Yoneda Lemma

Summary

The Yoneda lemma states that for a locally small category $\mathcal{A}$, any functor $X:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, and any $A\in \mathcal{A}$, there is a natural bijection $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\cong X(A)$. Natural transformations from a representable functor are determined by a single element — what a representable “sees” in $X$ is exactly the data of $X$ at the representing object.

Overview

The Yoneda lemma is arguably the single most important result in category theory. It says that an object $A$ is completely determined by the functor it represents: to know $A$ is to know $H_A=\mathcal{A}(-,A)$. This has the immediate corollary that the Yoneda embedding $A\mapsto H_A$ is fully faithful — the category $\mathcal{A}$ embeds into its presheaf category without loss of information.

Main Content

Theorem 4.2.1: Yoneda Lemma

Let $\mathcal{A}$ be a locally small category, $X:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ a functor, and $A\in \mathcal{A}$. There is a bijection

$$[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\stackrel{\sim }{\to }X(A)$$ $$\alpha \mapsto {\alpha }_A(1_A)$$

that is natural in both $A$ and $X$.

The inverse sends $x\in X(A)$ to the natural transformation $\tilde{x}:H_A\Rightarrow X$ with components

$${\tilde{x}}_B(f)=(Xf)(x)\text{for }f:B\to A\in \mathcal{A}(B,A)$$

How to read the bijection: A natural transformation $\alpha :H_A\Rightarrow X$ is a coherent family of functions ${\alpha }_B:\mathcal{A}(B,A)\to X(B)$ for all $B$. The lemma says this entire infinite family is determined by just one piece of data: the single element ${\alpha }_A(1_A)\in X(A)$ obtained by evaluating the $A$-component on the identity $1_A$.

Proof sketch: Given $\alpha :H_A\Rightarrow X$ and $f:B\to A$, naturality forces:

$${\alpha }_B(f)={\alpha }_B(f^{\ast }(1_A))=(Xf)({\alpha }_A(1_A))$$

so $\alpha$ is completely determined by ${\alpha }_A(1_A)$. Conversely, given $x\in X(A)$, defining ${\tilde{x}}_B(f)=(Xf)(x)$ yields a natural transformation by functoriality of $X$.

Covariant Yoneda Lemma

The covariant version: for $X:\mathcal{A}\to \mathbf{Set}$ and $A\in \mathcal{A}$:

$$[\mathcal{A},\mathbf{Set}](H^A,X)\stackrel{\sim }{\to }X(A)$$ $$\alpha \mapsto {\alpha }_A(1_A)$$

where $H^A=\mathcal{A}(A,-)$.

Yoneda Lemma and Representability

Corollary 4.2.3: Yoneda and Representations

A functor $F:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ is representable by $A$ if and only if $F(A)$ contains a universal element $u\in F(A)$ such that for all $B$ and $x\in F(B)$, there is a unique $f:B\to A$ with $(Ff)(u)=x$.

The natural isomorphism $\alpha :H_A\stackrel{\sim }{\to }F$ corresponds to $u={\alpha }_A(1_A)$.

Example: Yoneda for the Product (BCT, Ch. 4.2)

The product $A\times B$ in a category $\mathcal{A}$ represents the functor $F(C)=\mathcal{A}(C,A)\times \mathcal{A}(C,B)$. The universal element is the pair of projections $(p_1:A\times B\to A,p_2:A\times B\to B)\in F(A\times B)$.

The Yoneda bijection recovers the universal property: every pair of maps $(f:C\to A,g:C\to B)$ corresponds to a unique map $⟨f,g⟩:C\to A\times B$.

Example: Yoneda Applied to Hom-Sets (BCT, Ch. 4.2)

Taking $X=H_B=\mathcal{A}(-,B)$, the Yoneda lemma gives:

$$[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,H_B)\cong H_B(A)=\mathcal{A}(A,B)$$

Natural transformations $H_A\Rightarrow H_B$ are in bijection with maps $A\to B$. This is the key computation for full faithfulness of the Yoneda embedding.

Naturality of the Yoneda Bijection

The bijection $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\cong X(A)$ is natural in both $A$ (contravariantly) and $X$ (covariantly). This naturality is essential for the consequences in Yoneda Embedding and Consequences.

Connections

• The Yoneda lemma is the foundation for the Yoneda embedding (Yoneda Embedding and Consequences).
• Limits via representables (Limits via Representables): the cone functor $\mathrm{Cone}(-,D):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ is a presheaf, and the limit represents it.
• Density theorem (Limits in Presheaf Categories): every presheaf is a colimit of representables.
• In Adjunctions: the natural bijection $\mathcal{B}(FA,B)\cong \mathcal{A}(A,GB)$ is a Yoneda-type statement about representability.

See Also

• Representable Functors — What the Yoneda lemma is about
• Yoneda Embedding and Consequences — Full faithfulness and uniqueness of representing objects
• Limits via Representables — Limits as universal elements
• Limits in Presheaf Categories — Density theorem