The Yoneda Embedding and Its Uses

The Yoneda lemma is a statement about one hom-set at a time: natural transformations $H_A\to X$ correspond to elements of $X(A)$. Its force appears when the correspondence is applied systematically — to the question of when $X$ is representable, to the functor $H_{\bullet }$ that sends each object to its representable presheaf, and to the question of when two objects are isomorphic. Each application is a short corollary, and the three together are among the most-used tools in the subject.¹

Throughout, $\mathcal{A}$ is locally small, $H_A=\mathcal{A}(-,A)$ is the representable presheaf at $A$, and an arrow decorated with $\sim$ denotes an isomorphism.

Representations as universal elements

A representation of a presheaf $X$ is an object $A$ with a natural isomorphism $\alpha :H_A\stackrel{\sim }{\to }X$. By Yoneda, $\alpha$ is determined by the element $u={\alpha }_A(1_A)\in X(A)$; the corollary below characterizes which elements $u$ arise from isomorphisms.

Such a $u$ is called a universal element of $X$; pairs $(B,x)$ with $x\in X(B)$ are called elements of the presheaf $X$, and $u$ is the element every other one factors through uniquely.

The shape for every $x$ there is a unique map making it work is the $\forall \dots \exists !$ pattern of a universal property. The corollary says representability is a universal property in disguise, and conversely: every universal property can be phrased as the representability of a suitable set-valued functor. The covariant dual reads the same way, with $X:\mathcal{A}\to \mathbf{Set}$, $u\in X(A)$, and the unique map going $\bar{x}:A\to B$.

Free vector spaces. Fix a set $S$ and consider $X=\mathbf{Set}(S,U(-)):{\mathbf{Vect}}_k\to \mathbf{Set}$, the functor of functions from $S$ into the underlying set. Two familiar statements about $X$:²

• Hom-set form. There is a vector space $F(S)$ with ${\mathbf{Vect}}_k(F(S),V)\cong \mathbf{Set}(S,U(V))$ naturally in $V$; that is, $X\cong H^{F(S)}$, so $X$ is representable.
• Element form. There is a vector space $F(S)$ and a function $u:S\to U(F(S))$ such that every function $f:S\to U(V)$ factors as $U(\bar{f})\circ u$ for a unique linear map $\bar{f}:F(S)\to V$. That is, $u\in X(F(S))$ is a universal element.

The first looks weaker: it asserts only an isomorphism of functors, while the second exhibits a specific insertion-of-generators map $u$ through which everything factors. The corollary says the two are equivalent: every natural isomorphism ${\mathbf{Vect}}_k(F(S),V)\cong \mathbf{Set}(S,U(V))$ arises from a universal element by $g\mapsto U(g)\circ u$, and the word natural in the hom-set form already encodes all the explicit detail of the element form.

Adjunction units. The same dictionary applies to any adjunction $F⊣G$. For fixed $A$, the functor $X=\mathcal{A}(A,G(-))$ is representable by $F(A)$, and its universal element is the unit component ${\eta }_A\in \mathcal{A}(A,G(F(A)))$ — equivalently, ${\eta }_A$ is an initial object of the comma category $(A\Rightarrow G)$. The three descriptions of an adjunction (hom-set bijection, unit/counit, universal arrows) are three phrasings of one representability statement; the details are in the adjunctions module.

Non-uniqueness of the isomorphism. Representing objects are unique up to isomorphism (below), but representations are not unique on the nose. The forgetful functor $U:\mathbf{Grp}\to \mathbf{Set}$ has universal element $1\in U(\mathbb{Z})$: for every group $G$ and $x\in G$ there is a unique homomorphism $\mathbb{Z}\to G$ with $1\mapsto x$. But $-1\in U(\mathbb{Z})$ is also universal, and the two induced isomorphisms $H^{\mathbb{Z}}\stackrel{\sim }{\to }U$ are different. Universal elements correspond one-to-one with representations, so $U$ has exactly as many representations by $\mathbb{Z}$ as $\mathbb{Z}$ has generators: two.

The Yoneda embedding is full and faithful

The second corollary applies the lemma with $X$ itself a representable.

Informally: a map of presheaves $H_A\to H_{A^{\prime }}$ is the same thing as a map $A\to A^{\prime }$ in $\mathcal{A}$.

A full and faithful functor deserves the name embedding: by fullness and faithfulness, $\mathcal{A}$ is equivalent to the full subcategory of $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ whose objects are the representable presheaves. Every category, however oddly presented, sits inside a category of set-valued functors — a category with excellent properties (all limits and colimits, computed pointwise) that $\mathcal{A}$ itself may lack.

Full subcategories are the well-behaved ones: for objects in a full subcategory, map and isomorphism mean the same whether computed inside or outside. The lemma that transports these notions is worth recording.

The practical reading of fullness, in Barr & Wells' phrasing: to construct an arrow $A\to A^{\prime }$, it is enough to construct a natural transformation $H_A\to H_{A^{\prime }}$ — that is, to give for each object $T$ a function $\mathcal{A}(T,A)\to \mathcal{A}(T,A^{\prime })$, a variable element of $A^{\prime }$ for each variable element of $A$, naturally in $T$. This is among the most used techniques in the subject: many arrows (diagonal maps, evaluation maps, canonical comparisons) are easiest to define on generalized elements, and Yoneda guarantees a unique actual arrow inducing the definition.³

Isomorphism of representables

The third corollary is the slogan an object is determined by the maps into it.

Functors preserve isomorphism automatically, so the content is the direction $H_A\cong H_{A^{\prime }}\Rightarrow A\cong A^{\prime }$: if $\mathcal{A}(B,A)\cong \mathcal{A}(B,A^{\prime })$ naturally in $B$, then $A\cong A^{\prime }$. Reading $\mathcal{A}(B,A)$ as $A$ viewed from $B$, two objects that look the same from every viewpoint, compatibly, are the same.

The naturality requirement does real work. In $\mathbf{Grp}$, isolated isomorphisms of hom-sets carry only partial information:

• $H_A(1)\cong H_{A^{\prime }}(1)$ always holds (both are one-element sets) and says nothing;
• $H_A(\mathbb{Z})\cong H_{A^{\prime }}(\mathbb{Z})$ says the underlying sets of $A$ and $A^{\prime }$ are isomorphic, but the group structures could still differ;
• $H_A(\mathbb{Z}/p\mathbb{Z})\cong H_{A^{\prime }}(\mathbb{Z}/p\mathbb{Z})$ for all primes $p$ says $A$ and $A^{\prime }$ have the same number of elements of each prime order.

None of these alone forces $A\cong A^{\prime }$; the corollary applies only when the isomorphisms hold for all $B$ and cohere naturally. The category $\mathbf{Set}$ is unusual in this respect: $A\cong \mathbf{Set}(1,A)=H^1(A)$, so one shape of generalized element (shape $1$) already determines a set. In a general category no single shape suffices, and the corollary compensates by quantifying over all of them.

Uniqueness of representing objects and of adjoints. If a functor $X$ is isomorphic to both $H^A$ and $H^{A^{\prime }}$, then $H^A\cong H^{A^{\prime }}$, so $A\cong A^{\prime }$: representing objects are unique up to isomorphism. This licenses the in definitions by universal property. The tensor product is a case in point: there is, up to isomorphism, at most one vector space $T$ with $\mathrm{Bilin}(U,V;W)\cong {\mathbf{Vect}}_k(T,W)$ naturally in $W$, so the tensor product $U\otimes V$ is well defined. Uniqueness of adjoints follows the same way: if $F$ and $F^{\prime }$ are both left adjoint to $G$, then

for each $A$, so $F(A)\cong F^{\prime }(A)$, and the isomorphisms are natural in $A$, giving $F\cong F^{\prime }$.

Cayley's theorem as a special case

The Yoneda embedding generalizes Cayley's theorem from group theory, a point Barr & Wells make explicit.⁴ Cayley's theorem: every group $G$ embeds in the symmetric group on its underlying set, via $g\mapsto (x\mapsto gx)$.

View $G$ as a one-object category. A set-valued functor on it is a $G$-set, and the single representable functor is the regular representation$\underline{G}$: the underlying set of $G$ acted on by multiplication. The Yoneda embedding sends the one object of $G$ to $\underline{G}$ and each arrow $g$ to the natural transformation multiply by $g$. Faithfulness says distinct group elements give distinct permutations of $\underline{G}$; functoriality says composition is preserved. That is precisely an injective homomorphism from $G$ into the group of permutations of its underlying set — Cayley's theorem, with Yoneda's fullness added for free: every $G$-equivariant automorphism of the regular representation is multiplication by a group element.

The one-object case displays the general mechanism in miniature. An arbitrary category has many objects, so instead of one regular representation there is one representable presheaf per object, and instead of a permutation group there is the presheaf category; but the embedding works for the same reason, with the identity arrow playing the role of the group identity in Cayley's proof.

Presheaves as generalized objects

The embedding invites a change of attitude: identify $A$ with $H_A$ and regard arbitrary presheaves as generalized objects of $\mathcal{A}$. A presheaf $X$ assigns to each object $B$ a set $X(B)$ of $B$-shaped figures, exactly as $H_A$ assigns the $B$-shaped generalized elements of $A$; the difference is only that $X$ need not be realized by an actual object. The presheaf category is then a completion of $\mathcal{A}$: it has all limits and colimits even when $\mathcal{A}$ has few, and $\mathcal{A}$ sits inside it fully faithfully. In a later module this is sharpened: every presheaf is a colimit of representables, roughly as every positive integer is a product of primes, making $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ the free cocompletion of $\mathcal{A}$.

Hom-functors record the maps out of and into each object; the Yoneda lemma says a natural map out of a representable is a single element; and its corollaries say the passage from objects to representables loses nothing, so an object may be studied, or even defined, by its maps in.