Yoneda Lemma
The Yoneda lemma is a result in category theory.
Contents
Statement
Let be a locally small category, and let
be
a functor from
to Set, the category of sets.
Let
denote the functor
that
sends every object
to
and
that takes the arrow
to the function
given by
. In other words,
is the
hom functor
. Then there exists a bijection
between the set of natural transformations from
to
and the set
. In symbols,
Proof
Let be a natural transformation.
Then for each object
of
,
gives us an arrow
in the category Set, i.e., a
function
, such that the diagram
\[\begin{array}{ccc} h_A(A) & \!\stackrel{\phi_A}{\longrightarrow}\! & F(A) \\ \downarrow h_A f && \downarrow Ff \\ h_A(B) & \! \stackrel{\phi_B}{\longrightarrow}\! & F(B) \end{aray}\] (Error compiling LaTeX. Unknown error_msg)
commutes for any arrow in
.
In particular, we have
But
is the map
from
to
. We thus have
Thus for every object
of
, the morphism
is uniquely determined by the element
.
Thus the map
is an injection
from
to
.
It thus remains to be shown that for any , the maps
for every object
of
define a natural transformation.
But this is true, as for any objects
and
of
,
any morphism
of
, and any element
of
, we have
since
is a functor. Thus the diagram
commutes, so
is a natural transformation, as desired.
Therefore the map is
a bijection between
and
, as
desired.
Dual Statement
If we replace the category with its opposite category
, then we get the following
result: if
is a contravariant functor from
to Set, then
for every object
of
. This dual statement
is also sometimes known as the Yoneda lemma.