In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
The Yoneda lemma suggests that instead of studying the locally small category
l{C}
l{C}
Set
Set
l{C}
Set
l{C}
l{C}
l{C}
This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category
l{C}
l{C}
Yoneda's lemma concerns functors from a fixed category
l{C}
Set
l{C}
A
l{C}
Set
hA=Hom(A,-)
hA
X\inl{C}
Hom(A,X)
f\colonX\toY
Y\inl{C}
f\circ-
f
g
Hom(A,X)
f\circg
Hom(A,Y)
hA(f)=Hom(A,f),or
hA(f)(g)=f\circg
Yoneda's lemma says that:
Here the notation
Setl{C}
l{C}
Set
Given a natural transformation
\Phi
hA
F
F(A)
u=\PhiA(idA)
u
F(A)
\PhiX(f)=F(f)(u)
f\colonA\toX
F(X)
There is a contravariant version of Yoneda's lemma, which concerns contravariant functors from
l{C}
Set
hA=Hom(-,A),
X
Hom(X,A)
G
l{C}
Set
Nat(hA,G)\congG(A).
The bijections provided in the (covariant) Yoneda lemma (for each
A
F
l{C} x Setl{C}
Set
-(-)\colonl{C} x Setl{C}\toSet
-(-)\colon(A,F)\mapstoF(A)
(f,\Phi)
f\colonA\toB
lC
\Phi\colonF\toG
\PhiB\circF(f)=G(f)\circ\PhiA\colonF(A)\toG(B).
\operatorname{Nat}(\hom(-,-),-)\colonlC x \operatorname{Set}lC\to\operatorname{Set},
\operatorname{Nat}(\hom(-,-),-)\colon(A,F)\mapsto\operatorname{Nat}(\hom(A,-),F),
(f,\Phi)
\operatorname{Nat}(\hom(f,-),\Phi)=\operatorname{Nat}(\hom(B,-),\Phi)\circ\operatorname{Nat}(\hom(f,-),F)=\operatorname{Nat}(\hom(f,-),G)\circ\operatorname{Nat}(\hom(A,-),\Phi)
\Psi\colon\hom(A,-)\toF
\Phi\circ\Psi\circ\hom(f,-)\colon\hom(B,-)\toG
(\Phi\circ\Psi\circ\hom(f,-))C(g)=(\Phi\circ\Psi)C(g\circf) (g\colonB\toC).
The use of
hA
hA
The mnemonic "falling into something" can be helpful in remembering that
hA
A
hA
X
A
X
Since
\Phi
This diagram shows that the natural transformation
\Phi
\PhiA(idA)=u
f\colonA\toX
\PhiX(f)=(Ff)u.
u\inF(A)
An important special case of Yoneda's lemma is when the functor
F
l{C}
Set
hB
Nat(hA,hB)\congHom(B,A).
That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism
f\colonB\toA
Hom(f,-)
Mapping each object
A
l{C}
hA=Hom(A,-)
f\colonB\toA
Hom(f,-)
h\bullet
l{C}
Setl{C}
l{C}
Set
h\bullet
h\bullet\colonl{C}op\toSetl{C}.
h\bullet
l{C}op
Set
\{hA|A\inC\}
Setl{C
l{C}op
\{hA|A\inC\}
The contravariant version of Yoneda's lemma states that
Nat(hA,hB)\congHom(A,B).
h\bullet
l{C}
Set
h\bullet\colonl{C}\toSetl{Cop
l{C}
l{C}
Set
h\bullet
The Yoneda embedding is sometimes denoted by よ, the Hiragana kana Yo.[3]
See main article: Representable functor. The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner. That is,
Nat(hA,P)\congP(A)
See main article: End (category theory).
Given two categories
C
D
F,G:C\toD
Nat(F,G)=\intcHomD(Fc,Gc)
For any functors
K\colonCop\toSets
H\colonC\toSets
K\cong\intcKc x HomC(-,c), K\cong\intc
HomC(c,-) | |
(Kc) |
,
H\cong\intcHc x HomC(c,-), H\cong\intc
HomC(-,c) | |
(Hc) |
.
See main article: Preadditive category.
A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is why preadditive categories are viewed as generalizations of rings. Rings are preadditive categories with one object.
The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring
R
R
M\congHomR(R,M)
M
R
As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory. To see this, let
l{C}
*
G=Homl{C
In this context, a covariant functor
l{C}\toSet
X
G\toPerm(X)
Perm(X)
X
X
G
\alpha\colonX\toY
\alpha(g ⋅ x)=g ⋅ \alpha(x)
g
G
x
X
⋅
G
X
Y
Now the covariant hom-functor
Homl{C
G
F=Homl{C
Nat(Homl{C
G
G
Perm(G)
g
G
g
G
Perm(G)
Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda in the Gare du Nord station.[4] [5]