nLab Yoneda extension

Idea

The Yoneda extension of a functor $F : C \to D$ is a universal extension (Kan extension) along the Yoneda embedding $Y \colon C \to [C^{op},Set]$ of its domain category to a functor

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

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

Definition

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

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

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

$$\tilde F \;\coloneqq\; Lan_Y F \,.$$

Remarks

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

$$\hat F \;\coloneqq\; \tilde{Y \circ F} \;\colon\; [C^{op},Set] \to [D^{op}, Set] \,.$$

This is in fact a left adjoint to the restriction functor $F^\ast \;\colon \; [D^{op}, Set] \to [C^{op}, Set]$ which maps $H \mapsto H \circ F$. This is relevant, for instance, to restriction and extension of sheaves.

Formula

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

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

one finds for the Yoneda extension the formula

$$\begin{aligned} \tilde F (A) & \;\coloneqq\; (Lan_Y F)(A) \\ & \simeq \colim_{(Y(U) \to A) \in (Y,A) } F(U) \end{aligned} \,.$$

(Recall the notation for the comma category $(Y,A) \;\coloneqq\; (Y, const_A)$ whose objects are pairs $(U \in C, (Y(U) \to A) \in [C^{op}, Set] )$.

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

• (infinity,1)-Yoneda extension