The Yoneda extension functor associated to $F$ is the left Kan extension
-
1.
The Homotopy Category Functor. Let
\[ \iota \colon \mathbb {\Delta }\hookrightarrow \mathsf{Cats} \]be the functor given by $[n]\to \mathbb {n}$. Then the Yoneda extension
\[ \operatorname {\mathrm{Lan}}_{{\text{よ}}}(\iota )\colon \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}}\to \mathsf{Cats} \]of $\iota $ is given by the homotopy category functor $\mathsf{Ho}$ of
, .
-
2.
The Geometric Realisation Functor. Let
\[ \iota \colon \mathbb {\Delta }\hookrightarrow \mathsf{Top} \]be the functor given by $[n]\to \left\lvert \Delta ^{n}\right\rvert $. Then the Yoneda extension
\[ \operatorname {\mathrm{Lan}}_{{\text{よ}}}(\iota )\colon \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}}\to \mathsf{Top} \]of $\iota $ is given by the geometric realisation functor $\left\lvert -\right\rvert $ of
, .
-
3.
The Path Simplicial Category 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 Yoneda extension
\[ \operatorname {\mathrm{Lan}}_{{\text{よ}}}(\iota )\colon \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}}\to \mathsf{sCats} \]of $\iota $ is given by the path simplicial category functor $\mathsf{Path}$ of
, .
-
4.
The Barycentric Subdivision 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 Yoneda extension
\[ \operatorname {\mathrm{Lan}}_{{\text{よ}}}(\mathrm{sd})\colon \underbrace{\mathsf{PSh}(\mathbb {\Delta })}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{sSets}}\to \mathsf{sSets} \]of $\mathrm{sd}$ is given by the barycentric subdivision functor $\mathrm{Sd}$ of
.
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1.
Functoriality. The assignment $F\mapsto \operatorname {\mathrm{Lan}}_{{\text{よ}}}(F)$ defines a functor
\[ \operatorname {\mathrm{Lan}}_{{\text{よ}}}\colon \mathsf{Fun}(\mathcal{C},\mathcal{D})\to \mathsf{Fun}(\mathsf{PSh}(\mathcal{C}),\mathcal{D}). \] -
2.
Adjointness. We have an adjunction1
witnessed by a bijection
\[ \operatorname {\mathrm{Hom}}_{\mathcal{D}}([\operatorname {\mathrm{Lan}}_{{\text{よ}}}(F)](\mathcal{F}),D)\cong \operatorname {\mathrm{Nat}}(\mathcal{F},{\text{よ}}_{F}(D)), \]natural in $\mathcal{F}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$ and $D\in \operatorname {\mathrm{Obj}}(\mathcal{D})$.
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3.
Interaction With the Yoneda Embedding. We have a natural isomorphism of functors
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4.
As a Coend. We have
\begin{align*} [\operatorname {\mathrm{Lan}}_{{\text{よ}}}(F)](\mathcal{F}) & \cong \int ^{A\in \mathcal{C}}\operatorname {\mathrm{Nat}}(h_{A},\mathcal{F})\odot F(A)\\ & \cong \int ^{A\in \mathcal{C}}\mathcal{F}(A)\odot F(A) \end{align*}for each $\mathcal{F}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$.
-
5.
Interaction With Tensors of Presheaves With Functors. We have a natural isomorphism
\[ \operatorname {\mathrm{Lan}}_{{\text{よ}}}(F)\cong (-)\odot _{\mathcal{C}}F, \]natural in $F\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(\mathcal{C},\mathcal{D}))$.
-
6.
Interaction With Finite Limits. Let $F\colon \mathcal{C}\to \mathsf{Sets}$ be a functor. The following conditions are equivalent:
- 1Applying Item 2 of Proposition 12.3.1.1.4, we see that this adjunction has the form $\operatorname {\mathrm{Lan}}_{{\text{よ}}}(F)\dashv \operatorname {\mathrm{Lan}}_{F}({\text{よ}})$.
12.3.2 The Yoneda Extension Functor
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor with $\mathcal{C}$ small and $\mathcal{D}$ cocomplete.
Here are some examples of Yoneda extensions.
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor with $\mathcal{C}$ small and $\mathcal{D}$ cocomplete.
Proof of Proposition 12.3.2.1.3.
Item 1: Functoriality
This follows from , of .
Item 2: Adjointness
Omitted.
Item 3: Interaction With the Yoneda Embedding
This follows from , of .
Item 4: As a Coend
This follows from , of and Theorem 12.1.5.1.1.
Item 5: Interaction With Tensors of Presheaves With Functors
This follows from Item 4.
Item 6: Interaction With Finite Limits
See Proposition 3.2.15 of [[Unknown Reference: coend-calculus]].
