Yoneda Embedding and Consequences

Summary

The Yoneda embedding $H^{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$, $A\mapsto H_A=\mathcal{A}(-,A)$, is fully faithful: every category embeds into its presheaf category without loss of information. Consequently, representing objects are unique up to isomorphism, and the Yoneda embedding preserves limits.

Overview

The Yoneda lemma has immediate powerful consequences. The most fundamental is that the assignment $A\mapsto H_A$ is a full and faithful functor — the category $\mathcal{A}$ sits inside the much larger presheaf category $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ as a full subcategory. This is the categorical analogue of Cayley’s theorem (every group embeds into a symmetric group).

Main Content

The Yoneda Embedding

Definition: Yoneda Embedding

For a locally small category $\mathcal{A}$, the Yoneda embedding is the functor

$$H^{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}],A\mapsto H_A=\mathcal{A}(-,A)$$

On morphisms: for $f:A\to B$, the natural transformation $H^{\bullet }(f)=f_{\ast }:H_A\Rightarrow H_B$ has components $(f_{\ast }{)}_C:\mathcal{A}(C,A)\to \mathcal{A}(C,B)$, $g\mapsto f\circ g$ (postcomposition by $f$).

Full Faithfulness

Corollary 4.3.7: Yoneda Embedding is Fully Faithful (BCT, Ch. 4.3)

The Yoneda embedding $H^{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is fully faithful: for all $A,B\in \mathcal{A}$,

$$\mathcal{A}(A,B)\stackrel{\sim }{\to }[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,H_B)$$ $$f\mapsto f_{\ast }$$

is a bijection (natural in $A$ and $B$).

Proof: Apply the Yoneda lemma with $X=H_B$: $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,H_B)\cong H_B(A)=\mathcal{A}(A,B)$.

Interpretation: An object $A\in \mathcal{A}$ is completely determined (up to isomorphism) by knowing all the sets $\mathcal{A}(C,A)$ and how they transform. “To know an object is to know all maps into it.”

Uniqueness of Representations

Corollary 4.3.10: Representing Objects Are Unique (BCT, Ch. 4.3)

If $A,A^{\prime }\in \mathcal{A}$ both represent the same functor $F:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ (i.e., $H_A\cong F\cong H_{A^{\prime }}$), then $A\cong A^{\prime }$ in $\mathcal{A}$.

More precisely: if $H_A\cong H_{A^{\prime }}$ as functors ${\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, then $A\cong A^{\prime }$ in $\mathcal{A}$.

Proof: Since $H^{\bullet }$ is faithful, $H_A\cong H_{A^{\prime }}$ implies $A\cong A^{\prime }$.

Consequence: All universal properties define their objects uniquely up to isomorphism. Products, limits, free objects, tensor products — they are all representing objects and hence unique up to unique isomorphism (when the isomorphism is required to be compatible with the universal element).

Key Applications of Uniqueness

Example: Adjoints Are Unique (BCT, Ex. 4.3.13)

If $F⊣G$ and $F⊣G^{\prime }$, then $G\cong G^{\prime }$. Proof: for each $B$, both $GB$ and $G^{\prime }B$ represent the functor $\mathcal{B}(F-,B):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$. By uniqueness of representations, $GB\cong G^{\prime }B$ naturally.

Example: Tensor Product Is Unique (BCT, Ex. 4.3.14)

The tensor product $M{\otimes }_{R}N$ (if it exists) represents $\mathrm{Bilin}(M,N;-)$. Any two objects with this universal property are isomorphic.

The Yoneda Embedding Preserves Limits

Proposition 6.2.2: Representables Preserve Limits (BCT, Ch. 6.2)

For any $A\in \mathcal{A}$ and diagram $D:\mathcal{I}\to \mathcal{A}$:

$$\mathcal{A}(A,\lim D)\cong \underset{I\in \mathcal{I}}{\lim }\mathcal{A}(A,DI)$$

i.e., $H^A=\mathcal{A}(A,-)$ sends limits in $\mathcal{A}$ to limits in Set.

Equivalently: the Yoneda embedding $H^{\bullet }$ preserves limits.

This follows from the Yoneda lemma and the fact that cones into $D$ with vertex $A$ correspond to cones into $H^A\circ D$ with vertex $\{\ast \}$.

The Contravariant Yoneda Embedding

Dually, the contravariant Yoneda embedding

$$H_{\bullet }:{\mathcal{A}}^{\mathrm{op}}\to [\mathcal{A},\mathbf{Set}],A\mapsto H^A=\mathcal{A}(A,-)$$

is also fully faithful, and preserves limits (= colimits in $\mathcal{A}$).

Connections

• Density theorem (Limits in Presheaf Categories): Every presheaf $X\in [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is a colimit of representables — the Yoneda embedding is dense.
• Limits in presheaf categories are computed pointwise, and the Yoneda embedding is limit-preserving (Limits in Presheaf Categories).
• The cartesian closed structure of $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is related to the universal properties of representables (Cartesian Closed Categories).

See Also

• Yoneda Lemma — The theorem this builds on
• Representable Functors — What is being embedded
• Limits in Presheaf Categories — Density theorem and pointwise limits
• Adjoints and Limits — Representables preserve limits