1

Adjoint Functors:The Standard Theory and the Heteromorphic Theory

David EllermanUniversity of California at Riverside

2

Adjoints in CT and Foundations

"The notion of adjoint functor applies everything that we've learned up to now to unify and subsume all the different universal mapping properties that we have encountered, from free groups to limits to exponentials. But more importantly, it also captures an important mathematical phenomenon that is invisible without the lens of category theory. Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics." (Steve Awodey, Category Theory)

"The isolation and explication of the notion of adjointness is perhaps the most profound contribution that category theory has made to the history of general mathematical ideas." (Robert Goldblatt, Topoi)

"Nowadays, every user of category theory agrees that [adjunction] is the concept which justifies the fundamental position of the subject in mathematics." (Paul Taylor, Practical Foundations of Mathematics)

3

Standard Hom-Set Dfn of Adjoints

Let A and X be categories with functors F:X→A and G:A →X between them.

Then F and G are a pair of adjoint functors if there is an isomorphism natural in x and a:

x,a: HomA(Fx,a) HomX(x,Ga).

F is the left adjoint and G is the right adjoint. Maps associated across the isomorphism are adjoint

transposes or correlates of one another. Example: A = Groups; X = Sets; F = free group functor;

G = underlying set functor.

4

Universal Mapping Property of Unit

Adjoint transpose of 1Fx:Fx→Fx is the unit x:x→GFx and transpose of 1Ga:Ga→Ga is the counit a:FGa→a.

UMP of unit: given any f:x→Ga, there is a unique map g:Fx →a (g = adjoint transpose of f) such that: Ggx:x →GFx →Ga = f:x →Ga.

Free Group Example: Given any map f:x→Ga from set x to underlying set Ga of a group, there is a unique group homomorphism g:Fx→a such that the underlying set map Gg factors f through the unit x.

x Ga

Fx a

x

f

GFx

Gg

!g

5

Given any map g:Fx→a, there is a unique map f:x →Ga (f is adjoint transpose of g) such that aFf: Fx →FGa →a = g:Fx →a.

Example: Given any group map g:Fx→a, there is a unique set map f:x→Ga such that the group homomorphism Ff factors g through the counit a.

Universal Mapping Property of Counit

Fx a

x Ga

a

g

FGa

Ff

!f

6

Aspects of the Standard Treatment

All object-to-object morphisms are within one category or the other—only homomorphisms, no heteromorphisms (yet).

Unique factor maps for units and counits exist in the other category.

Apparent symmetry of two categories; at first there seems to be no directionality of adjoints.

7

Directionality of Adjoints

But on closer examination, there is a directionality in: HomA(Fx,a) HomX(x,Ga).

Both the maps Fx→a and x →Ga go from the x-related object to the a-related object. An object “x” in X appears by itself only as a domain and an object “a” in A appears by itself only as a codomain.

Thus there seems to be some type of directionality from X to A.

8

Using Adjoints to embed X and A in their Product

There is another way to present the standard theory that forms a bridge to the heteromorphic theory.

Consider the product category XA with the embedding X → XA defined by x (x,Fx) and the embedding A→ XA defined by a (Ga,a) [obvious on maps].

The image of X → XA is an isomorphic subcategory XX and the image of A→ XA is an isomorphic subcategory AA.

Thus (x,Fx) plays the role of “x” in the isomorphic copy of X and (Ga,a) plays the role of “a” in the copy of A:

x ↔ (x,Fx) and a ↔ (Ga,a).

9

Origin of Adjunctive Square Diagram

Move the two commutative triangles of the unit and counit UMPs together and fill it out to get a diagram in the product category.

Fx a

a

g

FGa

Ff

x Ga

x

f

GFx

Gg

x Ga

x

f

GFx

Gg

Fx a

a

g

FGa

Ff

(x,Fx)

(GFx,Fx) (Ga,a)

(1Ga,a)

(Gg,g)

(Ga,FGa)(f,Ff)

(x,1Fx) (f,g)

X

A

AdjunctiveSquareDiagram

10

Adjunctive Square Diagram

Commutative square in first (second) coordinate is UMP of unit (counit). Top-to-bottom maps from objects in category XX to objects in category AA

(but are still ordinary maps in the product category XA). They “foreshadow” (and represent) heteromorphisms.

Diagonal map (f,g) is just pair of adjoint correlates. Thus in the hom-set isomorphism of the adjunction, a third unique correlate has appeared from the x-object (x,Fx) to the a-object (Ga,a).

The adjunction iso: g ↔ f now has 3 terms: (Gg,g) ↔ (f,g) ↔ (f,Ff).

Top map (f,Ff) is in XX category and is the image of f. Bottom map (Gg,g) is in AA category and is image of g. Maps f and g are adjoint correlates.

(x,Fx)

(GFx,Fx) (Ga,a)

(1Ga,a)

(Gg,g)

(Ga,FGa)(f,Ff)

(x,1Fx) (f,g)

X

A

11

“Hetero” Adjunctive Square Replace each object and map on top and bottom

in the adjunctive square by the object and map it represents, e.g., (x,Fx) becomes x, (Ga,a) becomes “a”, etc.

Then the top-to-bottom maps (if they exist) are new CT creatures, heteromorphisms (or chimera morphisms) between objects of different categories. (hets = thick arrows)

Example: Given f:x→Ga as any map from set x to underlying set Ga of group a, take c:x→a as same map point-wise but with codomain as the group, and take g:Fx→a as f’s adjoint correlate. Take hx:x→Fx as the usual embedding of a set into the free group Fx on the set and take ea:Ga→a as the map from the underlying set of a group back to the group.

Then the diagram commutes!

x

Fx a

ea

g

Gaf

hx

c

12

Het-bifunctors

How to rigorously say “it commutes”, i.e., how to define composition between homo- and hetero-morphisms?

Composition between homomorphisms within category X is rigorously defined using the Hom-bifunctors HomX:XopX→Set and similarly for category A.

Composition for heteromorphisms from X to A is rigorously defined using a bifunctor called a Het-bifunctor Het:XopA→Set.

Given g':a →a' in A, composition with hets like c:x →a is defined by Het(x,g'):Het(x,a) →Het(x,a').

Given f':x' →x in X, composition with hets like c:x →a is defined by Het(f',a):Het(x,a) →Het(x',a).

Thus a mule c breeds with a horse g' to make another mule g'c in Het(x,a'), and a mule c breeds with a donkey f' to make another mule cf' in Het(x',a).

x' f' x g'a a'cX A

13

Representations of a functor

Given a functor H:C→Set from a category C to sets, it is representable if there is a representing object r C and a representation which is a natural isomorphism HomC(r,-) H(-).

Instead of being given a pair of adjoints F:X→A and G:A→X, suppose we are given only a bifunctor Het:XopA→Set.

For each object x X, Het defines a functor Het(x,-):A→Set, and for each object a A, Het defines a functor Het(-,a):Xop→Set.

14

Adjoints = Birepresentations of a Het-bifunctor

Suppose for each object x X that the functor Het(x,-) is represented by an object Fx so there is a natural isomorphism: HomA(Fx,a) Het(x,a). By varying x, a functor F:X→A is defined.

Suppose for each a A that the functor Het(-,a) is represented by an object Ga so there is a natural isomorphism: Het(x,a) HomX (x,Ga). By varying a, the functor G:A→X is defined.

If both representations exist, then we have the natural isomorphisms: HomA(Fx,a) Het(x,a) HomX(x,Ga).

That is what an adjunction really is! [See: Ellerman, David 2006. A Theory of Adjoint Functors—with some

Thoughts on their Philosophical Significance. In What is Category Theory? Giandomenico Sica ed., Milan: Polimetrica: 127-183.]

15

Heteromorphic Theory of Adjoints

Every bifunctor Het:XopA→Set that is represented both on the left (by F) and right (by G) gives rise to a pair of adjoint functors:

HomA(Fx,a) Het(x,a) HomX(x,Ga). The usual treatment, HomA(Fx,a) HomX(x,Ga), leaves

out the Het bifunctor term in the middle. Conversely, given a pair of adjoints, the “abstract” Het

bifunctor can always be defined using the isomorphic copies of X and A, i.e.,

Het:XXopAA→Set where Het((x,Fx),(Ga,a)) is the set of main diagonals (f,g) in the adjunctive square diagram where f and g are adjoint transposes.

Empirical claim: concrete hets can also be found in natura.

16

Het Units and Counits

In the isomorphisms:

HomA(Fx,a) Het(x,a) HomX(x,Ga)

Adjoint correlates of 1Fx are the usual unit x:x→GFx and the het or chimera unit hx:x→Fx, and

Adjoint correlates of 1Ga:Ga→Ga are the usual counit a:FGa→a and the het or chimera counit ea:Ga→a.

The chimera units and counits are the vertical maps that make the het adjunctive square commute.

x

Fx a

ea

g

Gaf

hx

c

17

Simpler UMPs for Het Units and Counits

Given a het c:x→a, there is a unique morphism g:Fx→a in A that factors c through the unit hx .

Given a het c:x→a, there is a unique morphism f:x→Ga in X that factors c through the counit ea.

x

a

ea

Gaf

c

x

Fx ag

hx

c

18

Half-Adjunctions

Since the usual adjunctive isomorphism,

HomA(Fx,a) HomX(x,Ga)leaves out the middle Het term, there is ordinarily no such thing as a “half-adjunction.”

But starting with a Het bifunctor, it might only be represented on one side so there could be half-adjunctions in the form HomA(Fx,a) Het(x,a) or Het(x,a) HomX(x,Ga).

Each full adjunction, of course, makes two half-adjunctions. The simple UMPs for the het units and counits involve only the

respective half-adjunctions. In a full adjunction, it is typical that only one of the half-

adjunctions is of interest while the other is a trivial piece of conceptual bookkeeping, e.g., the free group functor is the interesting part of the free-group/underlying-set adjunction.

19

Example: Limits in Set (1)

Let DD be any (small) category taken as a “diagram” category and let D:DD→Set be a functor giving a “diagram in Set” which can be taken as an object in the functor category SetDD.

A heteromorphism from an object x Set to an object D SetDD is a set of maps c = {cd:x→Dd}dDD indexed by the objects in DD such that for any morphism δ:d→d' in DD, the following triangle commutes.

A het from a set to a functor is usually called a “cone.”

x

Dd'

Dδ

Ddcd

cd'

20

Example: Limits in Set (2)

The limit LimD is formed by taking the product of all the sets Dd and then taking LimD as the subset of all the elements in the product so that the projections commute with the Dδ maps.

The construction is functorial and provides a right adjoint Lim(-):SetDD→Set.

The cone of projection maps from the set LimD to the functor D is the het counit eD:LimD→D.

21

Example: Limits in Set (3) The trivial left adjoint associates with set x the constant

diagram functor x:DD→Set that maps each d to x and each map δ to 1x. The chimera unit hx:x→x is the cone of 1x's.

Situation summarized by het adj. square. Many texts give het counit eD:LimD→D

as a universal cone and picture other cones c:x→D as uniquely factoring through universal cone. That is, the texts are using the interesting half-adjunction and the simpler het counit UMP without realizing it. x

D

eD

LimD!fc

cLimDx

d

d'

DδeD!fc

cd

cd'

=

Standard diagram = Half of het adj. square

x

x D

eD

gc

LimDfc

hx

c

22

Homs to Hets Methodology

There is rich theory of the heteromorphisms surrounding adjoints that is normally hidden from view (although it occasionally comes out as with the universal cones).

One methodology to find the het theory is find structure in the ordinary adjunctive square diagram (and its images) and then go to the het version of the diagram.

We start with the canonical anti-diagonal map in the ordinary (hom) adjunctive square diagram.

23

Zig-Zag Factorizations

On the product category XA, the twist functor (F,G) is defined by applying the two adjoints and reversing the order of the results, i.e., (F,G)(x,Fx) = (GFx,Fx).

Applying the twist functor to the diagonal (f,g) (maps which are adjoint transposes) yields the antidiagonal map (F,G)(f,g) = (Gg,Ff) so that everything commutes in the following adj. sq.

(x,Fx)

(GFx,Fx) (Ga,a)

(1Ga,a)

(Gg,g)

(Ga,FGa)(f,Ff)

(x,1Fx)(f,g)

(Gg,Ff)

The main diagonal can then be factored through both the left and right vertical maps using the anti-diagonal map, which is called the zig-zag factorization:

(f,g) = (1Ga,a)(Gg,Ff)(x,1Fx)

and which uses both of the UMPs for the unit and counit.

24

Het Zig-Zag Factorizations

Example 1: In the free group adjunction, zc:Fx→Ga is just gc:Fx→a with the codomain taken as the underlying set.

Example 2: In the limits adjunction, zc:x→LimD is the functor-to-set het (“cocone”) where each map is fc.

x

Fx a

ea

gc

Gafc

hx

c

zc

Apply the hom-to-het method to expect to find in natura anti-diagonal maps in het adj. squares.

These give the all-het zig-zag factorizations:

c = eazchx

which use both the het unit and het counit.

x

x D

eD

gc

LimDfc

hx

c

zc

25

Triangular Identities

Just as UMPs of adjoints are much simplified in het versions, so other results such as the triangular identities are also simplified.

Usual triangular identities:

FxFx:Fx→FGFx→Fx = 1Fx:Fx→Fx

GaGa:Ga→GFGa→Ga = 1Ga:Ga→Ga. Het triangular identities:

eFxzhx:Fx→GFx→Fx = 1Fx:Fx→Fx

zeahGa:Ga→FGa→Ga = 1Ga:Ga→Ga.

Ga

FGa a

ea

a

Ga1Ga

hGa zea

ea

zea hGa = 1Ga

x

Fx Fx

eFx

1Fx

GFxx

hx zhx

hx

eFx zhx = 1Fx

26

Adjunctive-Image Square

Why apply the twist functor to just the main diagonal? Apply it to the whole adjunctive square to get a new adjunctive-image square where the previous antidiagonal (Gf,Ff) becomes the main diagonal (and note that F has to be 1-1 on maps x→Ga and G has to be 1-1 on maps Fx→a). (GFx,Fx)

(GFx,FGFx) (Ga,FGa)

(1FGa,Ga)

(Gg,FGg)

(GFGa,FGa)(GFf,Ff)

(1GFx, Fx)(Gg,Ff)

This generates many new identities, and one can even take more images.

27

Impact on Basic CT Concepts

According to Eilenberg-MacLane, categories and functors were defined to treat natural transformations (nts). Then the naturalness and canonicalness of certain math constructions can be rigorously characterized. A nt is between two functors with the same domains and the same codomains.

But there is a certain canonicalness in any construction that is functorial. However it cannot be captured with ordinary nts since the domain and codomain of a functor are, in general, in different categories.

This constraint is relaxed with the machinery for treating heteromorphisms because then a notion of het natural transformation can be defined.

28

Het Natural Transformations

Given functors F:X→A and H:X→B (different codomains) and a bifunctor Het:AopB→Set, a het natural transformation (rel. to Het) φ:FH is given by a set of heteromorphisms {φx Het(Fx,Hx)} such that for any morphism j:x→x' in X, the following diagram commutes.

Fx'

Fj

Fx

Hx'

Hx

Hj

φx

φx'

29

Examples of Het N.T.s

Het-less CT can say that the unit :1XGF is natural but cannot say that the het unit h:1XF is natural—because the het unit is a het nt. Ditto for the unit :FG1A and the het counit e:G1A.

Any functor F:X→A trivially defines a bifunctor Het:XopA→Set by Het(x,a) = HomA(Fx,a) so that 1XF is a het nt. And Het(a,x) = HomA(a,Fx) yields the het nt F1X.

The extent to which one can find concrete hets from X to A so that there would be a het nt 1XF (or vice-versa) is an open question.

30

Summing Up The development of CT has missed the concept of object-to-

object heteromorphisms between categories even though such morphisms are in common mathematical practice and are as “real” as any homomorphisms within categories.

Introducing hets brings out much hidden structure particularly concerning the central notion of adjoint functors.

Basic Result: Adjoint functors between categories arise from the birepresentations within each category of the heteromorphisms between the categories.

The aspect of adjoints that accounts for their importance is the formulation of universal mapping properties (which enters through the representations), an aspect that is amplified, simplified, and clarified in the heteromorphic treatment.

The explicit treatment of hets may also broaden other horizons in CT and its generalizations, e.g., by generalizing the notion of natural transformation.