Subsection 12.1.4 (02HL): The Yoneda Embedding

The Yoneda embedding of $\mathcal{C}$¹ is the functor²

\[ {\text{よ}}_{\mathcal{C}}\colon \mathcal{C}\hookrightarrow \mathsf{PSh}(\mathcal{C}) \]

where

•
Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have

\[ {\text{よ}}_{\mathcal{C}}(A) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h_{A}. \]
•
Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms

\[ {\text{よ}}_{\mathcal{C}|A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)\to \operatorname {\mathrm{Nat}}(h_{A},h_{B}) \]

of ${\text{よ}}_{\mathcal{C}}$ at $(A,B)$ is given by

\[ {\text{よ}}_{\mathcal{C}|A,B}(f)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h_{f} \]

for each $f\in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, where $h_{f}$ is the representable natural transformation associated to $f$ of Definition 12.1.3.1.1.

¹Further Terminology: Also called the covariant Yoneda embedding to distinguish it from the contravariant Yoneda embedding of Theorem 12.2.5.1.1.
²Further Notation: Also written $h_{(-)}$, or simply ${\text{よ}}$.

The notation ${\text{よ}}$ for the Yoneda embedding was first introduced in [JS, ({O}p)lax Natural Transformations, Twisted Quantum Field Theories, and ``Even Higher'' {M}orita Categories]. The symbol ${\text{よ}}$ is the hiragana for yo, and comes from “Yoneda” in Nobuo Yoneda (米田信夫).

It is pronounced yo but without letting the “o” in yo sound like an o-u diphthong:

•
•
IPA transcription: [jo̞].

Let $\mathcal{C}$ be a category.

1.
Fully Faithfulness. The Yoneda embedding

\[ {\text{よ}}_{\mathcal{C}}\colon \mathcal{C}\hookrightarrow \mathsf{PSh}(\mathcal{C}) \]

is fully faithful.
2.
Preservation and Reflection of Isomorphisms. The Yoneda embedding

\[ {\text{よ}}_{\mathcal{C}}\colon \mathcal{C}\hookrightarrow \mathsf{PSh}(\mathcal{C}) \]

preserves and reflects isomorphisms, i.e. given $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the following conditions are equivalent:
1. (a)
  We have $A\cong B$.
2. (b)
  We have $h_{A}\cong h_{B}$.
3.
Density. The Yoneda embedding

\[ {\text{よ}}_{\mathcal{C}}\colon \mathcal{C}\hookrightarrow \mathsf{PSh}(\mathcal{C}) \]

is dense.
4.
Interaction With Density Comonads. We have
5.
Interaction With Codensity Monads. We have

\[ \operatorname {\mathrm{Ran}}_{{\text{よ}}}({\text{よ}})\cong \mathsf{Spec}\circ \mathsf{O}, \]

where $\mathsf{Spec}$ and $\mathsf{O}$ are the functors of .

Item 1: Fully Faithfulness

Let $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$. Applying the Yoneda lemma (Theorem 12.1.5.1.1) to the functor $h_{B}$ (i.e. in the case $\mathcal{F}=h_{B}$), we have

\[ \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)\cong \operatorname {\mathrm{Nat}}(h_{A},h_{B}), \]

and the natural isomorphism

\[ \xi _{A,B}\colon \mathcal{h}_{B}(A)\Rightarrow \operatorname {\mathrm{Nat}}(h_{A},h_{B}) \]

witnessing this bijection is given by

\begin{align*} \xi _{A,B}(g)_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h^{X}_{g}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g_{*} \end{align*}

for each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$ and each $g\in h^{X}_{B}$, i.e. we have $\xi _{A,B}={\text{よ}}_{\mathcal{C}|A,B}$. Thus ${\text{よ}}_{\mathcal{C}}$ is fully faithful.

Item 2: Preservation and Reflection of Isomorphisms

This follows from Chapter 11: Categories, Item 1 of Proposition 11.5.1.1.6 and Item 3 of Proposition 11.6.3.1.2.

Item 3: Density

Omitted.

Item 4: Interaction With Density Comonads

Omitted.

Item 5: Interaction With Codensity Monads

Omitted.

The Clowder Project

12.1.4 The Yoneda Embedding