nLab co-Yoneda lemma

Idea

What is sometimes called the co-Yoneda lemma or density formula 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

• $V$ be a closed symmetric monoidal category (for instance $V =$ Set with its Cartesian product);

• $\mathcal{C}$ a small $V$-enriched category.

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

Remark

(Yoneda reduction)

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

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

where on the left we have the hom-object in the enriched functor category between the functor represented by $c$ and the given functor $F$.

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

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

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 $F(c)$ in terms of a coend.

The natural candidate is the statement that:

Proposition

Every presheaf $F$ is a colimit of representables, in that

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

hence

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

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

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

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

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

This statement we call the co-Yoneda lemma.

Proof

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

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

By the definition of the coend, maps

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

are in natural bijection with families of maps

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

extranatural in $c$ and natural in $d$. Those are in natural bijection with families of maps

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

natural in $c$ and extranatural in $d$. These are in natural bijection with families of maps

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

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

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

(natural in $G$). By Yoneda again, this gives

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

Remark

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

For $X$ a topological space, $f \colon X \to\mathbb{R}$ a continuous function and $\delta(-,x_0)$ denoting the Dirac distribution, then

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

Remark

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

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

Namely, it corresponds to the family of maps

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

(extranatural in $c$ and natural in $d$) which in turn corresponds to the natural family

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

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

Example

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

$$\underset{c,d \in \mathcal{C})}{\coprod} \mathcal{C}(c,d) \times \mathcal{C}(c_0,c) \times F(d) \underoverset {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c \in \mathcal{C}}{\coprod} \mathcal{C}(c_0,c) \times F(c) \overset{coeq}{\longrightarrow} \overset{c\in \mathcal{C}}{\int} \mathcal{C}(c_0,c) \times F(c) \,.$$

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

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

where two such pairs

$$( c_0 \overset{f}{\to} c,\; x \in F(c) ) \,,\;\;\;\; ( c_0 \overset{g}{\to} d,\; y \in F(d) )$$

are regarded as equivalent if there exists

$$c \overset{\phi}{\to} d$$

such that

$$g = \phi \circ f \,, \;\;\;\;\;and\;\;\;\;\; x = \phi^\ast(y) \,.$$

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

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

$$( c_0 \overset{\phi}{\to} d, \; y \in F(d))$$

is identified, in the coequalizer, with the pair

$$(id_{c_0},\; \phi^\ast(y) \in F(c_0)) \,,$$

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

$$Hom(c_0,d)\times F(d)\longrightarrow F(c_0)$$

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 attributed to Kan, is called the co-Yoneda lemma.

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

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

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

Here is an outline of an explicit proof:

Proof

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

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

Proof

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

$$Q : \Pi_G : El(G) \to D$$

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

$$El(G) \to El(F)$$

i.e. functor which commute on the nose with the projections $\Pi_G$, $\Pi_F$ to the base category $D$).

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

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

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

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

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

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

References

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

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, lemma 1.6 of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)

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

• Saunders MacLane, Categories for the Working Mathematician,

where it is attributed to Daniel Kan.

See also:

• Tom Leinster, MO:a/20451

• Fosco Loregian, Prop. 2.2.1 in: Coend calculus, Cambridge University Press 2021 (arXiv:1501.02503, doi:10.1017/9781108778657, ISBN:9781108778657).