Noun
Yoneda lemma
(category theory) Given a category
C
{\displaystyle {\mathcal {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
{\displaystyle {\mathcal {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
{\displaystyle \alpha }
from H to F is determined by what
α
A
(
id
A
)
{\displaystyle \alpha _{A}({\mbox{id}}_{A})}
is.)
As a corollary of the Yoneda lemma, given a pair of contravariant hom functors
Hom
(
−
,
A
)
{\displaystyle {\mbox{Hom}}(-,A)}
and
Hom
(
−
,
B
)
{\displaystyle {\mbox{Hom}}(-,B)}
, then any natural transformation
α
{\displaystyle \alpha }
from
Hom
(
−
,
A
)
{\displaystyle {\mbox{Hom}}(-,A)}
to
Hom
(
−
,
B
)
{\displaystyle {\mbox{Hom}}(-,B)}
is determined by the choice of some function
f
:
A
→
B
{\displaystyle f:A\rightarrow B}
to map the identity
id
A
:
A
→
A
{\displaystyle {\mbox{id}}_{A}:A\rightarrow A}
to, by the component
α
A
:
Hom
(
A
,
A
)
→
Hom
(
A
,
B
)
{\displaystyle \alpha _{A}:{\mbox{Hom}}(A,A)\rightarrow {\mbox{Hom}}(A,B)}
of
α
{\displaystyle \alpha }
. This implies that the Yoneda functor is fully faithful, which in turn implies that Yoneda embeddings are possible.
Generalities The Yoneda lemma suggests that instead of studying the ( locally small ) category C, one should study the category of all functors of C into Set (the category of sets with functions as morphisms ). Source: Internet
The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories. Source: Internet