nLab
Yoneda lemma

Idea

The Yoneda lemma says that the set of morphisms from a representable presheaf $h_c$ into an arbitrary presheaf $F$ is in natural bijection with the set $F(c)$ assigned by $F$ to the representing object $c$.

The Yoneda lemma is an elementary but deep and central result in category theory and in particular in sheaf and topos theory. It is essential background behind the central concepts of representable functor, universal construction, and universal element.

Preliminaries

Recall that for $C$ a locally small category and $[C^{op}, Set] (= Set^{C^{op}} = Hom(C^{op},Set))$ the category of presheaves on $C$, there naturally is a functor

$$Y : C \to [C^op,Set]$$

– called the Yoneda embedding for reasons explained below – which sends $C$ to the category of presheaves over it: this is just the image of the Hom-functor

$$C(-,-) : C^op \times C \to Set$$

under the Hom-adjunction

$$Hom(C^{op} \times C, Set) \stackrel{\simeq}{\to} Hom(C, [C^{op}, Set])$$

in the closed symmetric monoidal category of categories.

Hence $Y$ sends any object $c \in C$ to the presheaf which assigns to any other object $d$ of $C$ the set of morphisms from $d$ into $c$:

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

Remarks

One way to appreciate the meaning of this and of what the Yoneda lemma has to say about it is to regard this in the context of space and quantity: thinking of the objects of $C$ as test spaces, presheaves on $C$ are generalized spaces modeled on $C$ which are characterized by the way one can map objects of $C$ into them.

The Yoneda lemma states that the functor $Y$ has good properties which make this interpretation consistent.

The Yoneda Lemma

Let $C$ be a locally small category, $[C^{op}, Set]$ the category of presheaves on $C$. Let $c \in C$ be an object.

The Yoneda lemma asserts that the set of morphisms from the presheaf represented by $c$ into any other presheaf $X$ is in natural bijection with the set $X(c)$ that this presheaf assigns to $c$.

Formally:

Proposition

There is a canonical isomorphism

$$[C^op,Set](C(-,c),X) \simeq X(c)$$

natural in $c$.

Here $[C^{op}, Set]$ denotes the functor category, also denoted $Set^{C^{op}}$ and $C(-,c)$ the representable presheaf. This is the standard notation used mostly in pure category theory and enriched category theory. In other parts of the literature it is customary to denote the presheaf represented by $c$ as $h_c$. In that case the above is often written

$$Hom(h_c, X) \simeq X(c)$$

or

$$Nat(h_c, X) \simeq X(c)$$

to emphasize that the morphisms of presheaves are natural transformations of the corresponding functors.

Proof

The proof is by chasing the element $Id_c \in C(c, c)$ around both legs of a naturality square for a transformation $\eta: C(-, c) \to X$:

$$\array{ C(c, c) & \stackrel{\eta_c}{\to} & X(c) & & & & Id_c & \mapsto & \eta_c(Id_c) & \stackrel{def}{=} & \xi \\ _\mathllap{C(f, c)} \downarrow & & \downarrow _\mathrlap{X(f)} & & & & \downarrow & & \downarrow _\mathrlap{X(f)} & & \\ C(b, c) & \underset{\eta_b}{\to} & X(b) & & & & f & \mapsto & \eta_b(f) & & }$$

What this diagram shows is that the entire transformation $\eta: C(-, c) \to X$ is completely determined from the single value $\xi \coloneqq \eta_c(Id_c) \in X(c)$, because for each object $b$ of $C$, the component $\eta_b: C(b, c) \to X(b)$ must take an element $f \in C(b, c)$ (i.e., a morphism $f: b \to c$) to $X(f)(\xi)$, according to the commutativity of this diagram.

The crucial point is that the naturality condition on any natural transformation $\eta : C(-,c) \Rightarrow X$ is sufficient to ensure that $\eta$ is already entirely fixed by the value $\eta_c(Id_c) \in X(c)$ of its component $\eta_c : C(c,c) \to X(c)$ on the identity morphism $Id_c$. And every such value extends to a natural transformation $\eta$.

More in detail, the bijection is established by the map

$$[C^{op}, Set](C(-,c),X) \stackrel{|_{c}}{\to} Set(C(c,c), X(c)) \stackrel{ev_{Id_c}}{\to} X(c)$$

where the first step is taking the component of a natural transformation at $c \in C$ and the second step is evaluation at $Id_c \in C(c,c)$.

The inverse of this map takes $f \in X(c)$ to the natural transformation $\eta^f$ with components

$$\eta^f_d := X(-)(f) : C(d,c) \to X(d) \,.$$

Remarks

In the light of the interpretation in terms of space and quantity mentioned above this says that for $X$ a generalized space modeled on $C$, and for $c$ a test space, morphisms from $c$ to $X$ with $c$ regarded as a generalized space are just the morphisms from $c$ into $X$.

Corollaries

The Yoneda lemma has the following direct consequences. As the Yoneda lemma itself, these are as easily established as they are useful and important.

corollary I: Yoneda embedding

The Yoneda lemma implies that the Yoneda embedding functor $Y : C \to [C^op,Set]$ really is an embedding in that it is a full and faithful functor, because for $c,d \in C$ it naturally induces the isomorphism of Hom-sets.

$$[C^{op},Set](C(-,c),C(-,d)) \simeq (C(-,d))(c) = C(c,d)$$

corollary II: uniqueness of representing objects

Since the Yoneda embedding is a full and faithful functor, an isomorphism of representable presheaves $Y(c) \simeq Y(d)$ must come from an isomorphism of the representing objects $c \simeq d$:

$$Y(c) \simeq Y(d) \;\; \Leftrightarrow \;\; c \simeq d$$

corollary III: universality of representing objects

A presheaf $X : C^{op} \to Set$ is representable precisely if the comma category $(Y,const_X)$ has a terminal object. If a terminal object is $(d, f : Y(d) \to X) \simeq (d, f \in X(d))$ then $X \simeq Y(d)$.

This follows from unwrapping the definition of morphisms in the comma category $(Y,const_X)$ and applying the Yoneda lemma to find

$$(Y,const_X)((c,f \in X(c)), (d, g \in X(d))) \simeq \{ u \in C(c,d) : X(u)(g) = f \} \,.$$

Hence $(Y,const_X)((c,f \in X(c), (d, g \in X(d))) \simeq pt$ says precisely that $X(-)(f) : C(c,d) \to X(c)$ is a bijection.

Interpretation

For emphasis, here is the interpretation of these three corollaries in words:

• corollary I says that the interpretation of presheaves on $C$ as generalized objects probeable by objects $c$ of $C$ is consistent: the probes of $X$ by $c$ are indeed the maps of generalized objects from $c$ into $X$;

• corollary II says that probes by objects of $C$ are sufficient to distinguish objects of $C$: two objects of $C$ are the same if they have the same probes by other objects of $C$.

• corollary III characterizes representable functors by a universal property and is hence the bridge between the notion of representable functor and universal constructions.

Generalizations

The Yoneda lemma tends to carry over to all important generalizations of the context of categories:

• There is an analog of the Yoneda lemma in enriched category theory. See enriched Yoneda lemma.

• In the context of modules (see also Day convolution) the Yoneda lemma becomes the important statement of Yoneda reduction, which identifies the bimodule $\hom_C(-, -)$ as a unit bimodule.

• There is a Yoneda lemma for bicategories.

• There is a Yoneda lemma for (∞,1)-categories.

• In functional programming, the Yoneda embedding is related to the continuation passing style transform.

• Formulation of the lemma in dependent type theory: A type theoretical Yoneda lemma at homotopytypetheory.org

Related constructions

• Yoneda reduction

• co-Yoneda lemma

• Brown representability theorem

• continuation-passing style

Necessity of naturality

The assumption of naturality is necessary for the Yoneda lemma to hold. A simple counter-example is given by a category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \mathbb{Z}_{\geq 0}$, the set of integers greater than or equal to $0$, in which $Hom(B,A) = \mathbb{Z}_{\geq 1}$, the set of integers greater than or equal to $1$, and in which composition is addition. Here it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is greater than or equal to $1$, so cannot have an inverse, since $0$ is the identity on $A$ and $B$).

A finite counter-example is given by the category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \{0, 1\}$, in which $Hom(B,A) = \{0, 2\}$, and composition is multiplication modulo 2. Here, again, it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is $0$, so cannot have an inverse, since $1$ is the identity on $A$ and $B$).

Applications

• The Yoneda lemma is the or a central ingredient in various reconstruction theorems, such as those of Tannaka duality. See there for a detailed account.

• In its incarnations as Yoneda reduction the Yoneda lemma governs the algebra of ends and coends and hence that of bimodules and profunctors.

• The Yoneda lemma is effectively the reason that Isbell conjugation exists. This is a fundamental duality that relates geometry and algebra in large part of mathematics.

References

The term Yoneda lemma originated in an interview of Nobuo Yoneda by Saunders Mac Lane at Paris Gare du Nord:

• Yoshiki Kinoshita, posting to catlist in 1996.

In Categories for the Working Mathematician MacLane writes that this happened in 1954.

Reviews and expositions include

• Saunders MacLane, The Yoneda Lemma, Mathematica Japonicae 47: 156 (1998)

• Tom Leinster, The Yoneda Lemma: What's it all about?

A discussion of the Yoneda lemma from the point of view of universal algebra is in

• Vaughan Pratt, The Yoneda lemma without category theory: algebra and applications (pdf).