Yoneda lemma – Wikipedia
Embedding of categories into functor categories
In mathematics, the Yoneda lemma is arguably the most important result in category theory.[1] 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 (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 and representation theory. It is named after Nobuo Yoneda.
Generalities[edit]
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 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[edit]
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:
Lemma (Yoneda) — Let
be a functor from a locally small category
to
. Then for each object
of
, the natural transformations
from
to
are in one-to-one correspondence with the elements of
. That is,
Moreover, this isomorphism is natural in
and
when both sides are regarded as functors from
to
.
Here the notation
denotes the category of functors from
to
.
Given a natural transformation
from
to
, the corresponding element of
is
;[a] and given an element
of
, the corresponding natural transformation is given by
which assigns to a morphism
a value of
.
Contravariant version[edit]
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[edit]
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.[b]
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[edit]
Since
is a natural transformation, we have the following commutative diagram:
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[edit]
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:
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 kana Yo.[2]
Representable functor[edit]
The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, 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[edit]
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[edit]
A preadditive category is a category where the morphism sets form abelian groups 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 over
, and the statement of the Yoneda lemma reduces to the well-known isomorphism
- for all right modules over .
Relationship to Cayley’s theorem[edit]
As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley’s theorem from group theory. To see this, let
be a category with a single object
such that every morphism is an isomorphism (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 permutations 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[edit]
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.[5][6]
See also[edit]
References[edit]
- Freyd, Peter (1964), Abelian categories, Harper’s Series in Modern Mathematics (2003 reprint ed.), Harper and Row, Zbl 0121.02103.
- Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001
- Loregian, Fosco (2015). “This is the (co)end, my only (co)friend”. arXiv:1501.02503 [math.CT].
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