Yoneda lemma

Inmathematics,theYoneda lemmais a fundamental result incategory theory.It is an abstract result onfunctorsof the typemorphisms into a fixed object.It is a vast generalisation ofCayley's theoremfromgroup theory(viewing a group as a miniature category with just one object and only isomorphisms). It allows theembeddingof anylocally small categoryinto acategory of functors(contravariantset-valued functors) defined on that category. It also clarifies how the embedded category, ofrepresentable functorsand theirnatural transformations,relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments inalgebraic geometryandrepresentation theory.It is named afterNobuo Yoneda.

Generalities

The Yoneda lemma suggests that instead of studying thelocally smallcategory${\mathcal {C}}$,one should study the category of all functors of${\mathcal {C}}$into$\mathbf {Set}$(thecategory of setswithfunctionsasmorphisms).$\mathbf {Set}$is a category we think we understand well, and a functor of${\mathcal {C}}$into$\mathbf {Set}$can be seen as a "representation" of${\mathcal {C}}$in terms of known structures. The original category${\mathcal {C}}$is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in${\mathcal {C}}$.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 aringby investigating themodulesover that ring. The ring takes the place of the category${\mathcal {C}}$,and the category of modules over the ring is a category of functors defined on${\mathcal {C}}$.

Formal statement

Yoneda's lemma concerns functors from a fixed category${\mathcal {C}}$to acategory of sets,$\mathbf {Set}$.If${\mathcal {C}}$is alocally small category(i.e. thehom-setsare actual sets and notproper classes), then each object$A$of${\mathcal {C}}$gives rise to a functor to$\mathbf {Set}$called ahom-functor.This functor is denoted:

$h_{A}=\mathrm {Hom} (A,-)$.

The (covariant) hom-functor$h_{A}$sends$X\in {\mathcal {C}}$to the set ofmorphisms$\mathrm {Hom} (A,X)$and sends a morphism$f\colon X\to Y$(where$Y\in {\mathcal {C}}$) to the morphism$f\circ -$(composition with$f$on the left) that sends a morphism$g$in$\mathrm {Hom} (A,X)$to the morphism$f\circ g$in$\mathrm {Hom} (A,Y)$.That is,

$h_{A}(f)=\mathrm {Hom} (A,f),{\text{ or}}$

$h_{A}(f)(g)=f\circ g$

Yoneda's lemma says that:

Lemma(Yoneda)—Let$F$be a functor from a locally small category${\mathcal {C}}$to$\mathbf {Set}$.Then for each object$A$of${\mathcal {C}}$,thenatural transformations$\mathrm {Nat} (h_{A},F)\equiv \mathrm {Hom} (\mathrm {Hom} (A,-),F)$from$h_{A}$to$F$are in one-to-one correspondence with the elements of$F(A)$.That is,

$\mathrm {Nat} (h_{A},F)\cong F(A).$

Moreover, this isomorphism isnaturalin$A$and$F$when both sides are regarded as functors from${\mathcal {C}}\times \mathbf {Set} ^{\mathcal {C}}$to$\mathbf {Set}$.

Here the notation$\mathbf {Set} ^{\mathcal {C}}$denotes the category of functors from${\mathcal {C}}$to$\mathbf {Set}$.

Given a natural transformation$\Phi$from$h_{A}$to$F$,the corresponding element of$F(A)$is$u=\Phi _{A}(\mathrm {id} _{A})$;and given an element$u$of$F(A)$,the corresponding natural transformation is given by$\Phi _{X}(f)=F(f)(u)$which assigns to a morphism$f\colon A\to X$a value of$F(X)$.

Contravariant version

There is a contravariant version of Yoneda's lemma,which concernscontravariant functorsfrom${\mathcal {C}}$to$\mathbf {Set}$.This version involves the contravariant hom-functor

$h^{A}=\mathrm {Hom} (-,A),$

which sends$X$to the hom-set$\mathrm {Hom} (X,A)$.Given an arbitrary contravariant functor$G$from${\mathcal {C}}$to$\mathbf {Set}$,Yoneda's lemma asserts that

$\mathrm {Nat} (h^{A},G)\cong G(A).$

Naturality

The bijections provided in the (covariant) Yoneda lemma (for each$A$and$F$) are the components of anatural isomorphismbetween two certain functors from${\mathcal {C}}\times \mathbf {Set} ^{\mathcal {C}}$to$\mathbf {Set}$.One of the two functors is the evaluation functor

$-(-)\colon {\mathcal {C}}\times \mathbf {Set} ^{\mathcal {C}}\to \mathbf {Set}$

$-(-)\colon (A,F)\mapsto F(A)$

that sends a pair$(f,\Phi )$of a morphism$f\colon A\to B$in${\mathcal {C}}$and anatural transformation$\Phi \colon F\to G$to the map

$\Phi _{B}\circ F(f)=G(f)\circ \Phi _{A}\colon F(A)\to G(B).$

This is enough to determine the other functor since we know what the natural isomorphism is. Under the second functor

$\operatorname {Nat} (\hom(-,-),-)\colon {\mathcal {C}}\times \operatorname {Set} ^{\mathcal {C}}\to \operatorname {Set},$

$\operatorname {Nat} (\hom(-,-),-)\colon (A,F)\mapsto \operatorname {Nat} (\hom(A,-),F),$

the image of a pair$(f,\Phi )$is the map

$\operatorname {Nat} (\hom(f,-),\Phi )=\operatorname {Nat} (\hom(B,-),\Phi )\circ \operatorname {Nat} (\hom(f,-),F)=\operatorname {Nat} (\hom(f,-),G)\circ \operatorname {Nat} (\hom(A,-),\Phi )$

that sends a natural transformation$\Psi \colon \hom(A,-)\to F$to the natural transformation$\Phi \circ \Psi \circ \hom(f,-)\colon \hom(B,-)\to G$,whose components are

$(\Phi \circ \Psi \circ \hom(f,-))_{C}(g)=(\Phi \circ \Psi )_{C}(g\circ f)\qquad (g\colon B\to C).$

Naming conventions

The use of$h_{A}$for the covariant hom-functor and$h^{A}$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 withAlexander Grothendieck's foundationalEGAuse the convention in this article.

The mnemonic "falling into something" can be helpful in remembering that$h_{A}$is the covariant hom-functor. When the letter$A$isfalling(i.e. a subscript),$h_{A}$assigns to an object$X$the morphisms from$A$into$X$.

Proof

Since$\Phi$is a natural transformation, we have the followingcommutative diagram:

Proof of Yoneda's lemma

This diagram shows that the natural transformation$\Phi$is completely determined by$\Phi _{A}(\mathrm {id} _{A})=u$since for each morphism$f\colon A\to X$one has

$\Phi _{X}(f)=(Ff)u.$

Moreover, any element$u\in F(A)$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$F$from${\mathcal {C}}$to$\mathbf {Set}$is another hom-functor$h_{B}$.In this case, the covariant version of Yoneda's lemma states that

$\mathrm {Nat} (h_{A},h_{B})\cong \mathrm {Hom} (B,A).$

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$f\colon B\to A$the associated natural transformation is denoted$\mathrm {Hom} (f,-)$.

Mapping each object$A$in${\mathcal {C}}$to its associated hom-functor$h_{A}=\mathrm {Hom} (A,-)$and each morphism$f\colon B\to A$to the corresponding natural transformation$\mathrm {Hom} (f,-)$determines a contravariant functor$h_{\bullet }$from${\mathcal {C}}$to$\mathbf {Set} ^{\mathcal {C}}$,thefunctor categoryof all (covariant) functors from${\mathcal {C}}$to$\mathbf {Set}$.One can interpret$h_{\bullet }$as acovariant functor:

$h_{\bullet }\colon {\mathcal {C}}^{\text{op}}\to \mathbf {Set} ^{\mathcal {C}}.$

The meaning of Yoneda's lemma in this setting is that the functor$h_{\bullet }$isfully faithful,and therefore gives an embedding of${\mathcal {C}}^{\mathrm {op} }$in the category of functors to$\mathbf {Set}$.The collection of all functors$\{h_{A}|A\in C\}$is a subcategory of$\mathbf {Set} ^{\mathcal {C}}$.Therefore, Yoneda embedding implies that the category${\mathcal {C}}^{\mathrm {op} }$is isomorphic to the category$\{h_{A}|A\in C\}$.

The contravariant version of Yoneda's lemma states that

$\mathrm {Nat} (h^{A},h^{B})\cong \mathrm {Hom} (A,B).$

Therefore,$h^{\bullet }$gives rise to a covariant functor from${\mathcal {C}}$to the category of contravariant functors to$\mathbf {Set}$:

$h^{\bullet }\colon {\mathcal {C}}\to \mathbf {Set} ^{{\mathcal {C}}^{\mathrm {op} }}.$

Yoneda's lemma then states that any locally small category${\mathcal {C}}$can be embedded in the category of contravariant functors from${\mathcal {C}}$to$\mathbf {Set}$via$h^{\bullet }$.This is called theYoneda embedding.

The Yoneda embedding is sometimes denoted by よ, thehiraganaYo.

Representable functor

Main article:Representable functor

The Yoneda embedding essentially states that for every (locally small) category, objects in that category can berepresentedbypresheaves,in a full and faithful manner. That is,

$\mathrm {Nat} (h^{A},P)\cong P(A)$

for a presheafP.Many common categories are, in fact, categories of pre-sheaves, and on closer inspection, prove to be categories ofsheaves,and as such examples are commonly topological in nature, they can be seen to betopoiin 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

Main article:End (category theory)

Given two categories$\mathbf {C}$and$\mathbf {D}$with two functors$F,G:\mathbf {C} \to \mathbf {D}$,natural transformations between them can be written as the followingend.

$\mathrm {Nat} (F,G)=\int _{c\in \mathbf {C} }\mathrm {Hom} _{\mathbf {D} }(Fc,Gc)$

For any functors$K\colon \mathbf {C} ^{op}\to \mathbf {Sets}$and$H\colon \mathbf {C} \to \mathbf {Sets}$the following formulas are all formulations of the Yoneda lemma.

$K\cong \int ^{c\in \mathbf {C} }Kc\times \mathrm {Hom} _{\mathbf {C} }(-,c),\qquad K\cong \int _{c\in \mathbf {C} }(Kc)^{\mathrm {Hom} _{\mathbf {C} }(c,-)},$

$H\cong \int ^{c\in \mathbf {C} }Hc\times \mathrm {Hom} _{\mathbf {C} }(c,-),\qquad H\cong \int _{c\in \mathbf {C} }(Hc)^{\mathrm {Hom} _{\mathbf {C} }(-,c)}.$

Preadditive categories, rings and modules

Main article:Preadditive category

Apreadditive categoryis a category where the morphism sets formabelian groupsand the composition of morphisms isbilinear;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 ofrings.Rings are preadditive categories with one object.

The Yoneda lemma remains true for preadditive categories if we choose as our extension the category ofadditivecontravariant 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 amodule categoryover 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 anabelian category,a much more powerful condition. In the case of a ring$R$,the extended category is the category of all rightmodulesover$R$,and the statement of the Yoneda lemma reduces to the well-known isomorphism

$M\cong \mathrm {Hom} _{R}(R,M)$for all right modules$M$over$R$.

Relationship to Cayley's theorem

As stated above, the Yoneda lemma may be considered as a vast generalization ofCayley's theoremfromgroup theory.To see this, let${\mathcal {C}}$be a category with a single object$*$such that every morphism is anisomorphism(i.e. agroupoidwith one object). Then$G=\mathrm {Hom} _{\mathcal {C}}(*,*)$forms agroupunder the operation of composition, and any group can be realized as a category in this way.

In this context, a covariant functor${\mathcal {C}}\to \mathbf {Set}$consists of a set$X$and agroup homomorphism$G\to \mathrm {Perm} (X)$,where$\mathrm {Perm} (X)$is the group ofpermutationsof$X$;in other words,$X$is aG-set.A natural transformation between such functors is the same thing as anequivariant mapbetween$G$-sets: a set function$\alpha \colon X\to Y$with the property that$\alpha (g\cdot x)=g\cdot \alpha (x)$for all$g$in$G$and$x$in$X$.(On the left side of this equation, the$\cdot$denotes the action of$G$on$X$,and on the right side the action on$Y$.)

Now the covariant hom-functor$\mathrm {Hom} _{\mathcal {C}}(*,-)$corresponds to the action of$G$on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with$F=\mathrm {Hom} _{\mathcal {C}}(*,-)$states that

$\mathrm {Nat} (\mathrm {Hom} _{\mathcal {C}}(*,-),\mathrm {Hom} _{\mathcal {C}}(*,-))\cong \mathrm {Hom} _{\mathcal {C}}(*,*)$,

that is, the equivariant maps from this$G$-set to itself are in bijection with$G$.But it is easy to see that (1) these maps form a group under composition, which is asubgroupof$\mathrm {Perm} (G)$,and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every$g$in$G$the equivariant map of right-multiplication by$g$.) Thus$G$is isomorphic to a subgroup of$\mathrm {Perm} (G)$,which is the statement of Cayley's theorem.

History

Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined bySaunders Mac Lanefollowing an interview he had with Yoneda in theGare du Nordstation.

See also

• Representation theorem
• Completions in category theory

References

• Freyd, Peter(1964),Abelian categories,Harper's Series in Modern Mathematics (2003 reprint ed.), Harper and Row,Zbl0121.02103.
• Mac Lane, Saunders(1998),Categories for the Working Mathematician,Graduate Texts in Mathematics,vol. 5 (2nd ed.), New York, NY:Springer-Verlag,ISBN0-387-98403-8,Zbl0906.18001
• Loregian, Fosco (2021).(Co)end Calculus.arXiv:1501.02503.doi:10.1017/9781108778657.ISBN9781108778657.
• Leinster, Tom (2014),Basic Category Theory,arXiv:1612.09375,doi:10.1017/CBO9781107360068,ISBN978-1-107-04424-1
• Category Theory,Oxford University Press, 17 June 2010,ISBN978-0-19-958736-0

• Yoneda lemmaat thenLab

External links

• Mizar systemproof:Wojciechowski, M. (1997). "Yoneda Embedding".Formalized Mathematics Journal.6(3):377–380.CiteSeerX10.1.1.73.7127.
• Beurier, Erwan; Pastor, Dominique (July 2019)."A crash course on Category Theory".