIk : X \to I, there exists h:YIh : Y \to I such that hf=k h\circ f = k.

0 X f Y k h I. \array{ 0 &\to& X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{k}} & \swarrow_{\mathrlap{\exists h}} \\ && I } \,.

By the formal dual of this prop..

In chain complexes

See homotopically injective object for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.

Examples

Injective modules

Let RR be a commutative ring and C=RModC = R Mod the category of RR-modules. We discuss injective modules over RR (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. 1, only on those monomorphisms which include an ideal into the base ring RR.

Proposition

(Baer's criterion)

If the axiom of choice holds, then a module QRModQ \in R Mod is an injective module precisely if for II any left RR-ideal regarded as an RR-module, any homomorphism g:IQg : I \to Q in CC can be extended to all of RR along the inclusion IRI \hookrightarrow R.

This is due to (Baer).

Sketch of proof

Let i:MNi \colon M \hookrightarrow N be a monomorphism in RModR Mod, and let f:MQf \colon M \to Q be a map. We must extend ff to a map h:NQh \colon N \to Q. Consider the poset whose elements are pairs (M,f)(M', f') where MM' is an intermediate submodule between MM and NN and f:MQf' \colon M' \to Q is an extension of ff, ordered by (M,f)(M,f)(M', f') \leq (M'', f'') if MM'' contains MM' and ff'' extends ff'. By an application of Zorn's lemma, this poset has a maximal element, say (M,f)(M', f'). Suppose MM' is not all of NN, and let xNx \in N be an element not in MM'; we show that ff' extends to a map M=x+MQM'' = \langle x \rangle + M' \to Q, a contradiction.

The set {rR:rxM}\{r \in R: r x \in M'\} is an ideal II of RR, and we have a module homomorphism g:IQg \colon I \to Q defined by g(r)=f(rx)g(r) = f'(r x). By hypothesis, we may extend gg to a module map k:RQk \colon R \to Q. Writing a general element of MM'' as rx+yr x + y where yMy \in M', it may be shown that

f(rx+y)=k(r)+f(y)f''(r x + y) = k(r) + f'(y)

is well-defined and extends ff', as desired.

Corollary

(Assume that the axiom of choice holds.) Let RR be a Noetherian ring, and let {Q j} jJ\{Q_j\}_{j \in J} be a collection of injective modules over RR. Then the direct sum Q= jJQ jQ = \bigoplus_{j \in J} Q_j is also injective.

Proof

By Baer’s criterion, theorem 1, it suffices to show that for any ideal II of RR, a module homomorphism f:IQf \colon I \to Q extends to a map RQR \to Q. Since RR is Noetherian, II is finitely generated as an RR-module, say by elements x 1,,x nx_1, \ldots, x_n. Let p j:QQ jp_j \colon Q \to Q_j be the projection, and put f j=p jff_j = p_j \circ f. Then for each x ix_i, f j(x i)f_j(x_i) is nonzero for only finitely many summands. Taking all of these summands together over all ii, we see that ff factors through

jJQ j= jJQ jQ\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q

for some finite JJJ' \subset J. But a product of injectives is injective, hence ff extends to a map R jJQ jR \to \prod_{j \in J'} Q_j, which completes the proof.

Proposition

Conversely, RR is a Noetherian ring if direct sums of injective RR-modules are injective.

This is due to Bass and Papp. See (Lam, Theorem 3.46).

Injective abelian groups

Let C=ModC = \mathbb{Z} Mod \simeq Ab be the abelian category of abelian groups.

Proposition

If the axiom of choice holds, then an abelian group AA is an injective object in Ab precisely if it is a divisible group, in that for all integers nn \in \mathbb{N} we have nG=Gn G = G.

This follows for instance using Baer's criterion, prop. 1.

An explicit proof is spelled out at Planet math – abelian group is divisible if and only if it is an injective object

Example

By prop. 4 the following abelian groups are injective in Ab.

The group of rational numbers \mathbb{Q} is injective in Ab, as is the additive group of real numbers \mathbb{R} 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.

Example

Not injective in Ab is for instance the cyclic group /n\mathbb{Z}/n\mathbb{Z} for n>1n \gt 1.

In toposes

In any topos, the subobject classifier Ω\Omega is an injective object, as is any power of Ω\Omega (Mac Lane-Moerdijk, IV.10).

Also one can define various notions of internally injective objects. These turn out to be equivalent:

Proposition

In any elementary topos \mathcal{E} with a natural numbers object, the following statements about an object II \in \mathcal{E} are equivalent.

  1. The functor [,I]: op[-, I] : \mathcal{E}^op \to \mathcal{E} maps monomorphisms in \mathcal{E} to epimorphisms.
  2. The functor [,I]: op[-, I] : \mathcal{E}^op \to \mathcal{E} maps monomorphisms in \mathcal{E} to morphisms for which any global element of the target locally (after change of base along an epimorphism) possesses a preimage.
  3. For any morphism p:A1p : A \to 1 in \mathcal{E}, the object p *Ip^*I has property 1. as an object of /A\mathcal{E}/A.
  4. For any morphism p:A1p : A \to 1 in \mathcal{E}, the object p *Ip^*I has property 2. as an object of /A\mathcal{E}/A.
  5. The interpretation of the statement “II is an injective object” using the stack semantics of \mathcal{E} holds.
Proof

The implications “1. \Rightarrow 2.”, “3 \Rightarrow 4.”, “3. \Rightarrow 1.”, and “4. \Rightarrow 2.” are trivial.

The equivalence “3. \Leftrightarrow 5.” follows directly from the interpretation rules of the stack semantics.

The implication “2. \Rightarrow 4.” employs the extra left-adjoint p !:/Ap_! : \mathcal{E}/A \to \mathcal{E} to p *:/Ap^* : \mathcal{E} \to \mathcal{E}/A, as in the usual proof that injective sheaves remain injective when restricted to smaller open subsets: We have that p *[,p *I] /A[,I] p !p_* \circ [-, p^*I]_{\mathcal{E}/A} \cong [-, I]_{\mathcal{E}} \circ p_!, the functor p !p_! preserves monomorphisms, and one can check that p *p_* reflects the property that global elements locally possess preimages. Details are in (Harting, Theorem 1.1).

The implication “4. \Rightarrow 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.

Proposition

Let \mathcal{E} be the topos of sheaves over a locale. Then an object II \in \mathcal{E} is internally injective (in any of the senses given by Proposition 5) if and only if II is injective as in Definition 1.

Proof

Let II be an externally injective object. Then II satisfies condition 2. of Proposition 5, even without having to pass to a cover.

Conversely, let II be an internally injective object. Let m:XYm : X \to Y be a monomorphism and let k:XIk : X \to I be an arbitrary morphism. We want to show that there exists an extension YIY \to I of kk along mm. To this end, consider the sheaf

F{k:om(Y,I)|km=k}. F \coloneqq \{ k' : \mathcal{H}om(Y,I) | k' \circ m = k \}.

One can check that FF is flabby (this is particularly easy using the internal language, details will be added later) and therefore has a global section.

Remark

The analogs of Proposition 5 and Proposition 6 for abelian group objects instead of unstructured objects hold as well, with mostly the same proofs. Condition 1. then refers to the functor [,X]:Ab() opAb()[-, X] : Ab(\mathcal{E})^op \to Ab(\mathcal{E}).

In topological spaces

In the category Top of all topological spaces, the injective objects are precisely the inhabited indiscrete spaces.

In the category of T 0T_0 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 T 0T_0 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 T 0T_0 reflections are continuous lattices under the Scott topology.

In Boolean algebras

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).

Properties

Preservation of injective objects

Lemma

Given a pair of additive adjoint functors

(LR):RL𝒜 (L \dashv R) \;\colon\; \mathcal{B} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{A}

between abelian categories such that the left adjoint LL is a left exact functor (thus automatically exact), then the right adjoint preserves injective objects.

Proof

Observe that an object is injective precisely if the hom-functor into it sends monomorphisms to epimorphisms, and that LL preserves monomorphisms by assumption of (left-)exactness. With this the statement follows via adjunction isomorphism

Hom 𝒜(,R(I))Hom (L(),I). Hom_{\mathcal{A}}(-,R(I))\simeq Hom_{\mathcal{B}}(L(-),I) \,.
Remark

Additivity of the left adjoint follows from the remaining assumptions, since exact functors preserve biproducts.

Existence of enough injectives

We discuss a list of classes of categories that have enough injectives according to def. 2.

Proposition

Every topos has enough injectives.

Proof

Every power object can be shown to be injective, and every object embeds into its power object by the “singletons” map.

Proposition

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.

Proof

By prop. 4 an abelian group is an injective \mathbb{Z}-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 \mathbb{Q} of rational numbers is divisible and hence the canonical embedding \mathbb{Z} \hookrightarrow \mathbb{Q} shows that the additive group of integers embeds into an injective \mathbb{Z}-module.

Now by the discussion at projective module every abelian group AA receives an epimorphism ( sS)A(\oplus_{s \in S} \mathbb{Z}) \to A from a free abelian group, hence is the quotient group of a direct sum of copies of \mathbb{Z}. Accordingly it embeds into a quotient A˜\tilde A of a direct sum of copies of \mathbb{Q}.

ker = ker ( sS) ( sS) A A˜ \array{ ker &\stackrel{=}{\to}& ker \\ \downarrow && \downarrow \\ (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q}) \\ \downarrow && \downarrow \\ A &\hookrightarrow& \tilde A }

Here A˜\tilde A 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 AA into a divisible abelian group, hence into an injective \mathbb{Z}-module.

Lemma

Given a pair of additive adjoint functors

(LR):RL𝒜 (L \dashv R) \;\colon\; \mathcal{B} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{A}

between abelian categories such that the left adjoint LL is

  1. an exact functor,

  2. a faithful functor.

Then if \mathcal{B} has enough injectives, also 𝒜\mathcal{A} has enough injectives.

Proof

Consider A𝒜A \in \mathcal{A}. By the assumption that \mathcal{B} has enough injectives, there is an injective object II \in \mathcal{B} and a monomorphism i:L(A)Ii \colon L(A) \hookrightarrow I. The adjunct of this is a morphism

i˜:AR(I) \tilde i \colon A \longrightarrow R(I)

and so it is sufficient to show that

  1. R(I)R(I) is injective in 𝒜\mathcal{A};

  2. i˜\tilde i is a monomorphism.

The first point is the statement of lemma 1.

For the second point, consider the kernel of i˜\tilde i as part of the exact sequence

ker(i˜)Ai˜R(I). ker(\tilde i)\longrightarrow A \stackrel{\tilde i}{\longrightarrow} R(I) \,.

By the assumption that LL is an exact functor, the image of this sequence under LL is still exact

L(ker(i˜))L(A)L(i˜)L(R(I)). L(ker(\tilde i)) \longrightarrow L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \,.

Now observe that L(i˜)L(\tilde i) is a monomorphism: this is because its composite L(A)L(i˜)L(R(I))ϵIL(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \stackrel{\epsilon}{\longrightarrow} I with the adjunction unit is (by the formula for adjuncts) the original morphism ii, which by construction is a monomorphism. Therefore the exactness of the above sequence means that L(ker(i˜))L(A)L(ker(\tilde i)) \to L(A) is the zero morphism; and by the assumption that LL is a faithful functor this means that already ker(i˜)Aker(\tilde i) \to A is zero, hence that ker(i˜)=0ker(\tilde i) = 0, hence that i˜\tilde i is a monomorphism.

Proposition

As soon as the category Ab of abelian groups has enough injectives, so does the abelian category RRMod of modules over some ring RR.

In particular if the axiom of choice holds, then RModR Mod has enough injectives.

Proof

Observe that the forgetful functor U:RModAbGpU\colon R Mod \to AbGp has both a left adjoint R !R_! (extension of scalars from \mathbb{Z} to RR) and a right adjoint R *R_* (coextension of scalars). Since it has a left adjoint, it is exact. Thus the statement follows via lemma 2 from prop. 8.

Proposition

For R=kR = k a field, hence RRMod = kkVect, every object is both injective as well as projective.

Proposition

The category of abelian sheaves Ab(Sh(C))Ab(Sh(C)) on any small site CC, hence the category of abelian groups in the sheaf topos over CC, has enough injectives.

A proof of can be found in Peter Johnstone’s book Topos Theory, p261.

Remark

This is in stark contrast to the situation for projective objects, which generally do not exist in categories of sheaves.

Corollary

The category of sheaves of modules over any sheaf of rings? on any small site also enough injectives.

Proof

Combining prop. 11 with prop. 8 (which relativizes to any topos).

This slick proof of this important fact was pointed out by Colin McLarty in an email to the categories list dated 10 Oct 2010.

Injective resolutions

Proposition

Let 𝒜\mathcal{A} be an abelian category. Then for every object X𝒜X \in \mathcal{A} there is an injective resolution, hence a chain complex

J =[J 0J n]Ch (𝒜) J^\bullet = [J^0 \to \cdots \to J^n \to \cdots] \in Ch_(\mathcal{A})

equipped with a a quasi-isomorphism of cochain complexes XJ X \stackrel{\sim}{\to} J^\bullet

X 0 0 J 0 J 1 J n . \array{ X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots \\ \downarrow && \downarrow && && \downarrow \\ J^0 &\to& J^1 &\to& \cdots &\to& J^n &\to& \cdots } \,.

References

The notion of injective modules was introduced in

  • R. Baer, Abelian groups that are direct summands of every containing abelian group , Bulletin AMS 46 no. 10 (1940) pp.800-806. (projecteuclid)

(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

  • N. Jacobsen, Basic Algebra II, W.H. Freeman and Company, 1980.

See also

  • T.-Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics 189, Springer Verlag (1999).

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.

  • 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.

For a detailed discussion of internal notions of injectivity see

  • Roswitha Harting, Locally injective abelian groups in a topos, Communications in Algebra 11 (4), 1983.

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)

Revised on July 12, 2016 11:05:01 by Urs Schreiber (131.220.184.222)