Limits and Colimits
4.3. Contravariant and multivariable adjoint functors
Kan’s original exploration of adjoint functors also considered adjoint relationships involving functors that are contravariant or have more than one variable [Kan58]. Indeed, these examples provided the central motivation for Kan’s discovery, at least according to an apocryphal account of a question he posed to Eilenberg in a homological algebra seminar at Columbia University [Mar09, §4.2]:
kan: You have explained how the tensor product can be defined in terms of the hom functor. Can the hom instead be defined in terms of the tensor product?
eilenberg: No, of course not. That’s absurd.
In fact, Kan’s intuition was correct. By the Yoneda lemma, any adjoint functor determines its adjoints up to natural isomorphism. Example 4.3.11 explains how to define the hom bifunctor from the tensor product.
Contravariant adjoint functors require little special consideration, arising simply as an application of the principle of duality. Definition 4.1.1 can be dualized in three ways, by replacingC,D, or bothCandDby their opposite categories. The latter dualization recovers the original notion of adjunction, withG:Dop→Copleft adjoint toF: Cop →Dopif and only ifFaG. In particular, any theorem about left adjoints has a dual theorem about right adjoints, which is a very useful duality principle for adjunctions. The other two dualizations lead to new types of adjoint functors.
Definition 4.3.1. A pair of contravariant functors F: Cop → D andG: Dop → C are mutually left adjointif there exists a natural isomorphism
D(Fc,d)C(Gd,c),
4.3. CONTRAVARIANT AND MULTIVARIABLE ADJOINT FUNCTORS 127
ormutually right adjointif there exists a natural isomorphism D(d,Fc)C(c,Gd).
Dualizing Definition 4.2.5, a pair of mutual left adjointsF:Cop →DandG: Dop→C come equipped with a pair of “counit” natural transformationsGF ⇒1CandFG ⇒ 1D, while a pair of mutual right adjointsF: Cop → DandG:Dop → Ccome equipped with a pair of “unit” natural transformations1C ⇒GFand1D ⇒ FG. The formulation of the triangle identities in each case is left to Exercise 4.3.i.
Mutual right adjoints between preorders form what is sometimes called anantitone Galois connection.
Example 4.3.2. For a fixed natural numbernand algebraically closed fieldk, consider the sets of elements in the polynomial ringk[x1, . . . ,xn]and in the vector spacekn. There are contravariant functors
P(k[x1, . . . ,xn])op −→V P(kn) and P(kn)op→−I P(k[x1, . . . ,xn])
between the posets of subsets of these sets defined forS ⊂k[x1, . . . ,xn]andT ⊂knby:
V(S)B(t1, . . . ,tn)∈kn
f(t1, . . . ,tn)=0, ∀f ∈S I(T)Bf ∈k[x1, . . . ,xn]
f(t1, . . . ,tn)=0, ∀(t1, . . . ,tn)∈T , and these define mutual right adjoints because
T ⊂V(S) if and only if S ⊂I(T). This Galois connection is a starting point of modern algebraic geometry.
The unit and counit encode relations
T ⊂V(I(T)) and S ⊂I(V(S)).
A set of pointsT ⊂kndefines a fixed pointT =V(I(T))if and only ifT is closed in the Zariski topology onkn =Spec(k[x1, . . . ,xn]), introduced in Example 1.3.7(iv). A set of polynomialsS ⊂k[x1, . . . ,xn]defines a fixed pointS =I(V(S))if and only ifS defines a radical ideal ofk[x1, . . . ,xn]; this isHilbert’s Nullstellensatz. Exercise 4.2.i now implies that the poset of Zariski closed subsets ofknis isomorphic to the opposite of the poset of radical ideals ofk[x1, . . . ,xn].
Example 4.3.3. Let Axiomσ be a set of axioms, i.e., sentences in a fixed first-order language whosesignatureσ, specifies a list of function, constant, and relation symbols to be used with the standard logical symbols. Let Structσbe a set ofσ-structures, i.e., sets with interpretations of the given constant, relation, and function symbols. For instance, the language of the natural numbers has a constant symbol “0,” a binary function symbol “+,”
and a binary relation symbol “≤.” A structure for this language is any set with a specified constant, binary function, and binary relation. Axioms include sentences such as
∀x,y,z,((x≤y)∧(y≤z))→(x≤z) or ∀x,x+0=x,
asserting transitivity of the relation “≤” and that the constant “0” serves as a unit for the binary function “+.”
Given a set ofσ-structures Mand a set of axiomsA, we writeM Aif each of the axioms in Aissatisfied by—meaning “is true in”—each of theσ-structures in M. For instance, the first displayed sentence is satisfied by a structure if and only if its interpretation of the relation “≤” is transitive.
Form the poset categoriesP(Axiomσ)andP(Structσ)ordered by inclusion. There are contravariant functors
P(Axiomσ)op −−−−−True in→P(Structσ) and P(Structσ)op−Satisfying−−−−−−→P(Axiomσ) which send a set of axioms to the set of σ-structures, called models, that satisfy those axioms and send a set ofσ-structures to the set of axioms that they satisfy. These are mutual right adjoints, forming what is called theGalois connection between syntax and semantics[Smi].
Before turning our attention to multivariable functors, we make a useful observation that applies equally to ordinary adjunctions:
Proposition 4.3.4. Consider a functorF:A → Bso that for eachb ∈Bthere exists an objectGb∈Atogether with an isomorphism
(4.3.5) B(Fa,b)A(a,Gb), natural ina∈A.
Then there exists a unique way to extend the assignmentG: obB → obA to a functor G: B→Aso that the family of isomorphisms(4.3.5)is also natural inb∈B.
In other words, ifF:A→Bis a functor admitting representations for each functor B(F−,b) :Aop→Set
for eachb∈B, then this data assembles into a right adjointFaG. Proof. Naturality of (4.3.5) in f:b→b0demands that the function
A(a,Gb)B(Fa,b)−→f∗ B(Fa,b0)A(a,Gb0)
equals post-composition by the yet-to-be-defined morphismG f:Gb→Gb0. This compos-ite function defines a natural transformationA(−,Gb)⇒ A(−,Gb0), which by the Yoneda lemma must equal post-composition by a unique morphismG f:Gb→Gb0inA. Unique-ness of this definition implies that the assignmentG: morB→morAis functorial, as we
saw in the proof of Theorem 1.5.9.4
The argument used to prove Proposition 4.3.4 can be extended to functors of many variables.
Proposition 4.3.6. Suppose that F: A×B → C is a bifunctor so that for each object a∈A, the induced functorF(a,−) :B→Cadmits a right adjointGa:C→B. Then:
(i) These right adjoints assemble into a unique bifunctorG: Aop×C →B, defined so thatG(a,c)=Ga(c)and so that the isomorphisms
C(F(a,b),c)B(b,G(a,c)) are natural in all three variables.
If furthermore for eachb∈B, the induced functorF(−,b) :A→Cadmits a right adjoint Hb:C→A, then:
4More concisely,Gis defined to be the unique restriction of the functor B B(F−,−) //
G
SetAop
A. y ==
along the Yoneda embedding so that the triangle commutes up to the specified natural isomorphisms (4.3.5).
4.3. CONTRAVARIANT AND MULTIVARIABLE ADJOINT FUNCTORS 129
(ii) There is a unique bifunctorH:Bop×C→Adefined so thatH(b,c)=Hb(c)and the isomorphisms
C(F(a,b),c)B(b,G(a,c))A(a,H(b,c)) are natural in all three variables.
(iii) In this case, for eachc∈C, the functorsG(−,c) :Aop → BandH(−,c) :Bop → A are mutual right adjoints.
Proof. Exercise 4.3.ii.
Definition 4.3.7. A triple of bifunctors
A×B−→F C, Aop×C−→G B, Bop×C−→H A equipped with a natural isomorphism
C(F(a,b),c)B(b,G(a,c))A(a,H(b,c)) defines atwo-variable adjunction.
Particularly, whenF:C×C→Cdefines some sort of monoidal product, its pointwise-defined right adjointsGandHare called itsleftandright closures, respectively. When these are isomorphic, the bifunctorFis calledclosed.
Example 4.3.8. The product bifunctor
Set×Set−→× Set
is closed: the operation calledcurryingin computer science defines a family of natural isomorphisms
{A×B−→f C}{A−→f CB}{B−→f CA}. Thus, the product and exponential bifunctors
Set×Set−→× Set, Setop×Set (−)
(−)
−−−−→Set, Setop×Set (−)
(−)
−−−−→Set define a two-variable adjunction.
Definition 4.3.9. Acartesian closed categoryis a categoryC with finite products in which the product bifunctor
C×C−→× C is closed.
In addition to Set, the categories Fin, Cat, SetBG, and, more generally, SetC are cartesian closed; this was proven forCatin Exercise 1.7.vii.
Example 4.3.10 (a convenient category of spaces). The category Top is not cartesian closed. However, a famous paper of Steenrod proves that there exists aconvenient category of topological spacesthat is complete, cocomplete, cartesian closed, and sufficiently large so as to contain the CW complexes [Ste67]. Steenrod’s convenient category is a subcategory of the categoryHausof Hausdorff spaces and continuous functions. Previous researchers had argued that a “convenient category of spaces” satisfying Steenrod’s list of conditions did not exist, but Steenrod notes:
The arguments are based on a blind adherence to the customary definitions of the standard operations. These definitions are suitable for the category of Hausdorff spaces, but they need not be for a subcategory. The categorical viewpoint enables us to defrost these definitions and bend them a bit.
What he means is that the “product” on a yet-to-be-defined subcategorycgHausshould be the categorical product of Definition 3.1.9, which, as it turns out, is not preserved by the full inclusioncgHaus,→Haus.
Objects incgHausarecompactly generated Hausdorff spaces: Hausdorff spacesX with the property that any subsetA⊂ Xthat intersects each compact subset K ⊂X in a closed subsetA∩Kis itself closed inX. The inclusioncgHaus,→Haushas a right adjoint k:Haus→cgHauscalledk-ification: the spacek(X)refines the topology onXby adding to the collection of closed sets those subsetsA⊂Xwhose intersections with all compact subsetsKare closed.
By Exercise 4.6.ii and the dual of Proposition 4.5.15, the presence of this adjoint implies thatcgHausis complete and cocomplete. The product oncgHausis thek-ification of the product onHaus; this latter product is preserved by the inclusionHaus,→Top. The pointwise right adjoint is given by the following construction offunction spacesMap(X,Y): the underlying set ofMap(X,Y)is the set of continuous mapsX→Y, and the topology is thek-ification of the compact-open topology. See [Ste67] for a proof.
Other categories admit a hom bifunctor that is not necessarily right adjoint to the cartesian product. The most famous of these are the “tensor a hom” adjunctions that inspired Kan:
Example 4.3.11. Consider the hom bifunctor
Abop×Ab−−−→Hom Ab,
whereHom(A,B)is the group of homomorphismsA→Bwith addition defined pointwise in the abelian groupB.5 Fixing the contravariant variable, there is an adjunction
(4.3.12) Ab
A⊗Z−//
⊥ Ab
Hom(A,−)oo Ab(A⊗ZB,C)Ab(B,Hom(A,C))
defining the tensor product. Once the objectsA⊗ZB∈Abhave been defined, Proposition 4.3.4 uses the isomorphisms (4.3.12) to extend this data into a functorA⊗Z−: Ab→Ab. Proposition 4.3.6(i) then extends this data into thetensor product bifunctor
Ab×Ab−−→⊗Z Ab
in such a way that the tensor product and the hom define a two-variable adjunction.
As Kan suspected, this process can be reversed. Given the tensor product bifunctor
⊗Z:Ab×Ab → Ab, the abelian groupHom(B,C)can be defined as a representation for the functor
(4.3.13) Abop−Ab−−−−−−−−(−⊗ZB,C)→Set Ab(A,Hom(B,C))Ab(A⊗ZB,C).
Propositions 4.3.4 and 4.3.6(i) then imply that there is a unique way to extend this con-structionHom : obAb×obAb→obAbinto a bifunctorHom :Abop×Ab→Abin such a way that the isomorphisms (4.3.13) are natural inBandC, as well asA.
Example 4.3.14. Temporarily, let Topdenote not the ordinary category of topological spaces but the cartesian closed category of spaces defined in Example 4.3.10. In this
5Recall from §1.1 that this is why the collection of morphisms between a fixed pair of objects in a general category is often denoted “Hom.” Here we use “Ab” for the mere set of homomorphisms, which in this instance plays a secondary role.
4.3. CONTRAVARIANT AND MULTIVARIABLE ADJOINT FUNCTORS 131
context, the two-variable adjunction specializes to define an adjunction:
Top
S1×− //
⊥ Top
Map(Soo 1,−)
whereS1 ∈Topis the unit circle. The spaceMap(S1,X)is thefree loop spaceonX; points inMap(S1,X)are loops in the spaceX.
The slice category ofTopunder the singleton space∗defines a convenient category Top∗ of based topological spaces. By Exercise 4.3.iv(ii), Top∗ admits a two-variable adjunction
Top∗×Top∗−→∧ Top∗, Topop∗ ×Top∗−Map−−−→∗ Top∗, Topop∗ ×Top∗−Map−−−→∗ Top∗, whereMap∗((X,x),(Y,y))denotes the based space of basepoint-preserving continuous func-tions(X,x)→(Y,y). The bifunctor∧is called thesmash product.
The space
ΩXBMap∗(S1,(X,x))
is thebased loop spaceon(X,x); the basepoint x∈ Ximplicit in the common notation.
The functorΩhas a left adjointΣX BS1∧X, which constructs thereduced suspension of the based space(X,x). This defines theloopsasuspension adjunctionbetween based topological spaces:
Top∗ Σ //
⊥ Top∗
oo Ω Top∗(ΣX,Y)Top∗(X,ΩY)
Definition 4.3.7 can be generalized to define multivariable adjunctions. Applying Proposition 4.3.6, ann-variable adjunctionis determined by a functorF:A1×· · ·×An→B admitting pointwise right adjoints when any(n−1)of its variables are fixed. See [CGR14] for more.
Exercises.
Exercise 4.3.i. Dualize Definition 4.2.5 to define mutual left adjoints and mutual right ad-joints as a pair of contravariant functors equipped with appropriate natural transformations.
Exercise 4.3.ii. Prove Proposition 4.3.6.
Exercise 4.3.iii. Show that the contravariant power set functorP:Setop→Setis mutually right adjoint to itself.
Exercise 4.3.iv. Define pointwise adjoints to the following bifunctors, giving rise to examples of two-variable adjunctions.
(i) There is a bifunctor
Setop∗ ×Set∗ Hom∗
−−−−→Set∗,
whereHom∗((X,x),(Y,y))is defined to be the set of pointed functions(X,x)→(Y,y), with the constant function at yserving as the basepoint. Define a two-variable adjunction determined by this bifunctor, the pointwise left adjoints toHom∗((X,x),−). (ii) Describe the left adjoint bifunctorSet∗×Set∗
−→∧ Set∗constructed in (i) in a suffi-ciently categorical way so thatSetcan be replaced by any cartesian closed category with pushouts and pullbacks.
(iii) The discussion in Example 4.3.11 extends to any category ModR of modules over acommutativeringR. IfRis not commutative, for a pair ofR-modulesAandB, Hom(A,B) is not necessarily an R-module, but it is still an abelian group. This construction defines a bifunctor
ModopR ×ModR
−−−→Hom Ab. Extend this data to a two-variable adjunction.
(iv) In a similar fashion, define a two-variable adjunction determined by the hom bifunctor ChopR ×ChR
−−−→Hom ModR,
where the ringRis commutative and the setHom(A•,B•)of chain homomorphisms A• → B• inherits an R-module structure with addition and scalar multiplication defined elementwise.