# nLab Yoneda extension

Yoneda lemma

## In higher category theory

#### Limits and colimits

limits and colimits

# Contents

## Idea

The Yoneda extension of a functor $F:C\to D$ is extension along the Yoneda embedding $Y:C\to \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ of its domain category to a functor

$\stackrel{˜}{F}:\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\to D\phantom{\rule{thinmathspace}{0ex}}.$\tilde F : [C^{op}, Set] \to D \,.

The Yoneda extension exhibits the presheaf category $\mathrm{PSh}\left(C\right)$ as the free cocompletion of $C$.

## Definition

For $C$ a small category and $F:C\to D$ a functor, its Yoneda extension

$\stackrel{˜}{F}:\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\to D$\tilde F : [C^{op},Set] \to D

is the left Kan extension ${\mathrm{Lan}}_{Y}F:\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\to D$ of $F$ along the Yoneda embedding $Y$:

$\stackrel{˜}{F}:={\mathrm{Lan}}_{Y}F\phantom{\rule{thinmathspace}{0ex}}.$\tilde F := Lan_Y F \,.

### Remarks

Often it is of interest to Yoneda extend not $F:C\to D$ itself, but the composition $Y\circ F:C\to \left[{D}^{\mathrm{op}},\mathrm{Set}\right]$ to get a functor entirely between presheaf categories

$\stackrel{^}{F}:=\stackrel{˜}{Y\circ F}:\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\to \left[{D}^{\mathrm{op}},\mathrm{Set}\right]\phantom{\rule{thinmathspace}{0ex}}.$\hat F := \tilde{Y \circ F} : [C^{op},Set] \to [D^{op}, Set] \,.

### Formula

Recalling the general formula for the left Kan extension of a functor $F:C\to D$ through a functor $p:C\to C\prime$

$\left(\mathrm{Lan}F\right)\left(c\prime \right)\simeq {colim}_{\left(p\left(c\right)\to c\prime \right)\in \left(p,c\prime \right)}F\left(c\right)$(Lan F)(c') \simeq \colim_{(p(c) \to c') \in (p,c')} F(c)

one finds for the Yoneda extension the formula

$\begin{array}{rl}\stackrel{˜}{F}\left(A\right)& :=\left(\mathrm{Lan}F\right)\left(A\right)\\ & \simeq {colim}_{\left(Y\left(U\right)\to A\right)\in \left(Y,A\right)}F\left(U\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} \tilde F (A) & := (Lan F)(A) \\ & \simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U) \end{aligned} \,.

(Recall the notation for the comma category $\left(Y,A\right):=\left(Y,{\mathrm{const}}_{A}\right)$ whose objects are pairs $\left(U\in C,\left(Y\left(U\right)\to A\right)\in \left[{C}^{\mathrm{op}},\mathrm{Set}\right]\right)$.

For the full extension $\stackrel{^}{F}:\left[{D}^{\mathrm{op}},\mathrm{Set}\right]\to \left[{C}^{\mathrm{op}}.\mathrm{Set}\right]$ this yields

$\begin{array}{rl}\stackrel{^}{F}\left(A\right)\left(V\right)& =\left({colim}_{\left(Y\left(U\right)\to A\right)\in \left(Y,A\right)}F\left(U\right)\right)\left(V\right)\\ & \simeq {colim}_{\left(Y\left(U\right)\to A\right)\in \left(Y,A\right)}F\left(U\right)\left(V\right)\\ & \simeq {colim}_{\left(Y\left(U\right)\to A\right)\in \left(Y,A\right)}{\mathrm{Hom}}_{D}\left(V,F\left(U\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} \hat F(A)(V) &= (\colim_{(Y(U) \to A) \in (Y,A) } F(U))(V) \\ &\simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U)(V) \\ &\simeq \colim_{(Y(U) \to A) \in (Y,A) } Hom_{D}(V,F(U)) \end{aligned} \,.

Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the Yoneda lemma.

## Properties

• The restriction of the Yoneda extension to $C$ coincides with the original functor: $\stackrel{˜}{F}\circ Y\simeq F$.

• The Yoneda extension commutes with small colimits in $C$ in that for $\alpha :A\to C$ a diagram, we have $\stackrel{˜}{F}\left(\mathrm{colim}\left(Y\circ \alpha \right)\right)\simeq \mathrm{colim}F\circ \alpha$ .

• Moreover, $\stackrel{˜}{F}$ is defined up to isomorphism by these two properties.

## Generalizations

Revised on October 16, 2012 08:05:41 by Urs Schreiber (82.169.65.155)