The functor of points

The goal of this post is to give a rapid introduction to the functor of points in algebraic geometry.

The basis for the functor of points in the Yoneda lemma, which we will recall and prove now. I'm going to disregard all ``size concerns" related to the categories in question, because thinking of such things is not particularly fun (in my opinion).

Let \(\mathsf{C}\) be a category, let \(A \in \text{Obj}(\mathsf{C})\) and denote \(h_A(-) = \text{Hom}(A, -)\), which we can think of as a functor \(h_A : \mathsf{C} \to \mathsf{Set}\) if we define \(h_A(f)(g) = f \circ g\). Recall that if \(f : B \to C\), then \(h_A(f)\) is a set map from \(\text{Hom}(A, B)\) to \(\text{Hom}(A, C)\). Thus, we are simply taking \(g \in \text{Hom}(A, B)\) as post-composing to create an element of \(\text{Hom}(A, C)\). We can also consider the assignment \(A \mapsto h_A\), going from \(\mathsf{C}\) to the category of functors from \(\mathsf{C}\) to \(\mathsf{Set}\). Denote this latter category \(\mathsf{Fun}(\mathsf{C}, \mathsf{Set})\).

Lemma (Yoneda, basic version). If \(F \in \text{Obj}(\mathsf{Fun}(\mathsf{C}, \mathsf{Set}))\) is a functor, then for any \(A \in \text{Obj}(\mathsf{C})\), there is a bijection from \(\text{Hom}(h_A, F)\), the set of morphism is the functor category (i.e. natural transformations) to \(F(A)\), which is of course an object in \(\mathsf{Set}\).

Proof. We will define this bijection explicitly. A natural transformation \(\eta : h_A \to F\) is described by the data of \(\eta(B) : h_A(B) = \text{Hom}(A, B) \to F(B)\) for \(B \in \text{Obj}(\mathsf{C})\) which are natural with respect to arrows. If \(f \in \text{Hom}(A, B)\), then \(F(f)\) goes from \(F(A)\) to \(F(B)\), and some \(a \in F(A)\) yields \(F(f)(a) \in F(B)\). This leads to a natural choice: we define \(\Phi_a(B)(f) = F(f)(a)\). To verify that this is a natural transformation, we require that for some \(f : B \to C\) in \(\mathsf{C}\), we have commutatitivty of the diagram:

\begin{equation} \begin{CD} \text{Hom}(A, B) @> \Phi_a(B) >> F(B) \\ @V f \circ VV @V F(f) VV \\ \text{Hom}(A, C) @> \Phi_a(C) >> F(C) \end{CD} \end{equation}

which follows from the fact that

\begin{equation} \Phi_a(C)(f \circ g) = F(f \circ g)(a) = F(f)(F(g)(a)) = F(f)(\Phi_a(B)(g)) \end{equation}

On the other hand, we may also define a map in the other direction in which \(\eta : h_A \to F\) is taken to \(\eta(A)(\text{id}_A)\). Notice immediately that if \(b = \eta(A)(\text{id}_A)\), then

\begin{equation} \Phi_b(B)(f) = F(f)(b) = (F(f) \circ \eta(A))(\text{id}_A) = \eta(B)(f \circ \text{id}_A) = \eta(B)(f) \end{equation}

and

\begin{equation} \Phi_a(A)(\text{id}_A) = F(\text{id}_A)(a) = a \end{equation}

Thus, we have the desired bijection.

Remark. There is also a contravariant version of Yoneda: if \(F\) is a contravariant functor, then we obtain \(\text{Hom}(h^A, F) \simeq F(A)\), where \(h^A(-) = \text{Hom}(-, A)\) is a contravariant functor defined in the obvious way. As immediate corollaries, we get

\begin{equation} h_B(A) = \text{Hom}(A, B) \simeq \text{Hom}(h_A, h_B) \ \ \ \ \ \text{and} \ \ \ \ \ h^B(A) = \text{Hom}(B, A) \simeq \text{Hom}(h^A, h^B) \end{equation}

In this particular case, one can show that the assignment defined above gives a fully faithful functor from \(\mathsf{C}\) to \(\mathsf{Fun}(\mathsf{C}, \mathsf{Set})\). In particular, object \(A\) is sent to \(h_A\) and arrow \(f : A \to B\) is sent to \(\Phi_f\), a natural transformation from \(h_A\) to \(h_B\) in which \(\Phi_f(C) : \text{Hom}(C, A) \to \text{Hom}(C, B)\) is the map where \(g : C \to A\) is taken to \(f \circ g : C \to B\). Clearly, \(f \mapsto \Phi_f\) satisfies \(\Phi_{f \circ g} = \Phi_f \circ \Phi_g\) and \(\Phi_{\text{id}} = \text{id}\). Thus,

Corollary. There is a fully faithful functor \(\mathsf{C} \to \mathsf{Fun}(\mathsf{C}, \mathsf{Set})\) (we call it the Yoneda functor and denote it by \(\mathcal{Y}\)). The essential image of the Yoneda functor is called the collection of "representable functors on \(\mathsf{C}\)". There is a similar contravariant Yoneda functor as well.

The functor of points is nothing more than the Yoneda lemma applied to schemes. Let \(X\) be a scheme, recall that an \(R\) -point for some ring \(R\) is a morphism \(\text{Spec}(R) \to X\). Of course, we also can send ring morphism \(\phi : R \to S\) to \(\text{Spec}(\phi) : \text{Spec}(S) \to \text{Spec}(R)\) and then pre-compose to obtain an \(S\) -point of \(X\). It is clear that this defines a functor from \(\mathsf{CRing}\) to \(\mathsf{Set}\) which we call the functor of points of \(X\). There is also the extended functor of points of \(X\), where we have arbitrary scheme \(T\) and consider a morphism \(T \to X\) and \(\phi : S \to T\) taking \(T \to X\) to \(S \to T \to X\) via pre-composition. This is exactly the contravariant functor \(h^X : \mathsf{Sch} \to \mathsf{Set}\) defined above, so Yoneda's lemma tells us immediately that we have a natural bijection

\begin{equation} \text{Hom}(h^X, h^Y) \simeq \text{Hom}(Y, X) \end{equation}

Let \(X : \mathsf{CRing} \to \mathsf{Set}\) denote the functor of points of \(X\). As it turns out, we also have natural bijections \(\text{Hom}(X(-), Y(-)) \simeq \text{Hom}(X, Y)\). Via the fully faithful contravariant \(\text{Spec}\) functor from \(\mathsf{CRing} \to \mathsf{AffSch}\), this reduces to showing that any natural transformation between \(h^X\) and \(h^Y\), both functors \(\mathsf{Sch} \to \mathsf{Set}\), is uniquely determined by the full subcategory \(\mathsf{AffSch}\). To this end, given a scheme \(T\), let \(T_i\) form an affine cover, we have inclusions \(T_i \to T\).

Suppose we are provided the data of arrows \(\eta(B) : \text{Hom}(B, X) \to \text{Hom}(B, Y)\), such that we have naturality in \(B\) (any scheme). Given some \(f : T \to X\), we can restrict to affines \(f_i = f|_{T_i}\) and note that we obtain a collection \(\eta(T_i)(f_i)\) of morphisms \(T_i \to Y\). We additionally may cover each intersection \(T_i \cap T_j\) in affines, \(T_{ij1}, \dots, T_{ij k_{ij}}\), and note that further restriction also yields \(\eta(T_{ijk})(f_{ijk})\) from restricting \(f_i\) and \(f_j\). By naturality, restricting \(g_i = \eta(T_i)(f_i)\) to \(T_{ijk}\) is the same as \(\eta(T_{ijk})(f_{ijk})\), same for \(g_j = \eta(T_j)(f_j)\). Thus, \(g_i\) and \(g_j\) agree on an open cover of \(T_i \cap T_j\), so the collection of \(\eta(T_i)(f_i)\) must glue to define a morphism \(g : T \to Y\). Obviously, restricting to \(T_i\) gives

\begin{equation} g|_{T_i} = f_i = \eta(T_i)(f_i) = \eta(T)(f)|_{T_i} \end{equation}

for all \(i\), which implies that \(g = \eta(T)(f)\). Thus, we have shown that \(\eta(T)(f)\) is determined entirely by the data of \(\eta(T_i)(f_i)\), where each \(f : T_i \to X\) is affine in the domain (as \(g\) is glued out of these subcomponents mapped under \(\eta(T_i)\)). Thus, we have injectivity of the map taking a natural transformation of the extended functor of points to the functor of points: if two natural transformations agree on affines, then the above argument shows that they will give the same value for all schemes.

On the other hand, if we are given a natural transformation \(\eta\) of functors of points. If \(T\) is a scheme, suppose we have two affine covers \(T_i\) and \(T'_i\). We can take a common affine refinement \(S_j\), which allows us to prove that the morphism obtained from gluing all of the \(\eta(T_i)(f|_{T_i})\) and all of the \(\eta(T'_i)(f|_{T'_i})\) is the same, where \(f : T \to X\) is some morphism of schemes.

To see that this construction is natural, let \(h : T \to S\) be a morphism of schemes, cover \(S\) in affines \(S_i\) and cover \(h^{-1}(S_i)\) is affines \(T_{ij}\) for each \(i\). By naturality, we have the following diagram:

\begin{equation} \begin{CD} \text{Hom}(S_i, X) @> \eta(S_i) >> \text{Hom}(S_i, Y) \\ @V \circ h|_{T_{ij}} VV @V \circ h|_{T_{ij}} VV \\ \text{Hom}(T_{ij}, X) @> \eta(T_{ij}) >> \text{Hom}(T_{ij}, Y) \end{CD} \end{equation}

If we let \(f_{i} : S_i \to X\) be the restrictions of \(f\), notice that we will have

\begin{equation} \eta(T)(f \circ h)|_{T_{ij}} = \eta(T_{ij})(f_i \circ h_{ij}) = \eta(S_i)(f_i) \circ h_{ij} = (\eta(S)(f) \circ h)|_{T_{ij}} \end{equation}

Hence we have \(\eta(T)(f \circ h) = \eta(S)(f) \circ h\). This completes the proof.

Here is a basic example: letting \(X = \mathbb{A}_{\mathbb{Z}}^n\) and \(Y = \mathbb{A}_{\mathbb{Z}}^m\), recall that a map \(\text{Spec}(R) \to \mathbb{A}_{\mathbb{Z}}^n = \text{Spec}(\mathbb{Z}[x_1, \dots, x_n])\) is the same as a ring map \(\mathbb{Z}[x_1, \dots, x_n] \to R\), which is the same as a point in \(R^n\). Therefore, a natural transformation \(X(-) \to Y(-)\) is the data of set morphisms \(R^n \to R^m\) such that we have

\begin{equation} \begin{CD} X(R) @>>> Y(R) \\ @V X(\phi)VV @V Y(\phi)VV \\ X(S) @>>> Y(S) \end{CD} \end{equation}

for ring morphism \(\phi : R \to S\). Notice that \(\mathbb{A}_{\mathbb{Z}}^n(\phi)\) takes \(\text{Spec}(R) \to \mathbb{A}_{\mathbb{Z}}^n(\phi)\) to \(\text{Spec}(S) \to \mathbb{A}_{\mathbb{Z}}^n(\phi)\) via pre-composition with \(\text{Spec}(\phi)\), which under the identification of \(R\) -points with \(R^n\), is the same as post-composition with \(\phi \times \cdots \times \phi : R^n \to S^n\). Suppose we set our morphism \(R^n \to R^m\) as \((r_1, \dots, r_n) \mapsto (f_1(r_1, \dots, r_n), \dots, f_m(r_1, \dots, r_n))\), where each \(f_j \in R[x_1, \dots, x_n]\) is a polynomial in \(n\) variables. Clearly, these morphisms are natural with respect to any ring morphism, hence defines a natural transformation. To see what the corresponding morphism of schemes \(f : \mathbb{A}^n_{\mathbb{Z}} \to \mathbb{A}^m_{\mathbb{Z}}\) ends up being, notice that we are simply supposed to apply our natural transformation to the identity morphism \(\mathbb{A}_{\mathbb{Z}}^n \to \mathbb{A}_{\mathbb{Z}}^n\), which gives rise to a \(\mathbb{A}_{\mathbb{Z}}^n \to \mathbb{A}_{\mathbb{Z}}^m\). In this context, this will be the morphism of spectra yielded when we pullback the identity \(\mathbb{Z}[x_1, \dots, x_n] \to \mathbb{Z}[x_1, \dots, x_n]\) by the map \(\mathbb{Z}[y_1, \dots, y_m] \to \mathbb{Z}[x_1, \dots, x_n]\) where \(y_j \mapsto f_j\).