nLab
Yoneda extension

Idea

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

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

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

Definition

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

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

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

$$\tilde{F}:={\mathrm{Lan}}_{Y}F.$$\tilde F := Lan_Y F \,.

Remarks

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

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

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

one finds for the Yoneda extension the formula

$$\begin{array}{rl}\tilde{F}(A)& :=(\mathrm{Lan}F)(A)\\ & \simeq {colim}_{(Y(U)\to A)\in (Y,A)}F(U)\end{array}.$$\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 $(Y,A):=(Y,{\mathrm{const}}_A)$ whose objects are pairs $(U\in C,(Y(U)\to A)\in [C^{\mathrm{op}},\mathrm{Set}])$.

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

$$\begin{array}{rl}\hat{F}(A)(V)& =({colim}_{(Y(U)\to A)\in (Y,A)}F(U))(V)\\ & \simeq {colim}_{(Y(U)\to A)\in (Y,A)}F(U)(V)\\ & \simeq {colim}_{(Y(U)\to A)\in (Y,A)}{\mathrm{Hom}}_D(V,F(U))\end{array}.$$\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 $\tilde{F}(\mathrm{colim}(Y\circ \alpha ))\simeq \mathrm{colim}F\circ \alpha$ .

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

Generalizations

• (infinity,1)-Yoneda extension