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Representable Functors and the Yoneda Lemma

The capstone. An object is completely determined by the maps into it — the Yoneda lemma makes this exact. We compute the Yoneda bijection, see why the embedding is fully faithful, and meet the idea of representing a functor that pervades modern algebra and geometry.

Representable functors

Fix an object A in a locally small category C. The assignment h^A = Hom(A, −) : C → Set is a functor (it sends f : X → Y to post-composition f∘− : Hom(A, X) → Hom(A, Y)). A functor F : C → Set is [[representable-functor|representable]] if it is naturally isomorphic to some Hom(A, −); the object A is said to represent F. Representability is a universal property in disguise: A represents F precisely when F comes with a universal element.

The Yoneda lemma

Here is the theorem the whole track points toward. For any functor F : C → Set and any object A, the [[yoneda-lemma|Yoneda lemma]] gives a bijection, natural in A and F: Nat( Hom(A, −), F ) ≅ F(A). A natural transformation out of the representable Hom(A, −) is the same data as a single element of F(A). The proof is a one-line computation once you ask the right question.

Yoneda lemma:   Nat( Hom(A,-), F )  ~=  F(A).

Proof (both directions explicit):

  ->  Given a natural transformation eta : Hom(A,-) => F,
      look at its component at A:
            eta_A : Hom(A,A) -> F(A).
      Feed it the identity:   u := eta_A(id_A)  in F(A).
      Send eta |-> u.

  <-  Given u in F(A), DEFINE eta by, for f : A -> X,
            eta_X(f) := F(f)(u)  in F(X).
      Naturality of eta is forced by functoriality of F.

These are mutually inverse:
  start from eta, get u = eta_A(id_A), rebuild eta'_X(f)=F(f)(u);
  naturality square of eta at f : A -> X says
        eta_X( f o id_A ) = F(f)( eta_A(id_A) ),
  i.e. eta_X(f) = F(f)(u) = eta'_X(f).  So eta' = eta.  QED.

Corollary (Yoneda embedding).  Take F = Hom(B,-):
        Nat( Hom(A,-), Hom(B,-) )  ~=  Hom(B,A).
  So natural transformations between representables are exactly
  the morphisms B -> A.  The functor  A |-> Hom(A,-)  is
  FULLY FAITHFUL  (contravariantly).  An object is determined,
  up to unique iso, by the functor it represents.
The Yoneda lemma proved in full, plus the embedding corollary — the formal core of “an object is its maps.”

The Yoneda embedding corollary is the philosophical punchline. The map A ↦ Hom(−, A) embeds C fully faithfully into the category of functors Cᵒᵖ → Set. Fully faithful means: two objects with the same maps-into-them (naturally) are isomorphic, and isomorphisms of objects correspond exactly to natural isomorphisms of their representables. To know an object is to know all maps into it. This is why universal properties work — they pin down the representable functor, and Yoneda says that pins down the object.

What you can do now

  1. Recognize that products, coproducts, limits, colimits, tensor products, free objects, and quotients are all answers to mapping problems — instances of one idea.
  2. Use adjunctions as a search heuristic: when you build a free / forgetful / completion / localization, look for its adjoint and the limit/colimit it must preserve.
  3. Read “the functor of points” in algebraic geometry — a scheme is studied through Hom(−, X) — as Yoneda at work, and understand why representability is a theorem worth proving.