In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Yoneda lemma is arguably the most important result in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
. 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
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
(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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
representation theory. It is named after
Nobuo Yoneda
was a Japanese mathematician and computer scientist.
In 1952, he graduated the Department of Mathematics, the Faculty of Science, the University of Tokyo, and obtained his Bachelor of Science. That same year, he was appointed Assistant Profess ...
.
Generalities
The Yoneda lemma suggests that instead of studying the
locally small category
, one should study the category of all functors of
into
(the
category of sets with
functions as
morphisms).
is a category we think we understand well, and a functor of
into
can be seen as a "representation" of
in terms of known structures. The original category
is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in
. Treating these new objects just like the old ones often unifies and simplifies the theory.
This approach is akin to (and in fact generalizes) the common method of studying a
ring by investigating the
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over that ring. The ring takes the place of the category
, and the category of modules over the ring is a category of functors defined on
.
Formal statement
Yoneda's lemma concerns functors from a fixed category
to the
category of sets,
. If
is a
locally small category (i.e. the
hom-sets are actual sets and not proper classes), then each object
of
gives rise to a natural functor to
called a
hom-functor. This functor is denoted:
:
.
The (
covariant) hom-functor
sends
to the set of
morphisms
and sends a morphism
(where
and
are objects in
) to the morphism
(composition with
on the left) that sends a morphism
in
to the morphism
in
. That is,
:
:
Yoneda's lemma says that:
Here the notation
denotes the category of functors from
to
.
Given a natural transformation
from
to
, the corresponding element of
is
; and given an element
of
, the corresponding natural transformation is given by
.
Contravariant version
There is a contravariant version of Yoneda's lemma, which concerns
contravariant functors from
to
. This version involves the contravariant hom-functor
:
which sends
to the hom-set
. Given an arbitrary contravariant functor
from
to
, Yoneda's lemma asserts that
:
Naming conventions
The use of
for the covariant hom-functor and
for the contravariant hom-functor is not completely standard. Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors. However, most modern algebraic geometry texts starting with
Alexander Grothendieck's foundational
EGA use the convention in this article.
The mnemonic "falling into something" can be helpful in remembering that
is the covariant hom-functor. When the letter
is falling (i.e. a subscript),
assigns to an object
the morphisms from
into
.
Proof
Since
is a natural transformation, we have the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
:
This diagram shows that the natural transformation
is completely determined by
since for each morphism
one has
:
Moreover, any element
defines a natural transformation in this way. The proof in the contravariant case is completely analogous.
The Yoneda embedding
An important special case of Yoneda's lemma is when the functor
from
to
is another hom-functor
. In this case, the covariant version of Yoneda's lemma states that
:
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
the associated natural transformation is denoted
.
Mapping each object
in
to its associated hom-functor
and each morphism
to the corresponding natural transformation
determines a contravariant functor
from
to
, the
functor category of all (covariant) functors from
to
. One can interpret
as a
covariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
:
:
The meaning of Yoneda's lemma in this setting is that the functor
is
fully faithful, and therefore gives an embedding of
in the category of functors to
. The collection of all functors
is a subcategory of
. Therefore, Yoneda embedding implies that the category
is isomorphic to the category
.
The contravariant version of Yoneda's lemma states that
:
Therefore,
gives rise to a covariant functor from
to the category of contravariant functors to
:
:
Yoneda's lemma then states that any locally small category
can be embedded in the category of contravariant functors from
to
via
. This is called the ''Yoneda embedding''.
The Yoneda embedding is sometimes denoted by よ, the
Hiragana
is a Japanese syllabary, part of the Japanese writing system, along with ''katakana'' as well as ''kanji''.
It is a phonetic lettering system. The word ''hiragana'' literally means "flowing" or "simple" kana ("simple" originally as contras ...
kana
The term may refer to a number of syllabaries used to write Japanese phonological units, morae. Such syllabaries include (1) the original kana, or , which were Chinese characters ( kanji) used phonetically to transcribe Japanese, the most ...
Yo.
Representable functor
The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be
represented by
presheaves
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, in a full and faithful manner. That is,
:
for a presheaf ''P''. Many common categories are, in fact, categories of pre-sheaves, and on closer inspection, prove to be categories of
sheaves, and as such examples are commonly topological in nature, they can be seen to be
topoi in general. The Yoneda lemma provides a point of leverage by which the topological structure of a category can be studied and understood.
In terms of (co)end calculus
Given two categories
and
with two functors
, natural transformations between them can be written as the following
end.
:
For any functors
and
the following formulas are all formulations of the Yoneda lemma.
:
:
Preadditive categories, rings and modules
A ''
preadditive category'' is a category where the morphism sets form
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s 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
, the extended category is the category of all right
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over
, and the statement of the Yoneda lemma reduces to the well-known isomorphism
:
for all right modules
over
.
Relationship to Cayley's theorem
As stated above, the Yoneda lemma may be considered as a vast generalization of
Cayley's theorem from
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
. To see this, let
be a category with a single object
such that every morphism is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(i.e. a
groupoid with one object). Then
forms a
group under the operation of composition, and any group can be realized as a category in this way.
In this context, a covariant functor
consists of a set
and a
group homomorphism , where
is the group of
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s of
; in other words,
is a
G-set. A natural transformation between such functors is the same thing as an
equivariant map between
-sets: a set function
with the property that
for all
in
and
in
. (On the left side of this equation, the
denotes the action of
on
, and on the right side the action on
.)
Now the covariant hom-functor
corresponds to the action of
on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with
states that
:
,
that is, the equivariant maps from this
-set to itself are in bijection with
. But it is easy to see that (1) these maps form a group under composition, which is a
subgroup of
, and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every
in
the equivariant map of right-multiplication by
.) Thus
is isomorphic to a subgroup of
, which is the statement of Cayley's theorem.
History
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.
See also
*
Representation theorem
Notes
References
* .
*
*
*
External links
*
Mizar system proof: {{cite journal , first=M. , last=Wojciechowski , title=Yoneda Embedding , journal=Formalized Mathematics journal , volume=6 , issue=3 , pages=377–380 , date=1997 , citeseerx=10.1.1.73.7127
Representable functors
Lemmas in category theory
Articles containing proofs