Last Updated
：2025/11/24
Yoneda lemma
Noun
(category
theory)
Given
a
category
𝒞
with
an
object
A,
let
H
be
a
hom
functor
represented
by
A,
and
let
F
be
any
functor
(not
necessarily
representable)
from
𝒞
to
Sets,
then
there
is
a
natural
isomorphism
between
Nat(H,F),
the
set
of
natural
transformations
from
H
to
F,
and
the
set
F(A).
(Any
natural
transformation
𝛼
from
H
to
F
is
determined
by
what
𝛼_A(
mbox
id_A)
is.)
Japanese Meaning
圏論におけるヨネダの補題（Yoneda lemma）とは、ある圏𝒞とその中の対象Aが与えられた場合、Aによって表現されるホム関手Hと、必ずしも表現可能でない任意の関手F（集合から圏𝒞への関手）との間で、自然変換の集合Nat(H, F)とF(A)の間に自然同型が存在するという主張である。
Sense（1）
(category
theory)
Given
a
category
𝒞
with
an
object
A,
let
H
be
a
hom
functor
represented
by
A,
and
let
F
be
any
functor
(not
necessarily
representable)
from
𝒞
to
Sets,
then
there
is
a
natural
isomorphism
between
Nat(H,F),
the
set
of
natural
transformations
from
H
to
F,
and
the
set
F(A).
(Any
natural
transformation
𝛼
from
H
to
F
is
determined
by
what
𝛼_A(
mbox
id_A)
is.)
Quizzes for review
(category theory) Given a category 𝒞 with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from 𝒞 to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation 𝛼 from H to F is determined by what 𝛼_A( mbox id_A) is.)
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See correct answer
Yoneda lemma
By the Yoneda lemma, any natural transformation from the Hom-functor represented by A to a functor F is uniquely determined by the element α_A(id_A) in F(A).
See correct answer
By the Yoneda lemma, any natural transformation from the Hom-functor represented by A to a functor F is uniquely determined by the element α_A(id_A) in F(A).
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English - English
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