The Yoneda Lemma

The Yoneda lemma, according to Ravil Vakil in Foundations of Algebraic Geometry, is an "important exercise you should do once in your life." In this post, we give a slight variant of the usual proof and explain the idea behind the lemma.

Presheaves

In category theory, a presheaf is a mild generalization of the usual geometric notion of a presheaf. Instead of attaching data to each open set of a topological space, we attach data to each object in a category. More precisely, a presheaf on a category $\mathscr{C}$ is a functor $\mathcal{F} \colon \mathscr{C}^{\rm op} \to \mathbf{Set}$. Morphisms between presheaves are defined to be natural transformations of these functors. Together, these form the category of presheaves on $\mathscr{C}$, denoted by $\mathbf{PSh}(\mathscr{C})$.

In the above example, $\mathrm{Hom}_{\mathbf{Open}(X)}(V\!, U)$ has at most one morphism, determining whether or not $V$ is contained in $U$. Using this as intuition for a general category $\mathscr{C}$, we similarly think of $\operatorname{Hom}_{\mathscr{C}}(V, U)$ as comprising all of the ways in which the structure of $V$ can "fit inside" the structure of $U$.

Given a presheaf $\mathcal{F}$ on $\mathscr{C}$, we think of each section over $U$ as a way of fitting the data of $U$ inside of $\mathcal{F}$, much like in the étale space viewpoint. Then for $f \in \operatorname{Hom}(V, U)$, the map $\mathcal{F}(f) \colon \mathcal{F}(U) \to \mathcal{F}(V)$ takes each occurrence of $U$ in $\mathcal{F}$ and "restricts" it to obtain a suboccurrence of $V$ in $\mathcal{F}$ (this is also the intuition suggested in nLab article on presheaves).

In this article, we will call the map $\mathcal{F}(f)$ the pullback by $f$ and denote it by $f^*$ when the presheaf $\mathcal{F}$ is understood.

Representable Presheaves

Let $\mathcal{F} \in \mathbf{PSh}(\mathscr{C})$ be a presheaf and $U \in \mathcal{F}$. Given a section $s \in \mathcal{F}(U)$, what is the minimal subpresheaf containing $s$? To construct this, we intuitively must "throw away" or "cut out" all sections of $\mathcal{F}$ which don't fit inside $s$.

In general, for every $f \in \operatorname{Hom}(V, U)$, we must include the pullback $f^* s$ as a section of the minimal subpresheaf. To formalize this notion, consider the functor $\operatorname{Hom}(-, U)$, which may be interpreted as the presheaf of morphisms to $\hspace{0.5pt}U$. By definition, the sections of this presheaf over an object $V \in \mathscr{C}$ are given by $\operatorname{Hom}(-, U)(V)= \operatorname{Hom}(V, U)$, and the pullback along a morphism $f \in \operatorname{Hom}(V, W)$ is given by

$$f^* \overset{\mathrm{def}}{=} \operatorname{Hom}(-, U)(f) \colon \operatorname{Hom}(W, U) \to \operatorname{Hom}(V, U); \quad\quad f^*s \overset{\mathrm{def}}{=} s \circ f.$$

Given $U \in \mathscr{C}$, let us call a pair $(\mathcal{G}, s)$, where $\mathcal{G}$ is a presheaf on $\mathscr{C}$ and $s \in \mathcal{G}(U)$ a presheaf on $\mathscr{C}$ with a distinguished section on $\,U$. These form a category $\mathbf{PSh}_{\bullet}(\mathscr{C}, U)$ where a morphism between objects $(\mathcal{G}, s)$ and $(\mathcal{H}, t)$ is defined to be a morphism of presheaves $\mathcal{G} \Rightarrow \mathcal{H}$ which maps $s \mapsto t$.

Proof. Let $\mathcal{F}$ be a presheaf with a distinguished section $s \in \mathcal{F}(U)$. Then there exists a morphism $\phi \colon \operatorname{Hom}(-, U) \Rightarrow \mathcal{F}$ given by $f \mapsto f^*s = [\mathcal{F}(f)] (s)$. In other words, $\phi$ maps a section $f \in \operatorname{Hom}(V, U)$ to the pullback of the distinguished section $s$, which is a once again section over $V$ (but in $\mathcal{F}$). To check that this defines a morphism of presheaves, consider a morphism $g \colon V \to W$ in $\mathscr{C}$. Then the diagram

$$\begin{CD} W @. @. \operatorname{Hom}(W,\, U) @>\phi>> \mathcal{F}(W) \\ @AgAA @. @Vg^*VV @Vg^*VV \\ V @. @.\operatorname{Hom}(V,\, U) @>\phi>> \mathcal{F}(V) \end{CD}$$

commutes, since for all $f \in \operatorname{Hom}(W, U)$ we have $g^*(f^*s) = (f \circ g)^*s = (g^* f)^* s$. Moreover, the identity $\operatorname{id}_U$ is mapped to the distinguished section $s$, since ${\operatorname{id}_U}^* s = s$. Hence, $\phi$ defines a morphism $(\operatorname{Hom}(-, U), \operatorname{id}_U) \to ( \mathcal{F}, s)$.

Conversely, to show uniqueness, suppose that $\psi \colon (\operatorname{Hom}(-, U),\, \operatorname{id}_U) \to ( \mathcal{F}, s)$ is a morphism of presheaves with distinguished sections on $U$. Then $\psi_V(f)$ is automatically determined for all sections $f \in \operatorname{Hom}(V, U)$, since

$$\psi_V(f) = \psi_V(f^*\operatorname{id}_U) = f^*\big(\psi_U(\operatorname{id}_U)\big) = f^*s.$$

We call the initial object $\operatorname{Hom}(-, U)$ of $\mathbf{PSh}_{\bullet}(\mathscr{C}, U)$ the presheaf represented by $\,U$ and denote it by $H_U$. In a sense, this is the "least specific" presheaf on $\mathscr{C}$ which contains a section over $U$. The map taking an object to the presheaf it represents is known as the Yoneda embedding, which we denote by $\!\text{よ}$.

Proof. Let $W \overset{g}{\to} V \overset{f}{\to} U$ be a sequence of morphisms. Then

$$\mathrm{id}_W \overset{\!\text{よ}(g)_W}{\longmapsto} (g = g^*\operatorname{id}_V) \overset{\!\text{よ}(f)_V}{\longmapsto} (g^*f = f \circ g).$$

By uniqueness of maps from initial objects, it follows that $\!\text{よ}(f \circ g) = \!\text{よ}(f) \circ \!\text{よ}(g)$.

Explicitly, $\!\text{よ}(f)$ is the pushforward along $f$, taking $s \in \mathrm{Hom}(W, V)$ and mapping it to $f_*s = f \circ s \in \mathrm{Hom}(W, U)$. Indeed, the pushforward commutes with pullbacks, making it into a morphism of presheaves mapping the identity section over $V$ to $f$; by definition, this must be the Yoneda embedding.

The Yoneda Lemma

We now state and prove the Yoneda lemma.

Proof. For any section $s \in \mathcal{F}(U)$, there exists a unique morphism $\phi \colon H_U \to \mathcal{F}$ such that $\phi_U(\operatorname{id}_U) = s$, since $(H_U, \operatorname{id}_U)$ is initial in $\mathbf{PSh}_{\bullet}(\mathscr{C}, U)$. It follows that the evaluation map $\mathrm{ev}_{U,\, \mathcal{F}}$ is a bijection.

For fixed $\mathcal{F}$, the map $U \mapsto \mathcal{F}(U)$ is functorial by the definition of a presheaf. Likewise, for fixed $U$, the map $\mathcal{F} \mapsto \mathcal{F}(U)$ is functorial by the definition of the presheaf category. Thus, $(U, \mathcal{F}) \mapsto \mathcal{F}(U)$ defines a bifunctor $\mathscr{B}_{\rm pairing}$.

On the other hand, the Yoneda embedding $U \mapsto H_U$ is a functor, and $\operatorname{Hom}_{\mathbf{Psh}(\mathscr{C})}(-, -)$ is a bifunctor, which implies that $(U, \mathcal{F}) \mapsto \operatorname{Hom}_{\mathbf{PSh}(\mathscr{C})}(H_U, \mathcal{F})$ also defines a bifunctor $\mathscr{B}_{\rm yoneda}$.

It remains to show that the two bifunctors are naturally isomorphic. For this, it suffices to check that they are natural on each factor. So, let $f \in \operatorname{Hom}(V, U)$. Then for all presheaf morphisms $\phi \in \operatorname{Hom}_{\mathbf{PSh}(\mathscr{C})}(H_U, \mathcal{F})$, we have

$$\mathrm{ev}_{V,\,\mathcal{F}} \Big( \big[\mathscr{B}_{\rm yoneda}(f, \operatorname{id}_{\mathcal{F}})\big] (\phi) \Big) = \mathrm{ev}_{V,\,\mathcal{F}}\big(\phi \circ \!\text{よ}(f)\big) = \phi_V\Big([\!\text{よ}(f)]_V (\operatorname{id}_V)\Big) = \phi_V(f).$$

At the same time, we have

$$\big[\mathscr{B}_{\rm pairing}(f, \operatorname{id}_\mathcal{F})\big] \big(\mathrm{ev}_{U, \,\mathcal{F}} (\phi)\big) = \big[\mathscr{B}_{\rm pairing}(f, \operatorname{id}_\mathcal{F})\big] \big(\phi_U(\operatorname{id}_U)\big) = f^*\phi_U(\operatorname{id}_U) = \phi_V(f).$$

Therefore, the bifunctor isomorphism is natural in the first factor. For the second factor, consider a presheaf morphism $\psi \colon \mathcal{F} \Rightarrow \mathcal{G}$. Then for all presheaf morphisms $\phi \in \operatorname{Hom}_{\mathbf{PSh}(\mathscr{C})}(H_U, \mathcal{F})$, we have

$$\mathrm{ev}_{U,\,\mathcal{G}} \Big( \big[\mathscr{B}_{\rm yoneda}(\operatorname{id}_{U}, \psi)\big] (\phi) \Big) = \mathrm{ev}_{U,\,\mathcal{G}} \big(\psi \circ \phi) = \big(\psi_U \circ \phi_U\big)(\operatorname{id}_U) .$$

At the same time, we have

$$\big[\mathscr{B}_{\rm pairing}(\operatorname{id}_U, \psi)\big] \big(\mathrm{ev}_{U, \mathcal{F}} (\phi)\big) = \big[\mathscr{B}_{\rm pairing}(\operatorname{id}_U, \psi)\big] \big(\phi_U(\operatorname{id}_U)\big) = \psi_U(\phi_U(\operatorname{id}_U)) .$$

Therefore, the bifunctor isomorphism is also natural in the second factor. It follows that the evaluation bifunctor $\mathrm{ev}$ is natural, which concludes the argument.

The Yoneda Embedding

The Yoneda lemma implies that there is an isomorphism

$$\mathrm{ev}_{U,\, V}\colon \operatorname{Hom}_{\mathbf{PSh}(\mathscr{C})}(H_U, H_V) \overset{\cong}{\longrightarrow} \operatorname{Hom}(U, V)$$

which implies that the Yoneda embedding functor is fully faithful. In particular, an isomorphism of representable presheaves in $\mathbf{PSh}(\mathscr{C})$ must be induced by an isomorphism of their representing objects in $\mathscr{C}$. This can expressed as the following principle, whose importance cannot be understated:

To understand an object $X$ in a category $\mathscr{C}$, it suffices to understand how $X$ relates to every other object in $\mathscr{C}$.