[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/yoneda-lemma-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki7\/yoneda-lemma-wikipedia\/","headline":"Yoneda lemma – Wikipedia","name":"Yoneda lemma – Wikipedia","description":"Embedding of categories into functor categories In mathematics, the Yoneda lemma is arguably the most important result in category theory.[1]","datePublished":"2016-09-23","dateModified":"2016-09-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki7\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e7b3edab7022ca9e2976651bc59c489513ee9019","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e7b3edab7022ca9e2976651bc59c489513ee9019","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki7\/yoneda-lemma-wikipedia\/","wordCount":20851,"articleBody":"Embedding of categories into functor categoriesIn 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.Table of ContentsGeneralities[edit]Formal statement[edit]Contravariant version[edit]Naming conventions[edit]Proof[edit]The Yoneda embedding[edit]Representable functor[edit]In terms of (co)end calculus[edit]Preadditive categories, rings and modules[edit]Relationship to Cayley’s theorem[edit]History[edit]See also[edit]References[edit]External links[edit]Generalities[edit]The Yoneda lemma suggests that instead of studying the locally small category C{displaystyle {mathcal {C}}}, one should study the category of all functors of C{displaystyle {mathcal {C}}} into Set{displaystyle mathbf {Set} } (the category of sets with functions as morphisms). Set{displaystyle mathbf {Set} } is a category we think we understand well, and a functor of C{displaystyle {mathcal {C}}} into Set{displaystyle mathbf {Set} } can be seen as a “representation” of C{displaystyle {mathcal {C}}} in terms of known structures. The original category C{displaystyle {mathcal {C}}} is contained in this functor category, but new objects appear in the functor category, which were absent and “hidden” in C{displaystyle {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 a ring by investigating the modules over that ring. The ring takes the place of the category C{displaystyle {mathcal {C}}}, and the category of modules over the ring is a category of functors defined on C{displaystyle {mathcal {C}}}.Formal statement[edit]Yoneda’s lemma concerns functors from a fixed category C{displaystyle {mathcal {C}}} to the category of sets, Set{displaystyle mathbf {Set} }. If C{displaystyle {mathcal {C}}} is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object A{displaystyle A} of C{displaystyle {mathcal {C}}} gives rise to a natural functor to Set{displaystyle mathbf {Set} } called a hom-functor. This functor is denoted:hA=Hom(A,\u2212){displaystyle h_{A}=mathrm {Hom} (A,-)}.The (covariant) hom-functor hA{displaystyle h_{A}} sends X{displaystyle X} to the set of morphisms Hom(A,X){displaystyle mathrm {Hom} (A,X)} and sends a morphism f:X\u2192Y{displaystyle fcolon Xto Y} (where X{displaystyle X} and Y{displaystyle Y} are objects in C{displaystyle {mathcal {C}}}) to the morphism f\u2218\u2212{displaystyle fcirc -} (composition with f{displaystyle f} on the left) that sends a morphism g{displaystyle g} in Hom(A,X){displaystyle mathrm {Hom} (A,X)} to the morphism f\u2218g{displaystyle fcirc g} in Hom(A,Y){displaystyle mathrm {Hom} (A,Y)}. That is,hA(f)=Hom(A,f),\u00a0or{displaystyle h_{A}(f)=mathrm {Hom} (A,f),{text{ or}}}hA(f)(g)=f\u2218g{displaystyle h_{A}(f)(g)=fcirc g}Yoneda’s lemma says that:Lemma\u00a0(Yoneda)\u00a0\u2014\u00a0Let F{displaystyle F} be a functor from a locally small category C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} }. Then for each object A{displaystyle A} of C{displaystyle {mathcal {C}}}, the natural transformations Nat(hA,F)\u2261Hom(Hom(A,\u2212),F){displaystyle mathrm {Nat} (h_{A},F)equiv mathrm {Hom} (mathrm {Hom} (A,-),F)} from hA{displaystyle h_{A}} to F{displaystyle F} are in one-to-one correspondence with the elements of F(A){displaystyle F(A)}. That is,Nat(hA,F)\u2245F(A).{displaystyle mathrm {Nat} (h_{A},F)cong F(A).}Moreover, this isomorphism is natural in A{displaystyle A} and F{displaystyle F} when both sides are regarded as functors from C\u00d7SetC{displaystyle {mathcal {C}}times mathbf {Set} ^{mathcal {C}}} to Set{displaystyle mathbf {Set} }.Here the notation SetC{displaystyle mathbf {Set} ^{mathcal {C}}} denotes the category of functors from C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} }.Given a natural transformation \u03a6{displaystyle Phi } from hA{displaystyle h_{A}} to F{displaystyle F}, the corresponding element of F(A){displaystyle F(A)} is u=\u03a6A(idA){displaystyle u=Phi _{A}(mathrm {id} _{A})};[a] and given an element u{displaystyle u} of F(A){displaystyle F(A)}, the corresponding natural transformation is given by \u03a6X(f)=F(f)(u){displaystyle Phi _{X}(f)=F(f)(u)} which assigns to a morphism f:A\u2192X{displaystyle fcolon Ato X} a value of F(X){displaystyle F(X)}.Contravariant version[edit]There is a contravariant version of Yoneda’s lemma, which concerns contravariant functors from C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} }. This version involves the contravariant hom-functorhA=Hom(\u2212,A),{displaystyle h^{A}=mathrm {Hom} (-,A),}which sends X{displaystyle X} to the hom-set Hom(X,A){displaystyle mathrm {Hom} (X,A)}. Given an arbitrary contravariant functor G{displaystyle G} from C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} }, Yoneda’s lemma asserts thatNat(hA,G)\u2245G(A).{displaystyle mathrm {Nat} (h^{A},G)cong G(A).}Naming conventions[edit]The use of hA{displaystyle h_{A}} for the covariant hom-functor and hA{displaystyle 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 with Alexander Grothendieck’s foundational EGA use the convention in this article.[b]The mnemonic “falling into something” can be helpful in remembering that hA{displaystyle h_{A}} is the covariant hom-functor. When the letter A{displaystyle A} is falling (i.e. a subscript), hA{displaystyle h_{A}} assigns to an object X{displaystyle X} the morphisms from A{displaystyle A} into X{displaystyle X}.Proof[edit]Since \u03a6{displaystyle Phi } is a natural transformation, we have the following commutative diagram:This diagram shows that the natural transformation \u03a6{displaystyle Phi } is completely determined by \u03a6A(idA)=u{displaystyle Phi _{A}(mathrm {id} _{A})=u} since for each morphism f:A\u2192X{displaystyle fcolon Ato X} one has\u03a6X(f)=(Ff)u.{displaystyle Phi _{X}(f)=(Ff)u.}Moreover, any element u\u2208F(A){displaystyle uin F(A)} 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 F{displaystyle F} from C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} } is another hom-functor hB{displaystyle h_{B}}. In this case, the covariant version of Yoneda’s lemma states thatNat(hA,hB)\u2245Hom(B,A).{displaystyle 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:B\u2192A{displaystyle fcolon Bto A} the associated natural transformation is denoted Hom(f,\u2212){displaystyle mathrm {Hom} (f,-)}.Mapping each object A{displaystyle A} in C{displaystyle {mathcal {C}}} to its associated hom-functor hA=Hom(A,\u2212){displaystyle h_{A}=mathrm {Hom} (A,-)} and each morphism f:B\u2192A{displaystyle fcolon Bto A} to the corresponding natural transformation Hom(f,\u2212){displaystyle mathrm {Hom} (f,-)} determines a contravariant functor h\u2219{displaystyle h_{bullet }} from C{displaystyle {mathcal {C}}} to SetC{displaystyle mathbf {Set} ^{mathcal {C}}}, the functor category of all (covariant) functors from C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} }. One can interpret h\u2219{displaystyle h_{bullet }} as a covariant functor:h\u2219:Cop\u2192SetC.{displaystyle 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\u2219{displaystyle h_{bullet }} is fully faithful, and therefore gives an embedding of Cop{displaystyle {mathcal {C}}^{mathrm {op} }} in the category of functors to Set{displaystyle mathbf {Set} }. The collection of all functors {hA|A\u2208C}{displaystyle {h_{A}|Ain C}} is a subcategory of SetC{displaystyle mathbf {Set} ^{mathcal {C}}}. Therefore, Yoneda embedding implies that the category Cop{displaystyle {mathcal {C}}^{mathrm {op} }} is isomorphic to the category {hA|A\u2208C}{displaystyle {h_{A}|Ain C}}.The contravariant version of Yoneda’s lemma states thatNat(hA,hB)\u2245Hom(A,B).{displaystyle mathrm {Nat} (h^{A},h^{B})cong mathrm {Hom} (A,B).}Therefore, h\u2219{displaystyle h^{bullet }} gives rise to a covariant functor from C{displaystyle {mathcal {C}}} to the category of contravariant functors to Set{displaystyle mathbf {Set} }:h\u2219:C\u2192SetCop.{displaystyle h^{bullet }colon {mathcal {C}}to mathbf {Set} ^{{mathcal {C}}^{mathrm {op} }}.}Yoneda’s lemma then states that any locally small category C{displaystyle {mathcal {C}}} can be embedded in the category of contravariant functors from C{displaystyle {mathcal {C}}} to Set{displaystyle mathbf {Set} } via h\u2219{displaystyle h^{bullet }}. This is called the Yoneda embedding.The Yoneda embedding is sometimes denoted by \u3088, 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,Nat(hA,P)\u2245P(A){displaystyle mathrm {Nat} (h^{A},P)cong P(A)}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 C{displaystyle mathbf {C} } and D{displaystyle mathbf {D} } with two functors F,G:C\u2192D{displaystyle F,G:mathbf {C} to mathbf {D} }, natural transformations between them can be written as the following end.Nat(F,G)=\u222bc\u2208CHomD(Fc,Gc){displaystyle mathrm {Nat} (F,G)=int _{cin mathbf {C} }mathrm {Hom} _{mathbf {D} }(Fc,Gc)}For any functors K:Cop\u2192Sets{displaystyle Kcolon mathbf {C} ^{op}to mathbf {Sets} } and H:C\u2192Sets{displaystyle Hcolon mathbf {C} to mathbf {Sets} } the following formulas are all formulations of the Yoneda lemma.K\u2245\u222bc\u2208CKc\u00d7HomC(\u2212,c),K\u2245\u222bc\u2208C(Kc)HomC(c,\u2212),{displaystyle Kcong int ^{cin mathbf {C} }Kctimes mathrm {Hom} _{mathbf {C} }(-,c),qquad Kcong int _{cin mathbf {C} }(Kc)^{mathrm {Hom} _{mathbf {C} }(c,-)},}H\u2245\u222bc\u2208CHc\u00d7HomC(c,\u2212),H\u2245\u222bc\u2208C(Hc)HomC(\u2212,c).{displaystyle Hcong int ^{cin mathbf {C} }Hctimes mathrm {Hom} _{mathbf {C} }(c,-),qquad Hcong int _{cin mathbf {C} }(Hc)^{mathrm {Hom} _{mathbf {C} }(-,c)}.}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 \u2014 in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring R{displaystyle R}, the extended category is the category of all right modules over R{displaystyle R}, and the statement of the Yoneda lemma reduces to the well-known isomorphismM\u2245HomR(R,M){displaystyle Mcong mathrm {Hom} _{R}(R,M)}\u00a0\u00a0\u00a0for all right modules M{displaystyle M} over R{displaystyle R}.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 C{displaystyle {mathcal {C}}} be a category with a single object \u2217{displaystyle *} such that every morphism is an isomorphism (i.e. a groupoid with one object). Then G=HomC(\u2217,\u2217){displaystyle G=mathrm {Hom} _{mathcal {C}}(*,*)} 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 C\u2192Set{displaystyle {mathcal {C}}to mathbf {Set} } consists of a set X{displaystyle X} and a group homomorphism G\u2192Perm(X){displaystyle Gto mathrm {Perm} (X)}, where Perm(X){displaystyle mathrm {Perm} (X)} is the group of permutations of X{displaystyle X}; in other words, X{displaystyle X} is a G-set. A natural transformation between such functors is the same thing as an equivariant map between G{displaystyle G}-sets: a set function \u03b1:X\u2192Y{displaystyle alpha colon Xto Y} with the property that \u03b1(g\u22c5x)=g\u22c5\u03b1(x){displaystyle alpha (gcdot x)=gcdot alpha (x)} for all g{displaystyle g} in G{displaystyle G} and x{displaystyle x} in X{displaystyle X}. (On the left side of this equation, the \u22c5{displaystyle cdot } denotes the action of G{displaystyle G} on X{displaystyle X}, and on the right side the action on Y{displaystyle Y}.)Now the covariant hom-functor HomC(\u2217,\u2212){displaystyle mathrm {Hom} _{mathcal {C}}(*,-)} corresponds to the action of G{displaystyle G} on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with F=HomC(\u2217,\u2212){displaystyle F=mathrm {Hom} _{mathcal {C}}(*,-)} states thatNat(HomC(\u2217,\u2212),HomC(\u2217,\u2212))\u2245HomC(\u2217,\u2217){displaystyle mathrm {Nat} (mathrm {Hom} _{mathcal {C}}(*,-),mathrm {Hom} _{mathcal {C}}(*,-))cong mathrm {Hom} _{mathcal {C}}(*,*)},that is, the equivariant maps from this G{displaystyle G}-set to itself are in bijection with G{displaystyle G}. But it is easy to see that (1) these maps form a group under composition, which is a subgroup of Perm(G){displaystyle 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{displaystyle g} in G{displaystyle G} the equivariant map of right-multiplication by g{displaystyle g}.) Thus G{displaystyle G} is isomorphic to a subgroup of Perm(G){displaystyle mathrm {Perm} (G)}, 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\u00a0ed.), Harper and Row, Zbl\u00a00121.02103.Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol.\u00a05 (2nd\u00a0ed.), New York, NY: Springer-Verlag, ISBN\u00a00-387-98403-8, Zbl\u00a00906.18001Loregian, Fosco (2015). “This is the (co)end, my only (co)friend”. arXiv:1501.02503 [math.CT].External links[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/yoneda-lemma-wikipedia\/#breadcrumbitem","name":"Yoneda lemma – Wikipedia"}}]}]