A Representation is a Universal Element
The Yoneda lemma is fundamental, and its reach is best seen through its consequences. Three of them
organize the present stage, and they are the three readings of the lemma identified at the close of
the previous one: a representation is a single universal element, the embedding of a category into
its presheaves loses nothing, and an object is pinned down by the maps it receives. We take them in
that order, beginning with the one that turns the abstract definition of a representation into a
concrete object one can point to.
Recall that a
representation
of a presheaf \(X\) is an object \(A\) together with a natural isomorphism
\(\alpha : H_A \xrightarrow{\sim} X\). Such a natural isomorphism is, by the
Yoneda lemma,
the same data as a single element of \(X(A)\) — the element \(\alpha_A(1_A)\) to which the
isomorphism corresponds. The question is which elements arise this way, and the answer names the
elements that deserve to be called universal.
Corollary: Representations are Universal Elements
Let \(\mathscr{A}\) be a
locally small
category and \(X : \mathscr{A}^{\mathrm{op}} \to \mathbf{Set}\) a presheaf. To give a
representation of \(X\) is to give an object \(A \in \mathscr{A}\) together with an element
\(u \in X(A)\) with the following property:
\[
\text{for each } B \in \mathscr{A} \text{ and each } x \in X(B), \text{ there is a unique map }
\bar{x} : B \to A \text{ with } \big(X(\bar{x})\big)(u) = x .
\]
An element \(u\) with this property is called a universal element of \(X\).
Proof.
By the Yoneda lemma, an element \(u \in X(A)\) corresponds to a natural transformation
\(\widetilde{u} : H_A \to X\), and a representation of \(X\) is exactly a choice of \(A\) for
which the corresponding \(\widetilde{u}\) is a
natural isomorphism.
A natural transformation is a natural isomorphism precisely when each of its components is a
bijection, so \(\widetilde{u}\) is an isomorphism if and only if, for every \(B \in \mathscr{A}\),
the component
\[
\widetilde{u}_B : H_A(B) = \mathscr{A}(B, A) \to X(B)
\]
is a bijection. By construction of the backward map in the Yoneda lemma,
\(\widetilde{u}_B(\bar{x}) = \big(X(\bar{x})\big)(u)\) for a map \(\bar{x} : B \to A\). The
assertion that this component is a bijection is exactly the assertion that for each
\(x \in X(B)\) there is a unique \(\bar{x} : B \to A\) with \(\big(X(\bar{x})\big)(u) = x\), which
is the displayed property. Requiring it for all \(B\) is requiring \(\widetilde{u}\) to be an
isomorphism, hence requiring \((A, u)\) to be a representation.
The pairs \((B, x)\) with \(B \in \mathscr{A}\) and \(x \in X(B)\) are, in the light of the previous
stage, the elements of the presheaf \(X\): such an \(x\) is a
generalized element
of \(X\) of shape \(H_B\). The corollary says that a representation singles out one element \(u\),
sitting at the representing object, through which every other factors uniquely. This is the meaning
that the word universal has carried throughout: a universal element is one from which all
others are reached in exactly one way.
The corollary was stated for a presheaf, a
contravariant
functor, where the universal element sits at \(A\) and every other element factors through a map
into \(A\). Every example below is instead a covariant functor \(X : \mathscr{A} \to
\mathbf{Set}\), for which the statement dualizes: a representation is an object \(A\) and an element
\(u \in X(A)\) such that for each \(B\) and each \(x \in X(B)\) there is a unique map \(\bar{x} : A
\to B\) — an arrow now out of the representing object — with \(\big(X(\bar{x})\big)(u) = x\).
The only change is the direction of the factoring map, forced by the change of variance; the content
is identical, and the familiar constructions of algebra take this covariant form.
The free-forgetful case in two voices
A single example shows the corollary doing its characteristic work, which is to reveal that a
statement one already knew in an explicit form is the same as a statement one knew in a slicker form.
For a fixed set \(S\), consider the functor sending a vector space \(V\) over a field \(k\) to the set
\(\mathbf{Set}\big(S, U(V)\big)\) of functions from \(S\) into the underlying set of \(V\). There are
two familiar and true statements about it.
The first says that there exist a vector space \(F(S)\) and an isomorphism
\[
\mathbf{Vect}_k\big(F(S), V\big) \;\cong\; \mathbf{Set}\big(S, U(V)\big) ,
\]
natural in \(V\) — the
free–forgetful adjunction.
The second says that there exist a vector space \(F(S)\) and a function
\(u : S \to U\big(F(S)\big)\) such that every function \(f : S \to U(V)\) factors as \(U(\bar{f})
\circ u\) for a unique linear map \(\bar{f} : F(S) \to V\) — the universal property of the basis
inclusion. The first is the compact statement that the functor is
representable;
the second is the explicit statement that \(u\) is a universal element.
The second looks at first to assert more than the first, for it specifies not merely that the two
functors are isomorphic but that the isomorphism arises in a particular way, through composition with
the single function \(u\). The corollary dispels the appearance: representations and universal
elements are in bijection, so every natural isomorphism of the first kind comes from a universal
element of the second kind, and nothing is hidden by the word natural except the explicit
element it conceals. The slick statement and the hands-on statement are one.
Universal elements of an adjunction and of a forgetful functor
The pattern is not confined to vector spaces. For any
adjunction
\(F \dashv G\) with \(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\), and
any fixed \(A \in \mathscr{A}\), the functor \(\mathscr{A}(A, G(-))\) is
representable,
represented by \(F(A)\). Read through the present corollary, the universal element of this
representation is the
unit
\(\eta_A : A \to G(F(A))\), and the universal property it satisfies is precisely the statement that
\(\eta_A\) is an initial object of the relevant comma category — the same fact met earlier as a
reformulation of adjointness, now seen again from the side of representability.
The smallest instance is the most familiar. For the forgetful functor \(U : \mathbf{Grp} \to
\mathbf{Set}\), the element \(1 \in U(\mathbb{Z})\) is a universal element: every group element
\(x \in U(G)\) is the image of \(1\) under a unique homomorphism \(\phi : \mathbb{Z} \to G\), because
a homomorphism out of the
infinite cyclic group
is fixed freely by where it sends the generator. So \(\mathbb{Z}\) represents the forgetful functor,
with universal element \(1\). The choice of \(1\) was not forced — the element \(-1\) generates
\(\mathbb{Z}\) just as well and furnishes a second universal element, hence a second representation.
That these two are genuinely different representations, not the same one described twice, is
guaranteed by the corollary's bijection between universal elements and representations; we return to
count them exactly once the embedding theorem is in hand.
The Yoneda Embedding is Full and Faithful
The second consequence redeems the promise of the previous stage. The
Yoneda embedding
sends each object \(A\) to the presheaf \(H_A\) and each map to post-composition with it. The word
embedding was used in anticipation; we now justify it by showing that the functor is
full and faithful,
so that a map between two representables is the same thing as a map between the objects they
represent.
Corollary: The Yoneda Embedding is Full and Faithful
For any locally small category \(\mathscr{A}\), the Yoneda embedding
\[
H_\bullet : \mathscr{A} \to [\mathscr{A}^{\mathrm{op}}, \mathbf{Set}]
\]
is full and faithful. Informally: for objects \(A, A' \in \mathscr{A}\), a natural
transformation \(H_A \to H_{A'}\) of presheaves is the same thing as a map \(A \to A'\) in
\(\mathscr{A}\).
Proof.
Full and faithful means that for each pair \(A, A'\) the function
\[
\mathscr{A}(A, A') \to [\mathscr{A}^{\mathrm{op}}, \mathbf{Set}](H_A, H_{A'}),
\qquad f \mapsto H_f ,
\]
sending a map to the natural transformation it induces, is a bijection. Apply the
Yoneda lemma
with the presheaf \(X\) taken to be \(H_{A'}\). It supplies a bijection
\[
[\mathscr{A}^{\mathrm{op}}, \mathbf{Set}](H_A, H_{A'}) \;\cong\; H_{A'}(A) = \mathscr{A}(A, A') ,
\]
whose forward direction sends a natural transformation \(\alpha\) to \(\widehat{\alpha} =
\alpha_A(1_A)\). It is therefore enough to check that this Yoneda bijection and the function
\(f \mapsto H_f\) are mutually inverse, for which it suffices to compute one composite. Given
\(f : A \to A'\), the transformation \(H_f\) is sent by the Yoneda bijection to
\[
\widehat{H_f} = (H_f)_A(1_A) = f \circ 1_A = f ,
\]
since the component of \(H_f\) at \(A\) is post-composition by \(f\). Thus the Yoneda bijection
followed by \(f \mapsto H_f\) returns \(f\); as the Yoneda map is already a bijection, the
function \(f \mapsto H_f\) is its inverse and hence a bijection itself. The embedding is full and
faithful.
The result was previewed at the end of the previous stage as the exercise of proving directly that
isomorphic representables force isomorphic objects; that exercise is now subsumed, its content
flowing from the lemma without further labour. The phrase full and faithful functor earns the
name embedding: such a functor makes its domain equivalent to a full subcategory of its
codomain, so \(\mathscr{A}\) is realized as the full subcategory of \([\mathscr{A}^{\mathrm{op}},
\mathbf{Set}]\) whose objects are the representables. An arbitrary category sits inside its presheaf
category as the representable presheaves, with all and only the maps it already had.
What a full and faithful functor preserves
Embeddings of this kind transport isomorphisms faithfully in both directions, a fact used repeatedly
below and worth isolating. We prove it in full rather than defer it, since the consequences that
follow depend on it.
Lemma: Full and Faithful Functors Reflect Isomorphism
Let \(J : \mathscr{A} \to \mathscr{B}\) be full and faithful, and let \(A, A' \in \mathscr{A}\).
Then:
(a) a map \(f\) in \(\mathscr{A}\) is an
isomorphism
if and only if \(J(f)\) is an isomorphism in \(\mathscr{B}\);
(b) for any isomorphism \(g : J(A) \to J(A')\) in \(\mathscr{B}\), there is a
unique isomorphism \(f : A \to A'\) in \(\mathscr{A}\) with \(J(f) = g\);
(c) the objects \(A\) and \(A'\) are isomorphic in \(\mathscr{A}\) if and only if
\(J(A)\) and \(J(A')\) are isomorphic in \(\mathscr{B}\).
Proof.
(a)
A functor preserves identities and composites, so if \(f\) has a two-sided inverse \(f^{-1}\)
then \(J(f)\) has the two-sided inverse \(J(f^{-1})\); this direction needs no hypothesis on
\(J\). Conversely, suppose \(J(f) : J(A) \to J(A')\) is an isomorphism, with inverse
\(h : J(A') \to J(A)\). Since \(J\) is full, \(h = J(k)\) for some \(k : A' \to A\). Then
\(J(k \circ f) = J(k) \circ J(f) = h \circ J(f) = 1_{J(A)} = J(1_A)\), and \(J\) faithful forces
\(k \circ f = 1_A\); symmetrically \(f \circ k = 1_{A'}\). So \(f\) is an isomorphism with
inverse \(k\).
(b)
Given an isomorphism \(g : J(A) \to J(A')\), fullness provides \(f : A \to A'\) with
\(J(f) = g\), and by part (a) this \(f\) is an isomorphism. If \(J(f) = J(f') = g\), then
faithfulness gives \(f = f'\), so \(f\) is unique.
(c)
If \(A \cong A'\) in \(\mathscr{A}\) then applying \(J\) to an isomorphism gives one between
\(J(A)\) and \(J(A')\), by the easy direction of (a). Conversely an isomorphism
\(J(A) \to J(A')\) is lifted to an isomorphism \(A \to A'\) by (b). Hence \(A \cong A'\) if and
only if \(J(A) \cong J(A')\).
Counting the representations of the forgetful functor
The deferred count can now be settled. The forgetful functor \(U : \mathbf{Grp} \to \mathbf{Set}\) is
represented by \(\mathbb{Z}\), so the
covariant hom-functor
\(H^{\mathbb{Z}}\) is naturally isomorphic to \(U\). How many such natural isomorphisms
\(H^{\mathbb{Z}} \xrightarrow{\sim} U\) are there? Since \(H^{\mathbb{Z}} \cong U\), they are as
numerous as the natural isomorphisms \(H^{\mathbb{Z}} \xrightarrow{\sim} H^{\mathbb{Z}}\), which by
the dual form of the embedding theorem are as numerous as the group isomorphisms
\(\mathbb{Z} \xrightarrow{\sim} \mathbb{Z}\). The integers have exactly two automorphisms as a group,
the identity and negation, corresponding to the two generators \(\pm 1\). There are therefore exactly
two natural isomorphisms \(H^{\mathbb{Z}} \xrightarrow{\sim} U\), hence — by the bijection between
representations and universal elements — exactly two universal elements of \(U(\mathbb{Z})\), namely
\(1\) and \(-1\). The two representations found earlier are all of them.
Objects Known by Their Representables
The third consequence turns the embedding into a principle of identity. If the passage from an object
to its representable loses nothing, then objects with the same representable must be the same object.
The precise statement collects both variances at once.
Corollary: Isomorphism of Representables
Let \(\mathscr{A}\) be a locally small category and \(A, A' \in \mathscr{A}\). Then
\[
H_A \cong H_{A'} \iff A \cong A' \iff H^A \cong H^{A'} ,
\]
where \(H_A, H_{A'}\) are the
contravariant
representables and \(H^A, H^{A'}\) the
covariant
ones.
Proof.
By duality the two stated equivalences are mirror images, so it is enough to prove the first,
\(H_A \cong H_{A'} \iff A \cong A'\). The Yoneda embedding \(H_\bullet\) was just shown to be full
and faithful, and a full and faithful functor reflects and preserves isomorphism of objects, as
established in the lemma above: \(A \cong A'\) if and only if \(H_\bullet(A) \cong H_\bullet(A')\),
that is, if and only if \(H_A \cong H_{A'}\). This is exactly the asserted equivalence.
One direction is the easy half and deserves separate mention, since it holds for any functor at all:
isomorphisms are preserved by every functor, so \(A \cong A'\) always implies \(H_A \cong H_{A'}\).
The force of the corollary is the converse — that isomorphic representables force the objects
themselves to be isomorphic. Writing \(\mathscr{A}(B, A)\) as \(A\) viewed from \(B\), the
converse says that two objects agreeing in every view are isomorphic: if a thing looks the same from
every vantage point, it is the same up to isomorphism. It is the categorical form of the maxim that a
creature looking, walking, and sounding like a duck is one — with the understanding that
being a duck means, here, being isomorphic to one, not being literally identical to it.
Categorically an object is known only up to isomorphism, and the corollary respects that: it concludes
\(A \cong A'\), never \(A = A'\).
How much a single view reveals
The corollary requires agreement in all views, taken compatibly — naturally in \(B\). No
single view suffices in general, and the category of groups shows how partial the information from
one vantage point can be. Suppose two groups \(A\) and \(A'\) satisfy \(H_A(B) \cong H_{A'}(B)\) for
some particular \(B\), meaning they admit the same number of homomorphisms from \(B\).
Viewed from the trivial group, \(H_A(1) = \mathbf{Grp}(1, A)\) is a one-element set for every \(A\),
so agreement there says nothing whatever. Viewed from the
infinite cyclic group,
\(H_A(\mathbb{Z})\) is the underlying set of \(A\), so agreement says only that \(A\) and \(A'\) have
the same number of elements — nothing about how they multiply. Viewed from a cyclic group of prime
order, \(H_A(\mathbb{Z}/p\mathbb{Z})\) counts the elements of \(A\) whose order divides \(p\), so
agreement for every prime says \(A\) and \(A'\) have the same number of elements of each prime order
— a real constraint, but still far short of isomorphism. Each view contributes a fragment; only the
demand that all of them agree, and agree naturally, assembles the fragments into the conclusion
\(A \cong A'\).
The exceptional simplicity of sets
Against this background the category of sets is strikingly unusual. For any set \(A\),
\[
A \cong \mathbf{Set}(1, A) = H_A(1) ,
\]
so the single view from the one-element set already recovers the whole set up to isomorphism. Two
sets that look the same from the one-point set are isomorphic — in bijection — because the only thing
a set carries, categorically, is the supply of maps into it from a point, and these are exactly its
elements. For a general category the corollary insists on all views; for sets, one view — the
generalized elements
of shape \(1\) — is enough. The contrast measures how much structure an object can hide from any
fixed probe: in \(\mathbf{Set}\), none; in \(\mathbf{Grp}\), a great deal, recoverable only by probing
from every shape at once.
Uniqueness from Universal Properties
The three consequences combine into a tool used constantly in practice: an object defined by a
universal property is determined up to isomorphism, and that isomorphism is itself canonical. The
reasoning is always the same. A universal property presents an object as the representing object of
some functor; isomorphic representables force isomorphic objects; so any two solutions of the same
universal property are isomorphic. We draw out two instances, both of which settle a uniqueness that
had earlier been asserted but not proved.
Left adjoints are unique
Suppose a functor \(G : \mathscr{B} \to \mathscr{A}\) has two left adjoints, \(F\) and \(F'\). For
each object \(A \in \mathscr{A}\), the two
adjunctions
give natural isomorphisms
\[
\mathscr{B}(F(A), B) \;\cong\; \mathscr{A}(A, G(B)) \;\cong\; \mathscr{B}(F'(A), B) ,
\]
natural in \(B\). The outer two functors are the
covariant representables
\(H^{F(A)}\) and \(H^{F'(A)}\), so the composite is a natural isomorphism
\(H^{F(A)} \cong H^{F'(A)}\). By the isomorphism-of-representables corollary, \(F(A) \cong F'(A)\).
The isomorphism is moreover natural in \(A\), so the two left adjoints are isomorphic as functors,
\(F \cong F'\). A left adjoint, when it exists, is unique up to natural isomorphism — the uniqueness
asserted when adjunctions were first introduced, now proved. Dually, right adjoints are unique.
The tensor product, identified at last
The second instance returns to a construction met two stages ago and supplies the uniqueness its
definition relied on. For vector spaces \(U\) and \(V\) over a field \(k\), the functor sending a
space \(W\) to the set of bilinear maps \(U \times V \to W\) is
representable:
there is a space \(T\) and a natural isomorphism
\[
\{\text{bilinear } U \times V \to W\} \;\cong\; \mathbf{Vect}_k(T, W), \qquad \text{natural in } W .
\]
When this construction was first encountered, the representing object was exhibited as the
tensor product
\(U \otimes V\), and its defining property was exactly this natural isomorphism. What could not be
said at that point is that the property pins the object down. The corollary now supplies it: any two
spaces representing the same functor are isomorphic, so up to isomorphism there is at most one space
\(T\) with
\[
\{\text{bilinear } U \times V \to W\} \;\cong\; \mathbf{Vect}_k(T, W), \qquad \text{natural in } W .
\]
Since such a space does exist, it is legitimate to speak of the tensor product of \(U\) and
\(V\): the definite article is earned by the uniqueness, and the uniqueness is an instance of the
Yoneda principle. The functor determines its representing object, if one exists, and the tensor
product is what that object is called.
Why a universal property is a specification
The pattern licenses a habit of definition that runs throughout modern mathematics and its
computational descendants. To define an object by a universal property is to specify the functor
it represents rather than to build the object by hand; the corollary guarantees that the
specification has at most one solution up to canonical isomorphism, so the construction details
may be forgotten once existence is known. A product, a quotient, a free object, a limit, a tensor
product, an optimal solution selected by an extremal condition — each is named by what it is
universal for, and each is interchangeable with any other object satisfying the same universal
property. In settings where objects are built compositionally, this is what allows a component to
be replaced by any implementation meeting the same interface without disturbing what surrounds
it: the universal property is the interface, and the Yoneda principle is the guarantee that the
interface determines the component up to the only kind of sameness that matters. Identity of
behaviour under all maps is identity of object.
A last observation points past the present stage. Every representable presheaf is one object seen
through its incoming maps, and the embedding shows these representables sitting inside the presheaf
category as a faithful copy of \(\mathscr{A}\). The presheaves that are not representable are the new
objects the enlargement adds, and a later development will show that every presheaf, representable or
not, is assembled from representables — built up from them in much the way that every positive integer
is built from primes. The representables are the atoms; the Yoneda lemma is the statement that each
atom is faithfully named by the object it represents. With that, the consequences of the lemma are in
hand, and the foundations laid across this stage are ready to carry the constructions that follow.