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).

Being injective is a property of an object; the corresponding structure is called an algebraic injective.

General definition

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

Definition

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

X → I j∈J↓ Z \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow \\ Z }

admit an extension

X → I j∈J↓ ↗ ∃ 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.

Definition

One says that a category CC 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:

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:A→Bf : A \to B such that 0→A→fB0 \to A \stackrel{f}{\to} B is exact.

Proof

By definition of abelian category every monomorphism A↪BA \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 0→A→B0 \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 op→𝒜𝒷Hom_C(-, I) : C^{op} \to \mathscr{Ab} is exact where 𝒜𝒷\mathscr{Ab} is the category of abelian groups;
for all morphisms f:X→Yf : X \to Y such that 0→X→Y0 \to X \to Y is exact and for all k:X→Ik : X \to I, there exists h:Y→Ih : Y \to I such that h∘f=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. , only on those monomorphisms which include an ideal into the base ring RR.

This is due to (Baer).

Sketch of proof

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

The set {r∈R:rx∈M′}\{r \in R: r x \in M'\} is an ideal II of RR, and we have a module homomorphism g:I→Qg \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:R→Qk \colon R \to Q. Writing a general element of M″M'' as rx+yr x + y where y∈M′y \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 f′f', as desired.

Proof

By Baer’s criterion, theorem , it suffices to show that for any ideal II of RR, a module homomorphism f:I→Qf \colon I \to Q extends to a map R→QR \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:Q→Q jp_j \colon Q \to Q_j be the projection, and put f j=p j∘ff_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

∏ j∈J′Q j=⨁ j∈J′Q j↪Q\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q

for some finite J′⊂JJ' \subset J. But a product of injectives is injective, hence ff extends to a map R→∏ j∈J′Q jR \to \prod_{j \in J'} Q_j, which completes the proof.

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

Injective abelian groups

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

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

Example

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

The group of rational numbers ℚ\mathbb{Q} is injective in Ab, as is the additive group of real numbers ℝ\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 ℤ/nℤ\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 ℰ\mathcal{E} with a natural numbers object, the following statements about an object I∈ℰI \in \mathcal{E} are equivalent.

The functor [−,I]:ℰ op→ℰ[-, I] : \mathcal{E}^op \to \mathcal{E} maps monomorphisms in ℰ\mathcal{E} to epimorphisms.
The functor [−,I]:ℰ op→ℰ[-, I] : \mathcal{E}^op \to \mathcal{E} maps monomorphisms in ℰ\mathcal{E} to morphisms for which any global element of the target locally (after change of base along an epimorphism) possesses a preimage.
For any morphism p:A→1p : A \to 1 in ℰ\mathcal{E}, the object p *Ip^*I has property 1. as an object of ℰ/A\mathcal{E}/A.
For any morphism p:A→1p : A \to 1 in ℰ\mathcal{E}, the object p *Ip^*I has property 2. as an object of ℰ/A\mathcal{E}/A.
The interpretation of the statement “II is an injective object” using the stack semantics of ℰ\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 !:ℰ/A→ℰp_! : \mathcal{E}/A \to \mathcal{E} to p *:ℰ→ℰ/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] ℰ/A≅[−,I] ℰ∘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 ℰ\mathcal{E} be the topos of sheaves over a locale. Then an object I∈ℰI \in \mathcal{E} is internally injective (in any of the senses given by Proposition ) if and only if II is injective as in Definition .

Proof

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

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

F≔{k′:ℋom(Y,I)|k′∘m=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.

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

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

The preceding lemma has the following variant:

Proposition

Let 𝒞\mathcal{C}, 𝒟\mathcal{D} be categories and L⊣R:𝒟→𝒞L\dashv R:\mathcal{D}\to\mathcal{C} be an adjunction. If LL maps monos to monos, then RR maps injectives to injectives.

Proof

Let I∈𝒟I\in\mathcal{D} be injective. Consider the following diagram in 𝒞\mathcal{C}:

A ↣m B f↓ R(I) \array{ A & \overset{m}{\rightarrowtail} & B \\ f\downarrow & & \\ R(I) & & }

Let θ:Hom 𝒞(X,R(Y))→≃Hom 𝒟(L(X),Y)\theta: Hom_\mathcal{C}(X,R(Y))\overset{\simeq}{\rightarrow}Hom_\mathcal{D}(L(X),Y) the natural bijection given by the adjunction. Consider now the following diagram in 𝒟\mathcal{D} where the assumptions ensure that L(m)L(m) is mono:

L(A) ↣L(m) L(B) θ(f)↓ I \array{ L(A) & \overset{L(m)}{\rightarrowtail} & L(B) \\ \theta(f)\downarrow & & \\ I & & }

Since II is injective, there there exists a filler θ(g):L(B)→I\theta(g):L(B)\to I which by the adjunction must come from a uniquely determined g:B→R(I)g:B\to R(I). But the naturality of the bijection with respect to composition says that

L(A)↣L(m)L(B)→θ(g)IA↣mB→gR(I) \frac{L(A)\overset{L(m)}{\rightarrowtail} L(B)\overset{\theta(g)}{\rightarrow}I}{A\overset{m}{\rightarrowtail} B\overset{g}{\rightarrow}R(I)}

correspond to each other under the bijection whence θ(g∘m)=θ(g)∘L(m)\theta(g\circ m)=\theta(g)\circ L(m) but from the commutativity of the second diagram we have θ(g)∘L(m)=θ(f)=θ(g∘m)\theta(g)\circ L(m)=\theta(f)=\theta(g\circ m). Since θ\theta is a bijection it follows that f=g∘mf=g\circ m which proves that R(I)R(I) is injective.

Existence of enough injectives

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

Proposition

Every topos has enough injectives.

Proof

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.

Full AC is much more than required, however; small violations of choice suffices.

Proof

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

Now by the discussion at projective module every abelian group AA receives an epimorphism (⊕ s∈Sℤ)→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 ℤ\mathbb{Z}. Accordingly it embeds into a quotient A˜\tilde A of a direct sum of copies of ℚ\mathbb{Q}.

ker →= ker ↓ ↓ (⊕ s∈Sℤ) ↪ (⊕ s∈Sℚ) ↓ ↓ 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 ℤ\mathbb{Z}-module.

Proof

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

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

and so it is sufficient to show that

R(I)R(I) is injective in 𝒜\mathcal{A};
i˜\tilde i is a monomorphism.

The first point is the statement of lemma .

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

ker(i˜)⟶A⟶i˜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.

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

Corollary

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

Proof

Combining prop. with prop. (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 X∈𝒜X \in \mathcal{A} there is an injective resolution, hence a chain complex

J •=[J 0→⋯→J 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 X→∼J •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 } \,.

projective object, projective presentation, projective cover, projective resolution
- projective module
injective object, injective presentation, injective envelope, injective resolution
- injective module
algebraically injective object
free object, free resolution
- free module
flat object, flat resolution
- flat module
orthogonality

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 in

Henri Cartan, Samuel Eilenberg, Homological Algebra, Princeton Univ. Press (1956), Princeton Mathematical Series 19 (1999) [ISBN:9780691049915, doi:10.1515/9781400883844, pdf]

Textbook accounts:

Peter Hilton, Urs Stammbach, p. 30 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)

(used, e.g., for the proof of Lemma);

Masaki Kashiwara, Pierre Schapira, sections 9.5, 14.1 of:_Categories and Sheaves_

Using tools from the theory of accessible categories, injective objects are discussed in

Jiri Rosicky, Injectivity and accessible categories (ps)

Baer’s criterion is discussed in many texts, for example

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