Yoneda extension


Yoneda lemma

Limits and colimits



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

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

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


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

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

is the left Kan extension Lan YF:[C op,Set]DLan_Y F : [C^{op}, Set] \to D of FF along the Yoneda embedding YY:

F˜:=Lan YF. \tilde F := Lan_Y F \,.


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

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

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


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

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

one finds for the Yoneda extension the formula

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

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

F^(A)(V) =(colim (Y(U)A)(Y,A)F(U))(V) colim (Y(U)A)(Y,A)F(U)(V) colim (Y(U)A)(Y,A)Hom D(V,F(U)). \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.


  • The restriction of the Yoneda extension to CC coincides with the original functor: F˜YF \tilde F \circ Y \simeq F .

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

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


Revised on December 16, 2017 05:09:54 by Mike Shulman (