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Adjoint Functors: The Standard Theory and the Heteromorphic Theory

Adjoint Functors: The Standard Theory and the Heteromorphic Theory. David Ellerman University of California at Riverside. Adjoints in CT and Foundations.

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Adjoint Functors: The Standard Theory and the Heteromorphic Theory

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  1. Adjoint Functors:The Standard Theory and the Heteromorphic Theory David Ellerman University 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. f x Ga x Gg GFx !g a Fx 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: Ggx: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.

  5. !f x Ga FGa Ff a Fx a g Universal Mapping Property of Counit • 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.

  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 XA with the embedding X → XA defined by x  (x,Fx) and the embedding A→ XA defined by a  (Ga,a) [obvious on maps]. • The image of X → XA is an isomorphic subcategory X and the image of A→ XA is an isomorphic subcategory A. • 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. f f x x Ga Ga FGa FGa Gg Gg x x a a Ff Ff GFx GFx Fx Fx a a g g X (f,Ff) (x,Fx) (Ga,FGa) (f,g) (x,1Fx) (1Ga,a) (Gg,g) (GFx,Fx) (Ga,a) A 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. Adjunctive Square Diagram

  10. X (f,Ff) (x,Fx) (Ga,FGa) (f,g) (x,1Fx) (1Ga,a) (Gg,g) (GFx,Fx) (Ga,a) A Adjunctive Square Diagram • Commutative square in first (second) coordinate is UMP of unit (counit). • Top-to-bottom maps from objects in category X to objects in category A (but are still ordinary maps in the product category XA). 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 X category and is the image of f. Bottom map (Gg,g) is in A category and is image of g. Maps f and g are adjoint correlates.

  11. f x Ga c hx ea g Fx a “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!

  12. c f' g' X x' x a a' A 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:XopX→Set and similarly for category A. • Composition for heteromorphisms from X to A is rigorously defined using a bifunctor called a Het-bifunctor Het:XopA→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).

  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:XopA→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:XopA→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:XopA→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. f x Ga c hx ea g Fx a 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.

  17. f x x Ga c c hx ea g Fx a a 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.

  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. cd Dd x Dδ cd' Dd' Example: Limits in Set (1) • Let D be any (small) category taken as a “diagram” category and let D:D→Set be a functor giving a “diagram in Set” which can be taken as an object in the functor category SetD. • A heteromorphism from an object x  Set to an object D  SetD is a set of maps c = {cd:x→Dd}dD indexed by the objects in D such that for any morphism δ:d→d' in D, the following triangle commutes. • A het from a set to a functor is usually called a “cone.”

  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(-):SetD→Set. • The cone of projection maps from the set LimD to the functor D is the het counit eD:LimD→D.

  21. fc !fc d cd x x LimD LimD c c !fc eD = x hx eD Dδ eD LimD gc cd' d' x D D Standard diagram = Half of het adj. square Example: Limits in Set (3) • The trivial left adjoint associates with set x the constant diagram functor x:D→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.

  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. (f,Ff) (x,Fx) (Ga,FGa) (f,g) (x,1Fx) (1Ga,a) (Gg,Ff) (Gg,g) (GFx,Fx) (Ga,a) Zig-Zag Factorizations • On the product category XA, 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. • 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. fc fc x x LimD Ga c c hx hx ea eD zc zc gc gc x Fx a D Het Zig-Zag Factorizations • 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. • 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.

  25. 1Ga x Ga x GFx Ga ea hx hGa hx eFx ea zhx zea 1Fx a FGa Fx a Fx eFx zhx = 1Fx zea hGa = 1Ga 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: FxFx:Fx→FGFx→Fx = 1Fx:Fx→Fx GaGa:Ga→GFGa→Ga = 1Ga:Ga→Ga. • Het triangular identities: eFxzhx:Fx→GFx→Fx = 1Fx:Fx→Fx zeahGa:Ga→FGa→Ga = 1Ga:Ga→Ga.

  26. (GFf,Ff) (GFx,Fx) (GFGa,FGa) (Gg,Ff) (1GFx, Fx) (1FGa,Ga) (Gg,FGg) (GFx,FGFx) (Ga,FGa) 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). • 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. φx Fx Hx Fj Hj Fx' Hx' φx' Het Natural Transformations • Given functors F:X→A and H:X→B (different codomains) and a bifunctor Het:AopB→Set, a het natural transformation (rel. to Het) φ:FH 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.

  29. Examples of Het N.T.s • Het-less CT can say that the unit :1XGF is natural but cannot say that the het unit h:1XF is natural—because the het unit is a het nt. Ditto for the unit :FG1A and the het counit e:G1A. • Any functor F:X→A trivially defines a bifunctor Het:XopA→Set by Het(x,a) = HomA(Fx,a) so that 1XF is a het nt. And Het(a,x) = HomA(a,Fx) yields the het nt F1X. • The extent to which one can find concrete hets from X to A so that there would be a het nt 1XF (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.

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