Formal statement Yoneda lemma

1 formal statement

1.1 general version
1.2 naming conventions
1.3 proof
1.4 yoneda embedding

formal statement
general version

yoneda s lemma concerns functors fixed category c category of sets, set. if c locally small category (i.e. hom-sets actual sets , not proper classes), each object of c gives rise natural functor set called hom-functor. functor denoted:

h

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=

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{\displaystyle h^{a}=\mathrm {hom} (a,-).}

the (covariant) hom-functor h sends x set of morphisms hom(a,x) , sends morphism f x y morphism



f
∘
−


{\displaystyle f\circ -}

(composition f on left) sends morphism g in hom(a,x) morphism f o g in hom(a,y). is,

h

a


:
f
⟼

h
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[

[

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∈

h
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{\displaystyle h^{a}\colon f\longmapsto \mathrm {hom} (a,f)=[\![\mathrm {hom} (a,x)\ni g\mapsto f\circ g\in \mathrm {hom} (a,y)]\!]}

.

let f arbitrary functor c set. yoneda s lemma says that:

for each object of c, natural transformations h f in one-to-one correspondence elements of f(a). is,

n
a
t

(

h

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,
f
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≅
f
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{\displaystyle \mathrm {nat} (h^{a},f)\cong f(a).}

moreover isomorphism natural in , f when both sides regarded functors set x c set. (here notation set denotes category of functors c set.)

given natural transformation



Φ


{\displaystyle \phi }

h f, corresponding element of f(a)



u
=

Φ

a


(


i
d


a


)


{\displaystyle u=\phi _{a}(\mathrm {id} _{a})}

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there contravariant version of yoneda s lemma, concerns contravariant functors c set. version involves contravariant hom-functor

h

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{\displaystyle h_{a}=\mathrm {hom} (-,a),}

which sends x hom-set hom(x,a). given arbitrary contravariant functor g c set, yoneda s lemma asserts that

n
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t

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h

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,
g
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≅
g
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a
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{\displaystyle \mathrm {nat} (h_{a},g)\cong g(a).}



naming conventions

the use of h covariant hom-functor , ha contravariant hom-functor not standard. many texts , articles either use opposite convention or unrelated symbols these 2 functors. however, modern algebraic geometry texts starting alexander grothendieck s foundational ega use convention in article.

the mnemonic falling can helpful in remembering ha contravariant hom-functor. when letter falling (i.e. subscript), ha assigns object x morphisms x a.

proof

the proof of yoneda s lemma indicated following commutative diagram:

this diagram shows natural transformation Φ determined




Φ

a


(


i
d


a


)
=
u


{\displaystyle \phi _{a}(\mathrm {id} _{a})=u}

since each morphism f : → x 1 has

Φ

x


(
f
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=
(
f
f
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u
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{\displaystyle \phi _{x}(f)=(ff)u.}

moreover, element u∈f(a) defines natural transformation in way. proof in contravariant case analogous.

the yoneda embedding

an important special case of yoneda s lemma when functor f c set hom-functor h. in case, covariant version of yoneda s lemma states that

n
a
t

(

h

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,

h

b


)
≅

h
o
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b
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{\displaystyle \mathrm {nat} (h^{a},h^{b})\cong \mathrm {hom} (b,a).}

that is, natural transformations between hom-functors in one-to-one correspondence morphisms (in reverse direction) between associated objects. given morphism f : b → associated natural transformation denoted hom(f,–).

mapping each object in c associated hom-functor h = hom(a,–) , each morphism f : b → corresponding natural transformation hom(f,–) determines contravariant functor h c set, functor category of (covariant) functors c set. 1 can interpret h covariant functor:

h

−


:



c



op


→


s
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c



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{\displaystyle h^{-}\colon {\mathcal {c}}^{\text{op}}\to \mathbf {set} ^{\mathcal {c}}.}

the meaning of yoneda s lemma in setting functor h faithful, , therefore gives embedding of c in category of functors set. collection of functors {h, in c} subcategory of set. therefore, yoneda embedding implies category c isomorphic category {h, in c}.

the contravariant version of yoneda s lemma states that

n
a
t

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h

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,

h

b


)
≅

h
o
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,
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{\displaystyle \mathrm {nat} (h_{a},h_{b})\cong \mathrm {hom} (a,b).}

therefore, h– gives rise covariant functor c category of contravariant functors set:

h

−


:


c


→


s
e
t





c




o
p





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{\displaystyle h_{-}\colon {\mathcal {c}}\to \mathbf {set} ^{{\mathcal {c}}^{\mathrm {op} }}.}

yoneda s lemma states locally small category c can embedded in category of contravariant functors c set via h–. called yoneda embedding.

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