category theory

Yoneda lemma

Contents

Idea

The Yoneda lemma says that the set of morphisms from a representable presheaf $y(c)$ into an arbitrary presheaf $X$ is in natural bijection with the set $X(c)$ assigned by $X$ 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 functors, universal constructions, and universal elements.

Statement and proof

Definition

(functor underlying the Yoneda embedding)

For $\mathcal{C}$ a locally small category we write

$[C^{op}, Set] \coloneqq Func(C^{op}, Set)$

for the functor category out of the opposite category of $\mathcal{C}$ into Set.

This is also called the category of presheaves on $\mathcal{C}$. Other notation used for it includes $Set^{C^{op}}$ or $Hom(C^{op},Set))$.

There is a functor

$\array{ C &\overset{y}{\longrightarrow}& [C^op,Set] \\ c &\mapsto& Hom_{\mathcal{C}}(-,c) }$

(called the Yoneda embedding for reasons explained below) from $\mathcal{C}$ to its category of presheaves, which sends each object to the hom-functor into that object, also called the presheaf represented by $c$.

Remark

(Yoneda embedding is adjunct of hom-functor)

The Yoneda embedding functor $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ from Def. is equivalently the adjunct of the hom-functor

$Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set$
$Hom(C^{op} \times C, Set) \stackrel{\simeq}{\to} Hom(C, [C^{op}, Set])$

in the closed symmetric monoidal category of categories.

Proposition

(Yoneda lemma)

Let $\mathcal{C}$ be a locally small category, with category of presheaves denoted $[\mathcal{C}^{op},Set]$, according to Def. .

For $X \in [\mathcal{C}^{op}, Set]$ any presheaf, there is a canonical isomorphism

$Hom_{[C^op,Set]}(y(c),X) \;\simeq\; X(c)$

between the hom-set of presheaf homomorphisms from the representable presheaf $y(c)$ to $X$, and the value of $X$ at $c$.

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 natural transformation $\eta: C(-, c) \to X$ (hence a homomorphism of presheaves):

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

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

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

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 $y \colon C \to [C^op,Set]$ 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 $X \colon C^{op} \to Set$ is representable precisely if the comma category $(y,const_X)$ has a terminal object. If a terminal object is $(d, g : y(d) \to X) \simeq (d, g \in X(d))$ 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 $X(-)(f) \colon C(c,d) \to X(c)$ 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:

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 $B$).

On the other hand, there have been examples of locally finite categories where naturality is not necessary. For example, (Lovász, Theorem 3.6 (iv)) states precisely that finite relational structures $A$ and $B$ are isomorphic if, and only if, $Hom(C,A) \cong Hom(C,B)$ for every finite relational structure $C$. Later (Pultr, Theorem 2.2) generalised the result to finitely well-powered, locally finite categories with (extremal epi, mono) factorization system.

The Yoneda lemma in semicategories

An interesting phenomenon arises in the case of semicategories i.e. “categories” (possibly) lacking identity morphisms:

the Yoneda lemma fails in general, since its validity in a semicategory $\mathcal{G}$ implies that $\mathcal{G}$ is in fact already a category because the Yoneda lemma permits to embed $\mathcal{G}$ into $PrSh(\mathcal{G})$ and the latter is always a category, the embedding then implying that $\mathcal{G}$ is itself a category!

But for regular semicategories $\mathcal{R}$ there is a unity of opposites in the category of all semipresheaves on $\mathcal{R}$ between the so called regular presheaves that are colimits of representables and presheaves satisfying the Yoneda lemma, whence the Yoneda lemma holds dialectically for regular presheaves!

For some of the details see at regular semicategory and the references therein.

Applications

For general references see any text on category theory, as listed in the references there.

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

In Categories for the Working Mathematician MacLane writes that this happened in 1954. Review and exposition:

• 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)

• Emily Riehl, Category Theory in Context. Chapter 2. Universal Properties, Representability, and the Yoneda Lemma pdf

• Marie La Palme Reyes, Gonzalo E. Reyes, and Houman Zolfaghari, Generic figures and their glueings: A constructive approach to functor categories, Polimetrica sas, 2004 (author page,pdf).

• Paolo Perrone, Notes on Category Theory with examples from basic mathematics, Chapter 2. (arXiv)

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).

A treatment of the Yoneda lemma for categories internal to an (∞,1)-topos is in

• Louis Martini, Yoneda’s lemma for internal higher categories, (arXiv:2103.17141)

Early Lovász-Type results include

• László Lovász, Operations with structures, Acta Mathematica Academiae Scientiarum Hungarica 18.3-4 (1967): 321-328.
• Aleš Pultr. Isomorphism types of objects in categories determined by numbers of morphisms, Acta Scientiarum Mathematicarum, 35:155–160, 1973.