nLab Yoneda embedding




For CC a locally small category, every object XX of CC induces a presheaf on CC: the representable presheaf h Xh_X represented by XX. This assignment extends to a functor C[C op,Set]C \to [C^{op}, Set] from CC to its category of presheaves. The Yoneda lemma implies that this functor is full and faithful and hence realizes CC as a full subcategory inside its category of presheaves.

Recall from the discussion at representable presheaf that the presheaf represented by an object XX of CC is the functor h X:C opSeth_X :C^{op} \to Set whose assignment is illustrated by

which sends each object UU to Hom C(U,X)Hom_C(U,X) and each morphism α:UU\alpha:U'\to U to the function

h Xα:Hom C(U,X)Hom C(U,X).h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).

Moreover, for f:XYf : X \to Y a morphism in CC, this induces a natural transformation h f:h Xh Yh_f : h_X \to h_Y, whose component on UU in XX is illustrated by

For this to be a natural transformation, we need to have the commuting diagram

h XU h fU h YU h Xα h Yα h XU h fU h YU\array{ h_X U & \stackrel{h_f U}{\rightarrow} & h_Y U \\ \mathllap{h_X\alpha\quad}{\downarrow} & {} & \mathrlap{\downarrow}{\quad h_Y\alpha} \\ h_X U' & \stackrel{h_f U'}{\rightarrow} & h_Y U' }

but this simply means that it doesn’t matter if we first “comb” the strands back to UU' and then comb the strands forward to YY, or comb the strands forward to YY first and then comb the strands back to UU'

which follows from associativity of composition of morphisms in CC.


The Yoneda embedding for CC a locally small category is the functor

Y:C[C op,Set] Y : C \to [C^{op}, Set]

from CC to the category of presheaves over CC which is the image of the hom-functor

Hom:C op×CSet Hom : C^{op} \times C \to Set

under the Hom adjunction

Hom(C op×C,Set)Hom(C,[C op,Set]) Hom(C^{op} \times C , Set) \simeq Hom(C, [C^{op}, Set])

in the closed symmetric monoidal category Cat.

Hence YY sends any object cCc \in C to the representable presheaf which assigns to any other object dd of CC the hom-set of morphisms from dd into cc:

Y(c):C opC(,c)Set. Y(c) \;\colon\; C^{op} \stackrel{C(-,c)}{\to} Set \,.

The Yoneda embedding is sometimes denoted by よ, the hiragana kana for “Yo”; see the references below.


We can also curry the Hom functor in the other variable, thus obtaining a contravariant functor

C op[C,Set] C^{op} \to [C, Set]

which is explicitly given by cC(c,)c \mapsto C(c,-). This is sometimes jokingly called the contravariant Yoneda embedding.

However, since C op(,c)=C(c,)C^{op}(-,c)=C(c,-), it is easy to see that the contravariant Yoneda embedding is just the Yoneda embedding Y:C op[(C op) op,Set]=[C,Set]Y: C^{\op} \to [(C^{op})^{op}, Set]=[C, Set] of C opC^{op}, and hence does not require special treatment.



(Yoneda embedding is a fully faithful functor)

For 𝒞\mathcal{C} any category, the functor

𝒞 Y [𝒞 op,Set] c Hom 𝒞(,c) \array{ \mathcal{C} &\overset{Y}{\hookrightarrow}& [\mathcal{C}^{op}, Set] \\ c &\mapsto& Hom_{\mathcal{C}}(-,c) }

is fully faithful.


We need to show that for c 1,c 2𝒞c_1, c_2 \in \mathcal{C} any two objects, we have that every morphism of presheaves between their represented presheaves

Hom 𝒞(,c 1)ϕHom 𝒞(,c 2) Hom_{\mathcal{C}}(-,c_1) \overset{\phi}{\longrightarrow} Hom_{\mathcal{C}}(-,c_2)

is of the form

ϕ=Hom 𝒞(,f) \phi \;=\; Hom_{\mathcal{C}}(-,f)

for a unique morphism

f:c 1c 2 f \;\colon\; c_1 \to c_2

in 𝒞\mathcal{C}. This follows by the Yoneda lemma, which says that morphisms ϕ\phi as above are identified with the elements in

Hom 𝒞(,c 2)(c 1)=Hom 𝒞(c 1,c 2). Hom_{\mathcal{C}}(-,c_2)(c_1) \;=\; Hom_{\mathcal{C}}(c_1,c_2) \,.

It is also limit preserving (= continuous functor), but does in general not preserve colimits.

The Yoneda embedding of a small category SS into the category of presheaves on SS gives a free cocompletion of SS.

If the Yoneda embedding of a category has a left adjoint, then that category is called a total category .



Early accounts:

  • Alexander Grothendieck, Section A.1 of: Technique de descente et théorèmes d’existence en géométrie algébriques. II. Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p. (numdam:SB_1958-1960__5__369_0)

For more see at Yoneda lemma the list of references given there.


It seems that the notation “よ” for the Yoneda embedding (the hiragana kana for “Yo”) was first used in

  • Theo Johnson-Freyd, Claudia Scheimbauer, p. 33 of (Op)lax natural transformations, twisted quantum field theories, and “even higher” Morita categories, (arxiv:1502.06526)

Their Latex code for the command reads as follows:



\DeclareFontShape{U}{min}{m}{n}{<-> udmj30}{}


Subsequent references that use this notation include:

Last revised on September 25, 2022 at 15:32:33. See the history of this page for a list of all contributions to it.