Limits and Colimits
4.5. Adjunctions, limits, and colimits
In this section, we explore the interaction between adjoint functors and limits and colimits. Our first result provides an adjoint characterization of completeness and cocom-pleteness of a categoryCwith regard to diagrams of a fixed shapeJ.
Proposition 4.5.1. A categoryCadmits all limits of diagrams indexed by a small category Jif and only if the constant diagram functor∆:C→CJadmits a right adjoint, and admits all colimits ofJ-indexed diagrams if and only if∆admits a left adjoint:
C ∆ //CJ.
colim
~~ ⊥
lim
__ ⊥
When these adjoints exist, they define the limit and colimit functors introduced in Proposition 3.6.1. Recall that the axiom of choice is needed to define the action of the limit or colimit functor on objects.
Proof. These dual statements follow immediately from the defining universal proper-ties of a limit and colimit. Forc∈CandF∈CJ, the hom-setCJ(∆c,F)is the set of natural transformations from the constantJ-diagram atcto the diagramF. This is precisely the set of cones overFwith summitc, as defined in 3.1.2. There is an objectlimF∈Ctogether with an isomorphism
CJ(∆c,F)C(c,limF)
that is natural inc∈Cif and only if this limit exists. If such natural isomorphisms exist for all diagramsF ∈CJ, then Proposition 4.3.4 applies to extend the objectslimF to a limit
functorlim :CJ→C.
Proposition 4.5.1 reveals that limits assemble into a right adjoint to the constant diagram functor. The reason that limits appear on the right is because limits are defined as representations for contravariant functors: the universal property of the limit ofF:J→C characterizes the functorC(−,limF).
Similarly, the value of a right adjoint functor G: D → C on an object d ∈ D is determined by a characterization of the contravariant representable functorC(−,Gd). Based on these observations, one might expect right adjoints to interact nicely with limits and indeed this is the case:
Theorem 4.5.2 (RAPL). Right adjoints preserve limits.
4.5. ADJUNCTIONS, LIMITS, AND COLIMITS 137
Proof. Consider a diagram K:J → D admitting a limit coneλ: limK ⇒ KinD, illustrated here in the case where the indexing categoryJis the poset of integers.
limK The statement asserts that this is a limit cone for the diagramGK: J→C. To prove this, consider another coneµ:c⇒GK. The legs of this cone transpose across the adjunctionFaGto define maps inD
Fc
which define a cone µ]: Fc ⇒ K overK, by naturality of the transposition relation, as expressed for instance by Lemma 4.1.3. There is a unique factorizationτ:Fc→limKof the coneµ]through the limit coneλ. The mapτtransposes to define a mapτ[:c→GlimK
which, by Lemma 4.1.3, defines a factorization of the coneµthrough the coneGλ. This factorization is clearly unique: another such factorizationc→GlimKwould transpose to
define a factorization ofµ]through λ, which, by the universal property of the limit cone λwould necessarily equalτ, andτ[ is the unique transpose of this map. This shows that Gλ:GlimK⇒GKhas the universal property required to define a limit cone forGK. Put more concisely, the argument just presented describes a series of natural isomor-phisms:
CJ(∆c,GK)DJ(F∆c,K)DJ(∆Fc,K)D(Fc,limJK)C(c,GlimJK), which, by the defining universal property of the limit, says thatGlimJKdefines a limit for the diagramGK.
Dually, of course:
Theorem 4.5.3 (LAPC). Left adjoints preserve colimits.
Corollaries are ubiquitous. For instance:
Corollary 4.5.4. For any function f: A → B, the inverse image f−1: PB → PA, a function between the power sets ofAandB, preserves both unions and intersections, while the direct image f∗: PA→PBonly preserves unions.
Proof. Example 4.1.8 describes adjunctions
PB f−1 //PA.
f∗
}} ⊥
f!
`` ⊥
Unions are colimits and intersections are limits in the poset categoryPA. Corollary 4.5.5. For any vector spacesU,V,W,
U⊗(V⊕W)(U⊗V)⊕(U⊗W).
Proof. The two-variable adjunction of Example 4.3.11 can be defined for the category of modules over any commutative ring, in particular for the category of vector spaces over a fieldk. It follows that the functorU⊗ −:Vectk→Vectkis left adjoint toHom(U,−)and consequently preserves the coproductV⊕W. This argument also proves that the tensor
product distributes over arbitrary direct sums.
Similarly, on account of the adjunctionsA× − a(−)Abetween the product and expo-nential onSetor its subcategoryFin, Theorems 4.5.2 and 4.5.3 supply proofs of many of the basic operations in arithmetic, first discussed in Example 1.4.9.
Corollary 4.5.6. For any setsA,B,C, there are natural isomorphisms
A×(B+C)(A×B)+(A×C) (B×C)ABA×CA AB+CAB×AC. Consequently, for any cardinalsα, β, γ, cardinal arithmetic satisfies the laws:
α×(β+γ)=(α×β)+(α×γ) (β×γ)α=βα×γα αβ+γ =αβ×αγ. Proof. The left adjointA× −preserves the coproductB+C, the right adjoint(−)A preserves the productB×C, and the functorA−: Setop → Set, which is mutually right adjoint to itself, carries coproducts inSetto products inSet. The laws of cardinal arithmetic follow by applying the cardinality functor| − |:Setiso→Card, which converts these natural isomorphisms to identities in the discrete category of cardinals.
4.5. ADJUNCTIONS, LIMITS, AND COLIMITS 139
The forgetful functors of Example 4.1.10 carry any limits that exist in the categories of groups, rings, modules, and so forth to corresponding limits of their underlying sets.
Indeed, in §5.6, we show that these forgetful functors create, and not merely preserve, all limits. The dual result, that left adjoints preserve colimits, is less directly useful in such contexts, characterizing only those colimit constructions involving “free” diagrams. For instance:
Corollary 4.5.7. The free group on the setXtYis the free product of the free groups on the setsXandY.
The free product of not-necessarily free groups will be constructed in §5.6 using language that also describes the construction of the coproducts in the categories of abelian groups, rings, and modules, among many others.
Theorems 4.5.2 and 4.5.3 also have important applications to homological algebra.
Corollary 4.5.8. For anyR-S bimoduleM, the tensor productM⊗S −is right exact.
Before proving Corollary 4.5.8, we should explain its statement. The term “right exact”
comes from homological algebra, which studies functors betweenabelian categories. An abstract definition is given in Definition E.5.1, but by a powerful result, Theorem E.5.2, it suffices to declare that a category isabelianif it is a full subcategory of category of modules that contains the zero object and is closed under direct sums, kernels, and cokernels.8
The following general definitions of left and right exactness makes sense for any functor, not necessarily between abelian categories.
Definition 4.5.9. A functor isright exactif it preserves finite colimits andleft exactif it preserves finite limits.
The following proposition connects Definition 4.5.9 to the notions of left and right exactness that are used in homological algebra.
Proposition 4.5.10. A functorF:A→ Bbetween abelian categories is left exact if and only if it preserves direct sums and kernels. For such functors, if
0→A→A0→A00 is an exact sequence inA, then
0→FA→FA0→FA00 is exact inB.
Proof. Direct sums and kernels are both finite limits, so if F is left exact, these constructions are preserved. Conversely, Theorem 3.4.12 proves that all finite limits can be built from finite products and equalizers. In an abelian category, the equalizer of a parallel pair of mapsf,g: A⇒A0is the kernel of the mapf−g:A→A0. A functor that preserves direct sums also preserves differences of maps, so ifFpreserves direct sums and kernels it preserves all finite limits.
Now a sequence
0 i //A j //A0 k //A00
is exact inAjust wheniis the kernel of j(equivalently, when jis a monomorphism), and when jis the kernel ofk. A left exact functor preserves these kernels and also the zero
object0.
8Azero objectis an object that is both initial and terminal; see Exercise 1.6.i. Direct sums are defined in Remark 3.1.27. As in Example 3.1.14, thekernelof a map f:A →A0is the equalizer of f with the zero homomorphism, while thecokernelis the coequalizer of this pair.
Immediately from Definition 4.5.9 and Theorems 4.5.2 and 4.5.3:
Corollary 4.5.11. For any adjoint functors between abelian categories, the left adjoint is right exact and the right adjoint is left exact. Moreover, both functors are additive.9
Corollary 4.5.8 is a special case:
Proof of Corollary 4.5.8. For anyR-S bimoduleM, there is a pair of adjoint func-tors
ModS M⊗S−//
⊥ ModR,
Homoo R(M,−)
where the right adjoint carries a leftR-moduleN to the leftS-module ofR-module ho-momorphisms M → N. As a left adjoint, Corollary 4.5.11 proves that M⊗S −is right
exact.
For a certain important class of adjoint functors, Theorems 4.5.2 and 4.5.3 take on a stronger form.
Definition 4.5.12. Areflective subcategoryof a categoryCis a full subcategoryDso that the inclusion admits a left adjoint, called thereflectororlocalization:
D ⊥ //C. oo L
Where possible, we identify the full subcategoryDwith its image inC, declining in particular to introduce notation for the inclusion functor. With this convention in mind, the components of the unit have the formc→ Lc; the mnemonic is that “an object looks at its reflection.” The following lemma implies that the components of the counitLddare isomorphisms. Via the counit, any object in a reflective subcategoryD,→Cis naturally isomorphic to its reflection back into that subcategory.
Lemma 4.5.13. Consider an adjunction C
F //
⊥ D
oo G
with counit:FG⇒1D. Then:
(i) Gis faithful if and only if each component ofis an epimorphism.
(ii) Gis full if and only if each component ofis a split monomorphism.
(iii) Gis full and faithful if and only ifis an isomorphism.
Dually, writingη: 1C⇒GFfor the unit:
(i) Fis faithful if and only if each component ofηis a monomorphism.
(ii) Fis full if and only if each component ofηis a split epimorphism.
(iii) Fis full and faithful if and only ifηis an isomorphism.
Proof. Exercise 4.5.vi.
Example 4.5.14. The following define reflective subcategories:
(i) Compact Hausdorff spaces define a reflective subcategory cHaus ,→ Topof the category of all topological spaces. The reflector is the functor β: Top → cHaus sending a space to its Stone–Čech compactification, which is constructed in Ex-ample 4.6.12. The universal property of the unit says that any continuous function
9A functor between abelian categories isadditiveif it preserves direct sums.
4.5. ADJUNCTIONS, LIMITS, AND COLIMITS 141
X → K from a space X to a compact Hausdorff spaceK extends uniquely to the Stone–Čech compactification:
X //
K βX
>>
(ii) Abelian groups define a reflective subcategoryAb ,→ Group of the category of groups. The reflector carries a groupGto itsabelianization, the quotientG/[G,G]by thecommutator subgroup, the normal subgroup generated by elementsghg−1h−1. The quotient maps G → G/[G,G] define the components of the unit. For any abelian groupA, there is an isomorphismA A/[A,A]; the commutator subgroup [A,A]is trivial if and only if the groupAis abelian. The adjunction asserts that a homomorphism from a groupGto an abelian groupAnecessarily factors through the abelianization ofG. A similar construction defines a left adjoint to the inclusion CRing,→Ringof commutative rings into the category of all rings.
(iii) The inclusionAbtf ,→Abof torsion-free abelian groups is reflective. The reflector sends an abelian groupAto the quotientA/T Aby its torsion subgroup. The quotient mapsA→ A/T Adefine the components of the unit. Any map fromAto a torsion-free group factors uniquely through this quotient homomorphism because any torsion elements ofAmust be contained in its kernel.
(iv) As described in Example 4.1.10(xi), for any ring homomorphismφ: R→ T, there exist adjoint functors
ModT
φ⊥∗ //ModR,
T⊗R−
oo
the right adjoint being restriction of scalars and the left adjoint being extension of scalars. The restriction of scalars functor is always faithful because both categories of modules admit faithful forgetful functors toAb, and is full if and only ifφ:R→T is an epimorphism. For such homomorphisms, restriction of scalars identifiesModT as a reflective subcategory ofModR.
Epimorphisms inRinginclude all surjections but are not limited to the surjec-tions. The localizations define another important class of epimorphisms. LetS ⊂R be a monoid under multiplication; that is, S is a multiplicatively closed subset of the ringR. ThelocalizationR→ R[S−1]is an initial object in the category whose objects are ring homomorphismsR→T that carry all of the elements ofS to units inT. For integral domains, the ringR[S−1]can be constructed as a field of fractions.
A similar construction exists for general commutative rings.
(v) Recall that apresheafon a spaceX is a contravariant functorO(X)op →Setfrom the poset of open subsets ofX to sets. A presheaf is asheaf if it preserves certain limits (see Definition 3.3.4). The sheaves define a reflective subcategoryShvXof the category of presheaves, with the left adjoint calledsheafification:
ShvX ⊥ //SetO(X)op
sheafify
oo
(vi) The category of small categories defines a reflective subcategory of the category Set∆opofsimplicial setsvia an adjunction that is constructed in Exercise 6.5.iv:
Cat
N //
⊥ Set∆op oo h
Here∆⊂Catis the full subcategory whose objects are the finite non-empty ordinals, in this context denoted by[0],[1],[2], . . ., and whose morphisms are all functors, i.e., order-preserving functions, between them. The embeddingN:Cat,→Set∆opcarries a small categoryCto itsnerve: NC: ∆op → Setsends[n] =0 → 1 → · · · → n to the set of functorsNCn :=Cat([n],C). The left adjointh:Set∆op → Catsends a simplicial set to itshomotopy category. Restricting to the objects[0],[1]∈∆, a simplicial setX:∆op →Sethas an underlying reflexive directed graph
X0oo ////X1.
The homotopy category hX is a quotient of the free category generated by this reflexive directed graph modulo relations that arise from elements of the setX2. In particular, the counit defines an isomorphismhNCCfor any categoryC, proving that the inclusion is full and faithful.
The following result explains our particular interest in reflective subcategories. The presence of a left adjoint to the inclusion of a full subcategory provides complete information about limits and colimits in that subcategory:
Proposition 4.5.15. IfD,→Cis a reflective subcategory, then:
(i) The inclusionD,→Ccreates all limits thatCadmits.
(ii) Dhas all colimits thatCadmits, formed by applying the reflector to the colimit inC.
By Theorems 4.5.2 and 4.5.3, ifDhas limits or colimits, they must be constructed in the way described in Proposition 4.5.15: limits are preserved by the inclusionD,→Cand colimits of diagrams inD, regarded as diagrams inC, are preserved by L:C → D. The real content of this result is that these (co)limits necessarily exist inD. A special case of this completes some unfinished business from §3.5:
Corollary 4.5.16. Catis complete and cocomplete.
Proof. By Proposition 3.3.9,Set∆op is complete and cocomplete, with limits and col-imits defined objectwise inSet. Proposition 4.5.15 implies that the reflective subcategory Catinherits these limits, defined objectwise in Set, and also these colimits, defined by applying the homotopy category functor to the colimit inSet∆op. Proposition 4.5.15(i) appears as Corollary 5.6.6, where this result is deduced as a special case of the more general Theorem 5.6.5.
Proof of 4.5.15(ii). For clarity, writei: D ,→ Cfor the inclusion, the right adjoint to the reflectorL. Consider a diagramF: J → D and letλ: iF ⇒ cbe a colimit cone for the diagramiFinC. By Theorem 4.5.3, the left adjointLsends this to a colimit cone Lλ: LiF⇒LcinD. Now the counit supplies a natural isomorphismLi1D. Composing with this natural isomorphism yields a colimit coneFLiF⇒Lcfor the original diagram
inD.
Exercises.
Exercise 4.5.i. When does the unique functor! :C→1have a left adjoint? When does it have a right adjoint?
Exercise 4.5.ii. Suppose the diagonal functor ∆: C → CJ admits both left and right adjoints. Describe the units and counits of these adjunctions.
Exercise 4.5.iii. Use Proposition 4.5.1 to prove that in any complete category limits commute with limits in the sense of the natural isomorphism of limit functors (3.8.2).