Lemma of Yoneda

The lemma of Yoneda , according to Nobuo Yoneda , is a mathematical statement from the subfield of category theory . It describes the set of natural transformations between a Hom functor and another functor .

The Yoneda lemma allows terms that are familiar from the category of sets to be transferred to any category.

motivation

Let there be a locally small category, Set the category of sets and a functor. For each object of the category one has the partial Hom functor , which is defined for objects and morphisms as follows: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle T\colon {\mathcal {C}}\rightarrow {\mbox{Set}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle H^{X}\colon {\mathcal {C}}\rightarrow {\mbox{Set}}}$ ${\displaystyle Y\in {\mbox{Ob}}({\mathcal {C}})}$ ${\displaystyle (f\colon Y\rightarrow Z)\in {\mbox{Mor}}({\mathcal {C}})}$

• ${\displaystyle H^{X}(Y):={\mbox{Hom}}_{\mathcal {C}}(X,Y)}$, where an alternative notation for is commonly used in this context . ${\displaystyle {\mbox{Hom}}_{\mathcal {C}}(X,Y)}$ ${\displaystyle {\mbox{Mor}}_{\mathcal {C}}(X,Y)}$

• ${\displaystyle H^{X}(f)\colon H^{X}(Y)\rightarrow H^{X}(Z),\,g\mapsto f\circ g}$.

One can now ask what natural transformations exist between the functors and from to set . The following Yoneda lemma gives an answer here. ${\displaystyle H^{X}}$ ${\displaystyle T}$ ${\displaystyle {\mathcal {C}}}$

statement

If a functor and an object are out , then there is a bijection of the set of all natural transformations into the set . ${\displaystyle T\colon {\mathcal {C}}\rightarrow {\mbox{Set}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \eta \mapsto \eta _{X}({\mbox{id}}_{X})}$ ${\displaystyle \eta \colon H^{X}\rightarrow T}$ ${\displaystyle T(X)}$

Please note that a natural transformation assigns a morphism to every object by definition , with certain compatibility conditions being met (see natural transformation ). In particular, one has a morphism in the category set (that is simply a mapping), so one can actually form as in the above lemma and get an element from . Hence the mapping is well defined; they are also called the Yoneda map or the Yoneda isomorphism . ${\displaystyle \eta \colon H^{X}\rightarrow T}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \eta _{Y}\colon H^{X}(Y)\rightarrow T(Y)}$ ${\displaystyle \eta _{X}\colon H^{X}(X)\rightarrow T(X)}$ ${\displaystyle \eta _{X}({\mbox{id}}_{X})}$ ${\displaystyle T(X)}$ ${\displaystyle {\mathcal {Y}}\colon \eta \mapsto \eta _{X}({\mbox{id}}_{X})}$

The proof is simple and illuminates the situation in the Yoneda lemma; therefore it is reproduced here: If a natural transformation is an object from and , that is, is a morphism , then the following diagram is commutative after the natural transformation is defined : ${\displaystyle \eta \colon H^{X}\rightarrow T}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f\in H^{X}(Y)}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle X\rightarrow Y}$

${\displaystyle {\begin{array}{ccc}H^{X}(X)={\mbox{Hom}}(X,X)&{\stackrel {\eta _{X}}{\longrightarrow }}&T(X)\\\downarrow _{H^{X}(f)}&&\downarrow _{T(f)}\\H^{X}(Y)={\mbox{Hom}}(X,Y)&{\stackrel {\eta _{Y}}{\longrightarrow }}&T(Y)\end{array}}}$

From this it follows . ${\displaystyle \eta _{Y}(f)=\eta _{Y}(f\circ {\mbox{id}}_{X})=(\eta _{Y}\circ H^{X}(f))({\mbox{id}}_{X})=(T(f)\circ \eta _{X})({\mbox{id}}_{X})=T(f)(\eta _{X}({\mbox{id}}_{X}))}$

Therefore it is already clearly established by and what the injectivity of the Yoneda mapping results from. This formula is also used for surjectivity . Namely , we define for each object in the picture by . Then you can check that this defines a natural transformation from to , which is mapped to under the Yoneda map . ${\displaystyle \eta _{Y}}$ ${\displaystyle T}$ ${\displaystyle \eta _{X}({\mbox{id}}_{X})}$ ${\displaystyle w\in T(X)}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \eta _{Y}\colon H^{X}(Y)\rightarrow T(Y)}$ ${\displaystyle \eta _{Y}(f)\,:=\,T(f)(w)}$ ${\displaystyle \eta }$ ${\displaystyle H^{X}}$ ${\displaystyle T}$ ${\displaystyle w}$

Remarks

• In particular, the Yoneda lemma shows that the natural transformations between functors and form a set, because the class of natural transformations between and is bijectively related to a set, namely , and is therefore itself one. ${\displaystyle H^{X}}$ ${\displaystyle T}$ ${\displaystyle H^{X}}$ ${\displaystyle T}$ ${\displaystyle T(X)}$
• Maps of the type presented above lead to the concept of the representability of functors. ${\displaystyle \eta _{Y}\colon H^{X}(Y)\rightarrow T(Y),\eta _{Y}(f)\,:=\,T(f)(w)}$
• If you have additional structures on the morphism sets ( enriched categories ), as in the case of Abelian categories , for example , you can replace the target category Set of the Hom functor with a corresponding category, for example the category Ab of the Abelian groups. In order to come back to the situation considered here, one only has to switch back the forget function . ${\displaystyle {\mbox{Ab}}\rightarrow {\mbox{Set}}}$

Yoneda embedding

Yoneda embedding is treated here as a simple application of the Yoneda lemma. The Yoneda embedding is used in the definition of the Ind objects and Pro objects .

If the category is locally small, then denote the category of functors with natural transformations as morphisms. Note that the natural transformations between two functors and according to the Yoneda lemma form a set, so there is actually a category. Next was with the dual category referred. In this situation, define the functor by the following data: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle [{\mathcal {C}},{\mbox{Set}}]}$ ${\displaystyle H^{X}}$ ${\displaystyle H^{X}}$ ${\displaystyle H^{Y}}$ ${\displaystyle {\mathcal {C}}^{op}}$ ${\displaystyle H^{*}\colon {\mathcal {C}}^{op}\rightarrow [{\mathcal {C}},{\mbox{Set}}]}$

• ${\displaystyle H^{*}(X):=H^{X}}$, the functors are the objects in . ${\displaystyle H^{X}}$ ${\displaystyle [{\mathcal {C}},{\mbox{Set}}]}$
• For a morphism, let it be defined by , where . Then there is a natural transformation, i.e. a morphism in . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle H^{*}(f)\colon H^{X}\rightarrow H^{Y}}$ ${\displaystyle H^{*}(f)_{Z}\colon H^{X}(Z)\rightarrow H^{Y}(Z),\,g\mapsto g\circ f}$ ${\displaystyle Z\in {\mbox{Ob}}({\mathcal {C}})}$ ${\displaystyle H^{*}(f)}$ ${\displaystyle [{\mathcal {C}},{\mbox{Set}}]}$

It is easy to check that this actually defines a functor . The dual category is selected on the left-hand side, since otherwise it would run "in the wrong direction". It applies now ${\displaystyle H^{*}\colon {\mathcal {C}}^{op}\rightarrow [{\mathcal {C}},{\mbox{Set}}]}$ ${\displaystyle f}$

• Yoneda embedding : The functor is a fully faithful embedding . ${\displaystyle H^{*}\colon {\mathcal {C}}^{op}\rightarrow [{\mathcal {C}},{\mbox{Set}}]}$

If you swap the roles of and , you get a fully faithful embedding . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{op}}$ ${\displaystyle {\mathcal {C}}\rightarrow [{\mathcal {C}}^{op},{\mbox{Set}}]}$

The proof is an application of the Yoneda lemma. For full fidelity it must be shown that the images

${\displaystyle H_{X,Y}^{*}\colon {\mbox{Mor}}_{{\mathcal {C}}^{op}}(X,Y)\rightarrow {\mbox{Mor}}_{[{\mathcal {C}},{\mbox{Set}}]}(H^{*}X,H^{*}Y),\,f\mapsto H^{*}(f)}$

are bijective. For , that is, for a natural transformation , is , that is, the Yoneda mapping defines a mapping ${\displaystyle \eta \in {\mbox{Mor}}_{[{\mathcal {C}},{\mbox{Set}}]}(H^{*}X,H^{*}Y)}$ ${\displaystyle \eta \colon H^{X}\rightarrow H^{Y}}$ ${\displaystyle {\mathcal {Y}}(\eta )\in H^{Y}(X)={\mbox{Hom}}_{\mathcal {C}}(Y,X)={\mbox{Hom}}_{{\mathcal {C}}^{op}}(X,Y)}$

${\displaystyle {\mathcal {Y}}_{X,Y}:{\mbox{Mor}}_{[{\mathcal {C}},{\mbox{Set}}]}(H^{*}X,H^{*}Y)\rightarrow {\mbox{Mor}}_{{\mathcal {C}}^{op}}(X,Y),\eta \mapsto \eta _{X}({\mbox{id}}_{X})}$.

Since this mapping is bijective after Yoneda lemma and because all the following applies: , ${\displaystyle f\in {\mbox{Mor}}_{{\mathcal {C}}^{op}}(X,Y)}$ ${\displaystyle {\mathcal {Y}}_{X,Y}H_{X,Y}^{*}(f)={\mathcal {Y}}_{X,Y}(H^{*}(f))=H^{*}(f)_{X}({\mbox{id}}_{X})={\mbox{id}}_{X}\circ f=f}$

is and therefore also bijective. Therefore is completely faithful. ${\displaystyle H_{X,Y}^{*}={\mathcal {Y}}_{X,Y}^{-1}}$ ${\displaystyle H^{*}}$

In order to see that there is even an embedding, the injectivity of the functor must be shown on the class of the objects (see article faithful functor ). If and are two different objects out , then the following applies because a morphism cannot have two different domains of definition, and it follows from this , that is . Hence there is also an embedding. ${\displaystyle H^{*}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mbox{Ob}}({\mathcal {C}}^{op})}$ ${\displaystyle {\mbox{Hom}}_{\mathcal {C}}(X,X)\cap {\mbox{Hom}}_{\mathcal {C}}(Y,X)=\emptyset }$ ${\displaystyle H^{X}\not =H^{Y}}$ ${\displaystyle H^{*}(X)\not =H^{*}(Y)}$ ${\displaystyle H^{*}}$

literature

• Horst Schubert : Categories (Heidelberg Pocket Books; Vol. 15-16). Springer, Berlin 1970 (2 vols.).