Item 3 (02K2) — The Clowder Project

12.3.1 Foundations

let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

Definition 12.3.1.1.1 ▶ The Restricted Yoneda Embedding Associated to a Functor

The restricted Yoneda embedding associated to $F$ is the functor

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

defined as the composition

\[ \mathcal{D} \xhookrightarrow {{\text{よ}}_{\mathcal{D}}} \mathsf{PSh}(\mathcal{D}) \xrightarrow {F^{\mathsf{op},*}} \mathsf{PSh}(\mathcal{C}). \]

Remark 12.3.1.1.2 ▶ Unwinding Definition 12.3.1.1.1

In detail, the restricted Yoneda embedding associated to $F$ is the functor

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

where

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

\begin{align*} {\text{よ}}_{F}(A) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h_{A}\circ F^{\mathsf{op}}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h^{F(-)}_{A}. \end{align*}
•
Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{D})$, the action on morphisms

\[ {\text{よ}}_{F|A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{D}}(A,B)\to \operatorname {\mathrm{Nat}}(h^{F(-)}_{A},h^{F(-)}_{B}) \]

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

\begin{align*} {\text{よ}}_{F|A,B}(f) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h^{F(-)}_{f}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h_{f}\star \operatorname {\mathrm{id}}_{F^{\mathsf{op}}}\end{align*}

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

Example 12.3.1.1.3 ▶ Examples of Restricted Yoneda Embeddings

Here are some examples of restricted Yoneda embeddings.

1.
The Nerve Functor. Let

\[ \iota \colon \mathbb {\Delta }\hookrightarrow \mathsf{Cats} \]

be the functor given by $[n]\to \mathbb {n}$. Then the restricted Yoneda embedding

\[ {\text{よ}}_{\iota }\colon \mathsf{Cats}\to \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}} \]

of $\iota $ is given by the nerve functor $\mathrm{N}_{\bullet }$ of , .
2.
The Singular Simplicial Set Associated to a Topological Space. Let

\[ \iota \colon \mathbb {\Delta }\hookrightarrow \mathsf{Top} \]

be the functor given by $[n]\to \left\lvert \Delta ^{n}\right\rvert $. Then the restricted Yoneda embedding

\[ {\text{よ}}_{\iota }\colon \mathsf{Top}\to \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}} \]

of $\iota $ is given by the singular simplicial set functor $\mathrm{Sing}_{\bullet }$ of , .

3.
The Coherent Nerve Functor. Let

\[ \iota \colon \mathbb {\Delta }\hookrightarrow \mathsf{sCats} \]

be the functor given by $[n]\to \mathsf{Path}(\Delta ^{n})$, where $\mathsf{Path}(\Delta ^{n})$ is the simplicial category of , . Then the restricted Yoneda embedding

\[ {\text{よ}}_{\iota }\colon \mathsf{sCats}\to \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}} \]

of $\iota $ is given by the coherent nerve functor $\mathrm{N}^{\mathrm{hc}}_{\bullet }$ of , .

Kan’s $\mathrm{Ex}$ Functor. Let

\[ \mathrm{sd}\colon \mathbb {\Delta }\hookrightarrow \mathsf{sSets} \]

be the functor given by $[n]\to \mathrm{Sd}(\Delta ^{n})$, where $\mathrm{Sd}(\Delta ^{n})$ is the barycentric subdivision of $\Delta ^{n}$ of . Then the restricted Yoneda embedding

\[ {\text{よ}}_{\mathrm{sd}}\colon \mathsf{sSets}\to \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}} \]

of $\mathrm{sd}$ is given by Kan’s $\mathrm{Ex}$ functor of .

Proposition 12.3.1.1.4 ▶ Properties of the Restricted Yoneda Embedding

let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

1.
Interaction With Fully Faithfulness. The following conditions are equivalent:
1. (a)
  The restricted Yoneda embedding ${\text{よ}}_{F}$ is fully faithful.
2. (b)
  The functor $F$ is dense (, ).
2.
As a Left Kan Extension. We have a natural isomorphism of functors