Yoneda Lemma

Yoneda Embedding

Given a locally small category $\mathcal{C}$, we have the category of presheaves on $\mathcal{C}$, written as $[\mathcal{C}^{\text{op}},\text{Set}]=\text{Hom}(\mathcal{C}^\text{op},\text{Set})$.

A functor, called Yoneda embedding $Y:\mathcal{C}\rightarrow \text{Hom}(\mathcal{C}^\text{op},\text{Set})$, is defined by $$ A \mapsto Y_A = \text{Hom}_\mathcal{C}(\cdot, A). $$

This requires that the category $\mathcal{C}$ is locally small, so the the functor $\text{Hom}_\mathcal{C}(\cdot,A)$ when applied to an object $B\in\mathcal{C}$ gives $\text{Hom}_\mathcal{C}(B,A)$, which is a set.

Yoneda Lemma

Statement

For $\mathcal{C}$ a locally small category, $A\in\mathcal{C}$ and $F\in [\mathcal{C}^\text{op},\text{Set}]$,

  1. There is a bijection $$ \theta_{F,A}:\text{Hom}_{[\mathcal{C}^\text{op},\text{Set}]}(Y_A,F) \rightarrow FA. $$
  2. This is natural in $A$., i.e., $\theta_{F,\cdot}: \text{Hom}_{[\mathcal{C}^\text{op},\text{Set}]}(Y_{(\cdot)},F) \Rightarrow F(\cdot)$ is a natural transformation.
  3. If $\mathcal{C}$ is small, this is natural in $F$, i.e., $\theta_{(\cdot,A)}$ is a natural isomorphism $\theta_{(\cdot,A)}:\text{Hom}_{[\mathcal{C}^\text{op},\text{Set}]}(Y_A,\cdot) \Rightarrow \text{ev}_A(\cdot)$, where $\text{ev}_A:[\mathcal{C}^{\text{op}},\text{Set}]\rightarrow \text{Set}$ is the evaluation functor.

Proof

We only prove the first claim since other claims can be verified straightforwardly.

Given a natural transformation $\alpha \in \text{Hom}_{[\mathcal{C}^\text{op},\text{Set}]}(Y_A,F)$, notice that its component $\alpha_A$ on $A$ takes $1_A$ to $\alpha_A (1_A) \in FA$. The commutative diagram coming from the naturality of $\alpha$ $$ \array{ Y_A A & \stackrel{\alpha_A}{\to} & FA & & & & 1_A & \mapsto & \alpha_A(1_A) & \\ _{Y_A f} \downarrow & & \downarrow _{Ff} & & & & \downarrow & & \downarrow _{Ff} & & \\ Y_A B & \underset{\alpha_B}{\to} & FB & & & & f & \mapsto & \alpha_B(f) & & } $$ shows that $\alpha_A(1_A)$ determines all other components, so the map is defined by $\theta_{F,A}(\alpha) = \alpha_A(1_A)$. This is just taking the component on $A$ and evaluate on $1_A$.

The inverse is defined by, for $a\in FA $, $a\mapsto \eta^a$. $\eta^a:Y_A \Rightarrow F$, with the component $\eta^a_f = Ffa$ where $f\in \text{Mor}(\mathcal{A})$. The naturality is easy to verify. Verification of bijection: $$ \theta_{F,A}\circ \eta (a) = \theta_{F,A} (\eta^a) = \eta_A^a(1_A) = F (1_A \circ 1_A) a = 1_{FA} a = a $$ where we identified $A$ with $1_A$, and $$ \eta\circ\theta_{F,A}(\alpha)=\eta(\alpha(1_A)) = F(\cdot)(\alpha(1_A)) $$ Now on a component $B$ this becomes, for $f:A\rightarrow B$, $$ \eta_f\circ\theta_{F,A}(\alpha_B)=\eta_f(\alpha_A(1_A)) = F(f)(\alpha_A(1_A))=\alpha_B Y_A f (1_A) = \alpha_B(f) $$ i.e. $$ \eta\circ\theta_{F,A}(\alpha_B) = \alpha_B. $$

Corollaries

  1. The Yoneda embedding is fully faithful.
  2. $Y_A \cong Y_B \Leftrightarrow A\cong B$