co-Yoneda lemma


Yoneda lemma

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What is sometimes called the co-Yoneda lemma is a basic fact about presheaves (a basic fact of topos theory): it says that every presheaf is a colimit of representables and more precisely that it is the “colimit over itself of all the representables contained in it”.

One might think of this as related by duality to the Yoneda lemma, hence the name.

Every presheaf is a colimit of representables

Throughout, let

For c,dObj(𝒱)c,d \in Obj(\mathcal{V}) two objects, we write 𝒞(c,d)V\mathcal{C}(c,d) \in V for the hom-object.


(Yoneda reduction)

Recall that the (enriched) Yoneda lemma says that for F:𝒞 opVF \colon \mathcal{C}^{op} \to V a VV-enriched functor out of the opposite category of 𝒞\mathcal{C}, hence a VV-valued presheaf on 𝒞\mathcal{C} and c𝒞c \in \mathcal{C} an object of 𝒞\mathcal{C}, there is a natural isomorphism in VV

[𝒞 op,V](𝒞(,c),F)F(c), [\mathcal{C}^{op},V](\mathcal{C}(-,c), F) \simeq F(c) \,,

where on the left we have the hom-object in the enriched functor category between the functor represented by cc and the given functor FF.

Using the expression of these hom-ebjects on the left as ends, this reads

c𝒞V(𝒞(c,c),F(c))F(c). \int_{c' \in \mathcal{C}} V(\mathcal{C}(c',c), F(c')) \simeq F(c) \,.

In this form the Yoneda lemma is also referred to as Yoneda reduction.

Under abstract duality an end turns into a coend, so a co-Yoneda lemma ought to be a similarly fundamental expression for F(c)F(c) in terms of a coend.

The natural candidate is the statement that:


Every presheaf FF is a colimit of representables, in that

F(c) cC𝒞(c,c)F(c) F(c) \simeq \int^{c' \in C} \mathcal{C}(c,c')\otimes F(c')


F() c𝒞Y(c)F(c), F(-) \simeq \int^{c' \in \mathcal{C}} Y(c')\otimes F(c') \,,

where YY denotes the Yoneda embedding. In module-language, using the tensor product of functors, this reads

F(c)𝒞(c,) 𝒞F. F(c) \simeq \mathcal{C}(c,-)\otimes_{\mathcal{C}} F \,.

Yet another way to state this is as a colimit over the comma category (Y,F)(Y,F), for YY the Yoneda embedding:

Fcolim (UF)(Y,F)Y(U). F \simeq colim_{(U \to F) \in (Y,F)} Y(U) \,.

This statement we call the co-Yoneda lemma.


To show that a presheaf F:𝒞 opVF \colon \mathcal{C}^{op} \to V is canonically presented as a colimit of representables, we exhibit a natural isomorphism

cF(c)𝒞(,c)F \int^{c} F(c) \otimes \mathcal{C}(-, c) \;\cong\; F

By the definition of the coend, maps

cF(c)×𝒞(,c)G() \int^c F(c) \times \mathcal{C}(-, c) \to G(-)

are in natural bijection with families of maps

F(c)𝒞(d,c)G(d) F(c) \otimes \mathcal{C}(d, c) \to G(d)

extranatural in cc and natural in dd. Those are in natural bijection with families of maps

F(c)V(𝒞(d,c),G(d)) F(c) \to V(\mathcal{C}(d, c), G(d))

natural in cc and extranatural in dd. These are in natural bijection with families of maps

F(c)Nat(𝒞(,c),G)G(c) F(c) \to Nat(\mathcal{C}(-, c), G) \cong G(c)

(natural in cc), where the isomorphism is by the Yoneda lemma. Thus we have exhibited a natural isomorphism

Nat( cF(c)×𝒞(,c),G)Nat(F,G) Nat(\int^c F(c) \times \mathcal{C}(-, c), G) \cong Nat(F, G)

(natural in GG). By Yoneda again, this gives

cF(c)×𝒞(,c)F. \int^c F(c) \times \mathcal{C}(-, c) \cong F \,.

The statement of the co-Yoneda lemma in prop. 1 is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):

For XX a topological space, f:Xf \colon X \to\mathbb{R} a continuous function and δ(,x 0)\delta(-,x_0) denoting the Dirac distribution, then

xXδ(x,x 0)f(x)=f(x 0). \int_{x \in X} \delta(x,x_0) f(x) = f(x_0) \,.

If one follows the Yoneda-lemma argument at the end of the proof of prop. 1, one arrives at the explicit isomorphism

cF(c)×𝒞(,c)F. \int^c F(c) \times \mathcal{C}(-, c) \to F \,.

Namely, it corresponds to the family of maps

F(c)×𝒞(d,c)F(d) F(c) \times \mathcal{C}(d, c) \to F(d)

(extranatural in cc and natural in dd) which in turn corresponds to the natural family

𝒞(d,c)hom(F(c),F(d)) \mathcal{C}(d, c) \to \hom(F(c), F(d))

associated with the structure of the functor F:𝒞 opVF \colon \mathcal{C}^{op} \to V.


Let V=V = Set, and recall the definition of the coend as a coequalizer

c,d𝒞)𝒞(c,d)×𝒞(c 0,c)×F(d)AAAAAAAAc,dρ (d,c)(c)c,dρ (c,d)(d)c𝒞𝒞(c 0,c)×F(c)coeqc𝒞𝒞(c 0,c)×F(c). \underset{c,d \in \mathcal{C})}{\coprod} \mathcal{C}(c,d) \times \mathcal{C}(c_0,c) \times F(d) \underoverset {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c \in \mathcal{C}}{\coprod} \mathcal{C}(c_0,c) \times F(c) \overset{coeq}{\longrightarrow} \overset{c\in \mathcal{C}}{\int} \mathcal{C}(c_0,c) \times F(c) \,.

This says that the coend is the set of equivalence classes of pairs

(c 0c,xF(c)), ( c_0 \overset{}{\to} c,\; x \in F(c) ) \,,

where two such pairs

(c 0fc,xF(c)),(c 0gd,yF(d)) ( c_0 \overset{f}{\to} c,\; x \in F(c) ) \,,\;\;\;\; ( c_0 \overset{g}{\to} d,\; y \in F(d) )

are regarded as equivalent if there exists

cϕd c \overset{\phi}{\to} d

such that

g=ϕf,andx=ϕ *(y). g = \phi \circ f \,, \;\;\;\;\;and\;\;\;\;\; x = \phi^\ast(y) \,.

(Because then the two pairs are the two images of the pair (f,y)(f,y) under the two morphisms being coequalized.)

But now considering the case that c=c 0c = c_0 and f=id c 0f = id_{c_0}, so that g=ϕg= \phi shows that any pair

(c 0ϕd,yF(d)) ( c_0 \overset{\phi}{\to} d, \; y \in F(d))

is identified, in the coequalizer, with the pair

(id c 0,ϕ *(y)F(c 0)), (id_{c_0},\; \phi^\ast(y) \in F(c_0)) \,,

hence with ϕ *(y)F(c 0)\phi^\ast(y)\in F(c_0), and that this coequalizing operation is the action

Hom(c 0,d)×F(d)F(c 0) Hom(c_0,d)\times F(d)\longrightarrow F(c_0)

of morphisms on elements of the presheaf by pullback.

MacLane’s co-Yoneda lemma

In a brief uncommented exercise on MacLane, p. 62
the following statement, which is atrributed to Kan, is called the co-Yoneda lemma.

For DD a category, Set the category of sets, K:DSetK : D \to Set a functor, let (*K)(* \darr K) be the comma category of elements xKdx \in K d, let Π:(*K)D\Pi: (* \darr K) \to D be the projection (xKd)d(x \in K d) \mapsto d and let for each aDa \in D the functor Δ a:(*K)D\Delta_a: (* \darr K) \to D be the diagonal functor sending everything to the constant value aa.

The co-Yoneda lemma in the sense of Kan/MacLane is the statement that there is a natural isomorphism of functor categories

[D,Set](K,D(a,))[(*K),D](Δ a,Π). [D,Set](K, D(a, -)) \cong [(*\darr K), D](\Delta_a, \Pi).

Here is an outline of an explicit proof:


A natural transformation ϕ:KD(a,)\phi: K \to D(a, -) assigns to each element xKcx \in K c an element ϕ c(x)D(a,c)\phi_c(x) \in D(a, c), i.e., an arrow ϕ c(x):ac\phi_c(x): a \to c. We define a corresponding transformation ψ:Δ aΠ\psi: \Delta_a \to \Pi which assigns to each object (c,xKc)(c, x \in K c) in (*K)(*\darr K) the morphism ϕ c(x):ac=Π(c,x)\phi_c(x): a \to c = \Pi(c, x). It is easy to check that the naturality condition on ϕ\phi corresponds to the naturality condition on ψ\psi, and that the correspondence is bijective.

Here is a more conceptual proof in terms of comma categories:


Set classifies discrete fibrations, in the sense that a functor G:DSetG : D \to Set classifies the discrete fibration

Q:Π G:El(G)D Q : \Pi_G : El(G) \to D

and natural transformations α:GF\alpha : G \to F correspond to maps of fibrations

El(G)El(f) El(G) \to El(f)

i.e. functor which commute on the nose with the projections Π G\Pi_G, Π F\Pi_F to the base category DD).

This applies in particular to F=hom(a,)F = hom(a,-). Notice the category of elements El(hom(a,))El(hom(a,-)) is the co-slice (aD)(a \downarrow D), with its usual projection Π\Pi to DD.

However, the comma category (aD)(a \downarrow D) is the “lax pullback” (really, the comma object, the discussion at 2-limit) appearing in

(aD) Π D Id * a D \array{ (a \downarrow D) &\stackrel{\Pi}{\to}& D \\ \downarrow &\Uparrow& \downarrow^{Id} \\ * &\stackrel{a}{\to}& D }

and so a fibration map El(G)(aD)El(G) \to (a \downarrow D) corresponds exactly to a lax square

El(G) Π G D Id * a D. \array{ El(G) &\stackrel{\Pi_G}{\to}& D \\ \downarrow &\Uparrow& \downarrow^{Id} \\ * &\stackrel{a}{\to}& D } \,.

This yields the co-Yoneda lemma in the sense of MacLane’s exercise.


For enrichment over V=Top cg */V = Top^{\ast/}_{cg} (pointd compactly generated topological spaces) the co-Yoneda lemma in the sense of every presheaf is a colimit of representables appears for instance as

The co-Yoneda lemma in the sense of MacLane appears as a brief uncommented exercise on p. 63 of

where it is atributed to Daniel Kan.

Revised on June 2, 2016 05:08:05 by Urs Schreiber (