Adjoint Functors: The Standard Theory and the Heteromorphic Theory

1
Adjoint FunctorsThe 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 FX?A and
  GA ?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 1FxFx?Fx is the unit
  ?xx?GFx and transpose of 1GaGa?Ga is the counit
  ?aFGa?a.
• UMP of unit given any fx?Ga, there is a unique
  map gFx ?a (g adjoint transpose of f) such
  that Gg?xx ?GFx ?Ga fx ?Ga.
• Free Group Example Given any map fx?Ga from set
  x to underlying set Ga of a group, there is a
  unique group homomorphism gFx?a such that the
  underlying set map Gg factors f through the unit
  ?x.

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Universal Mapping Property of Counit

• Given any map gFx?a, there is a unique map fx
  ?Ga (f is adjoint transpose of g) such that ?aFf
  Fx ?FGa ?a gFx ?a.
• Example Given any group map gFx?a, there is a
  unique set map fx?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 otheronly 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.

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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).

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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
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Adjunctive Square Diagram

• 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.

• 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).

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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 fx?Ga as any map from set x to
  underlying set Ga of group a, take cx?a as same
  map point-wise but with codomain as the group,
  and take gFx?a as fs adjoint correlate. Take
  hxx?Fx as the usual embedding of a set into the
  free group Fx on the set and take eaGa?a as the
  map from the underlying set of a group back to
  the group.
• Then the diagram commutes!

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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
  HomXXop?X?Set and similarly for category A.
• Composition for heteromorphisms from X to A is
  rigorously defined using a bifunctor called a
  Het-bifunctor HetXop?A?Set.
• Given g'a ?a' in A, composition with hets like
  cx ?a is defined by Het(x,g')Het(x,a)
  ?Het(x,a').
• Given f'x' ?x in X, composition with hets like
  cx ?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).

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Representations of a functor

• Given a functor HC?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 FX?A
  and GA?X, suppose we are given only a bifunctor
  HetXop?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.

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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 FX?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 GA?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
  Functorswith some Thoughts on their
  Philosophical Significance. In What is Category
  Theory? Giandomenico Sica ed., Milan
  Polimetrica 127-183.

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Heteromorphic Theory of Adjoints

• Every bifunctor HetXop?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.,
• HetXop?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.

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Het Units and Counits

• In the isomorphisms
• HomA(Fx,a) ? Het(x,a) ? HomX(x,Ga)
• Adjoint correlates of 1Fx are the usual unit
  ?xx?GFx and the het or chimera unit hxx?Fx, and
• Adjoint correlates of 1GaGa?Ga are the usual
  counit ?aFGa?a and the het or chimera counit
  eaGa?a.

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

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Simpler UMPs for Het Units and Counits

• Given a het cx?a, there is a unique morphism
  gFx?a in A that factors c through the unit hx .
• Given a het cx?a, there is a unique morphism
  fx?Ga in X that factors c through the counit ea.

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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.

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Example Limits in Set (1)

• Let D be any (small) category taken as a
  diagram category and let DD?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 cdx?Ddd?D
  indexed by the objects in D such that for any
  morphism dd?d' in D, the following triangle
  commutes.
• A het from a set to a functor is usually called a
  cone.

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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 Dd 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 eDLimD?D.

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Example Limits in Set (3)

• The trivial left adjoint associates with set x
  the constant diagram functor ?xD?Set that maps
  each d to x and each map d to 1x. The chimera
  unit hxx??x is the cone of 1x's.

• Situation summarized by het adj. square.
• Many texts give het counit eDLimD?D as a
  universal cone and picture other cones cx?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.

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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.

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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.
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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, zcFx?Ga
  is just gcFx?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.

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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?xFx?FGFx?Fx 1FxFx?Fx
• G?a?GaGa?GFGa?Ga 1GaGa?Ga.
• Het triangular identities
• eFxzhxFx?GFx?Fx 1FxFx?Fx
• zeahGaGa?FGa?Ga 1GaGa?Ga.

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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.

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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.

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Het Natural Transformations

• Given functors FX?A and HX?B (different
  codomains) and a bifunctor HetAop?B?Set, a het
  natural transformation (rel. to Het) fF?H is
  given by a set of heteromorphisms fx ?
  Het(Fx,Hx) such that for any morphism jx?x' in
  X, the following diagram commutes.

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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 h1X?F
  is naturalbecause the het unit is a het nt.
  Ditto for the unit ?FG?1A and the het counit
  eG?1A.
• Any functor FX?A trivially defines a bifunctor
  HetXop?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.

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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.