In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Yoneda lemma is a fundamental result in
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
.
It is an abstract result on
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s of the type ''morphisms into a fixed object''. It is a vast generalisation of
Cayley's theorem
In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric gro ...
from
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
(viewing a group as a miniature category with just one object and only isomorphisms). It also generalizes the information-preserving relation between a term and its
continuation-passing style transformation from
programming language theory
Programming language theory (PLT) is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of formal languages known as programming languages. Programming language theory is clos ...
. It allows the
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
of any
locally small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
into a
category of functors (
contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of
representable functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets an ...
s and their
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s, 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 which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
and
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
. 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 Professo ...
.
Generalities
The Yoneda lemma suggests that instead of studying the
locally small
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
category
, one should study the category of all functors of
into
(the
category of sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
with
functions as
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s).
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
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
by investigating the
modules
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computer science and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
...
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 a
category of sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
,
. If
is a
locally small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
(i.e. the
hom-set
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...
s are actual sets and not
proper classes), then each object
of
gives rise to a functor to
called a
hom-functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
. This functor is denoted:
:
.
The (
covariant) hom-functor
sends
to the set of
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s
and sends a morphism
(where
) 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
which assigns to a morphism
a value of
.
Contravariant version
There is a contravariant version of Yoneda's lemma, which concerns
contravariant 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 ...
s 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
:
Naturality
The bijections provided in the (covariant) Yoneda lemma (for each
and
) are the components of a
natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
between two certain functors from
to
.
One of the two functors is the evaluation functor
:
:
that sends a pair
of a morphism
in
and a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to the map
:
This is enough to determine the other functor since we know what the natural isomorphism is. Under the second functor
:
:
the image of a pair
is the map
:
that sends a natural transformation
to the natural transformation
, whose components are
:
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
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
'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
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
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 ...
:
:
The meaning of Yoneda's lemma in this setting is that the functor
is
fully faithful
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor.
Formal definitions
Explicitly, let ''C'' and ' ...
, 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 language, Japanese syllabary, part of the Japanese writing system, along with ''katakana'' as well as ''kanji''.
It is a phonetic lettering system. The word ''hiragana'' means "common" or "plain" kana (originally also "easy", ...
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 (: sheaves) 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 da ...
, 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 mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion ...
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
End, END, Ending, or ENDS may refer to:
End Mathematics
*End (category theory)
* End (topology)
* End (graph theory)
* End (group theory) (a subcase of the previous)
* End (endomorphism) Sports and games
*End (gridiron football)
*End, a division ...
.
:
For any functors
and
the following formulas are all formulations of the Yoneda lemma.
:
:
Preadditive categories, rings and modules
A ''
preadditive category
In mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.
That is, an Ab-category C is a category such that
every h ...
'' 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 commu ...
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
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-function see Sigma additivity
* Additive category, a preadditive category with fin ...
'' 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
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the rin ...
'' 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
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
The motivating prototypical example of an abelian category is the category o ...
, a much more powerful condition. In the case of a ring
, the extended category is the category of all right
modules
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computer science and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
...
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
In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric gro ...
from
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
. 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 or morphism 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 the ...
(i.e. a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
with one object). Then
forms a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
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
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
, where
is the group of
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
s of
; in other words,
is a
G-set
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under func ...
. A natural transformation between such functors is the same thing as an
equivariant map
In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ...
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
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
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
Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg.
Early life and education
Mac Lane was born in Norwich, Connecticut, near w ...
following an interview he had with Yoneda in the
Gare du Nord
The Gare du Nord (; ), officially Paris Nord, is one of the seven large mainline railway station termini in Paris, France. The station is served by trains that run between the capital and northern France via the Paris–Lille railway, as well ...
station.
See also
*
Representation theorem
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.
Examples
Algebra
* Cayley's theorem states that every group i ...
*
Completions in category theory In category theory, a branch of mathematics, there are several ways (completions) to enlarge a given category in a way somehow analogous to a completion in topology. These are (ignoring the set-theoretic matters for simplicity):
*free cocompletion, ...
*
2-Yoneda lemma
In mathematics, especially category theory, the 2-Yoneda lemma is a generalization of the Yoneda lemma to 2-categories. Precisely, given a contravariant pseudofunctor F on a category ''C'', it says: for each object x in ''C'', the natural functor ( ...
Notes
References
* .
*
*
*
*
*
External links
*
Mizar system
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used ...
proof:
*
{{Category theory
Representable functors
Lemmas in category theory
Articles containing proofs