# 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 $Y : C \to [C^{op},Set]$ of its domain category to a functor

$\tilde F : [C^{op}, Set] \to D \,.$

The Yoneda extension exhibits the presheaf category $PSh(C)$ as the free cocompletion of $C$.

## Definition

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

$\tilde F : [C^{op},Set] \to D$

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

$\tilde F := Lan_Y F \,.$

### Remarks

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

$\hat F := \tilde{Y \circ F} : [C^{op},Set] \to [D^{op}, Set] \,.$

This is in fact a left adjoint to the direct image or restriction functor $F_* : [D^{op}, Set] \to [C^{op}, Set]$ which maps $H \mapsto H \circ F$; see restriction and extension of sheaves.

### Formula

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

$(Lan F)(c') \simeq \colim_{(p(c) \to c') \in (p,c')} F(c)$

one finds for the Yoneda extension the formula

\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 $(Y,A) := (Y, const_A)$ whose objects are pairs $(U \in C, (Y(U) \to A) \in [C^{op}, Set] )$.

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

\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: $\tilde F \circ Y \simeq F$.

• The Yoneda extension commutes with small colimits in $C$ in that for $\alpha : A \to C$ a diagram, we have $\tilde F (colim (Y \circ \alpha)) \simeq colim F \circ \alpha$ .

• Moreover, $\tilde F$ is defined up to isomorphism by these two properties.

## Generalizations

Revised on April 16, 2014 03:18:09 by Adeel Khan (132.252.249.12)