For a topological space $X$, the category $\operatorname{Open}(X)$ with the open cover topology forms a site. The objects are open subsets $U \subseteq X$, morphisms are inclusions $V \hookrightarrow U$, and covering families of $U$ are open covers $\{U_i\}$ with $\bigcup U_i = U$.

This is the prototypical site. All concepts for general sites reduce to classical notions when restricted to this case.

For a scheme $X$, the small etale site $X_{\text{et}}$ is defined as follows:

• Objects: Etale morphisms $U \to X$ (equivalently, etale $X$-schemes).
• Morphisms: $X$-morphisms $U \to V$ (which are automatically etale since etale is stable under composition and the morphisms are over $X$).
• Covering families: Families $\{U_i \to U\}$ of etale morphisms such that $\coprod U_i \to U$ is surjective.

The category $X_{\text{et}}$ is essentially small (there is a set of isomorphism classes of etale $X$-schemes), so set-theoretic issues are mild.

The big fppf site $(\mathbf{Sch}/S)_{\text{fppf}}$ has:

• Objects: All $S$-schemes (or a suitable essentially small subcategory, e.g., schemes of bounded cardinality).
• Morphisms: $S$-morphisms.
• Covering families: Families $\{U_i \to U\}$ where each $U_i \to U$ is flat and locally of finite presentation, and $\coprod U_i \to U$ is surjective.

The big site allows us to define presheaves and sheaves that take values on all schemes, such as the functor of points of a group scheme.

Given a simplicial scheme $X_\bullet$ (a functor $\Delta^{\mathrm{op}} \to \mathbf{Sch}$), one can define an etale site on $X_\bullet$ whose objects are etale morphisms to the components $X_n$, with appropriate covering conditions. The cohomology of sheaves on this site computes the cohomological descent spectral sequence.

This is fundamental for the construction of etale cohomology via hypercoverings and for the theory of algebraic stacks.

For each object $U \in \mathcal{C}$, the representable presheaf (or Yoneda presheaf) is

$h_U = \operatorname{Hom}_{\mathcal{C}}(-, U): \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}.$

It sends $V \mapsto \operatorname{Hom}(V, U)$ and a morphism $f: W \to V$ to precomposition $g \mapsto g \circ f$.

In the scheme-theoretic setting with $\mathcal{C} = \mathbf{Sch}/S$, the representable presheaf $h_X$ for an $S$-scheme $X$ sends $T \mapsto \operatorname{Hom}_S(T, X)$. This is the functor of points of $X$.

The additive group scheme $\mathbb{G}_a$ over a base $S = \operatorname{Spec}(R)$ represents the presheaf

$\mathbb{G}_a(T) = \Gamma(T, \mathcal{O}_T)$

on $\mathbf{Sch}/S$, assigning to each $S$-scheme $T$ its ring of global sections. For $T = \operatorname{Spec}(A)$ (an affine $S$-scheme, i.e., an $R$-algebra $A$), $\mathbb{G}_a(T) = A$ as a set (with its additive group structure).

This is representable: $\mathbb{G}_a = \operatorname{Spec}(R[t])$, and $\operatorname{Hom}_S(\operatorname{Spec}(A), \operatorname{Spec}(R[t])) = \operatorname{Hom}_{R\text{-alg}}(R[t], A) \cong A$.

The multiplicative group scheme $\mathbb{G}_m$ represents

$\mathbb{G}_m(T) = \Gamma(T, \mathcal{O}_T)^\times$

the group of invertible global sections. It is representable: $\mathbb{G}_m = \operatorname{Spec}(R[t, t^{-1}])$.

For $T = \operatorname{Spec}(A)$, we have $\mathbb{G}_m(T) = A^\times$. The cohomology $H^1_{\text{et}}(X, \mathbb{G}_m) \cong \operatorname{Pic}(X)$ is one of the most important computations in etale cohomology.

For a positive integer $n$ invertible on the base $S$, the group scheme of $n$-th roots of unity $\mu_n$ represents

$\mu_n(T) = \{f \in \Gamma(T, \mathcal{O}_T)^\times \mid f^n = 1\}.$

It is representable: $\mu_n = \operatorname{Spec}(R[t]/(t^n - 1))$, which is a finite etale group scheme when $n$ is invertible on $S$. Over an algebraically closed field, $\mu_n(\overline{k}) \cong \mathbb{Z}/n\mathbb{Z}$.

The Kummer sequence $1 \to \mu_n \to \mathbb{G}_m \xrightarrow{(\cdot)^n} \mathbb{G}_m \to 1$ is exact in the etale topology and gives the fundamental exact sequence in etale cohomology.

Let $\mathcal{C} = \mathbf{Aff}_k = (\mathbf{CAlg}_k)^{\mathrm{op}}$ be the category of affine $k$-schemes. The affine line $\mathbb{A}^1_k = \operatorname{Spec}(k[t])$ represents the forgetful functor:

$h_{\mathbb{A}^1}(\operatorname{Spec}(A)) = \operatorname{Hom}(\operatorname{Spec}(A), \mathbb{A}^1) = \operatorname{Hom}_{k\text{-alg}}(k[t], A) \cong A.$

By Yoneda, a natural transformation $h_{\mathbb{A}^1} \to h_{\mathbb{A}^1}$ corresponds to an element of $h_{\mathbb{A}^1}(\mathbb{A}^1) = k[t]$. So endomorphisms of $\mathbb{A}^1$ as a scheme correspond to polynomials $f(t) \in k[t]$. This recovers the geometric fact that polynomial maps $\mathbb{A}^1 \to \mathbb{A}^1$ are classified by polynomials.

The Grassmannian $\operatorname{Gr}(r, n)$ represents the functor

$\operatorname{Gr}(r, n)(T) = \{\text{locally free quotients } \mathcal{O}_T^n \twoheadrightarrow \mathcal{E} \text{ of rank } r\} / \cong$

By Yoneda, a morphism $T \to \operatorname{Gr}(r, n)$ corresponds to a rank-$r$ quotient bundle of the trivial rank-$n$ bundle on $T$. The universal such quotient lives on $\operatorname{Gr}(r, n)$ itself.

This "moduli" perspective is the starting point for the theory of algebraic stacks: many moduli problems define presheaves (or fibered categories) that may not be representable by schemes but can be represented by stacks.

Consider the presheaf $\mathcal{M}_{1,1}$ on $\mathbf{Sch}/\mathbb{Z}$ that assigns to $T$ the set of isomorphism classes of elliptic curves over $T$:

$\mathcal{M}_{1,1}(T) = \{(E \to T, e: T \to E) \mid E/T \text{ smooth genus-1 curve, } e \text{ section}\} / \cong$

This presheaf is not representable by a scheme (not even by an algebraic space), because elliptic curves can have nontrivial automorphisms ($j=0$ and $j=1728$ have extra automorphisms).

However, $\mathcal{M}_{1,1}$ is representable as a Deligne-Mumford stack (or more precisely, a fibered category with descent data). This is one of the primary motivations for the theory of algebraic stacks.

On the big Zariski site of $S = \operatorname{Spec}(R)$, a quasi-coherent module $M$ on $S$ defines a presheaf of $\mathcal{O}$-modules by

$\widetilde{M}(\operatorname{Spec}(A)) = M \otimes_R A$

for any $R$-algebra $A$. This is a sheaf for the Zariski, etale, and even fpqc topology (by faithfully flat descent for modules).

On the small etale site $X_{\text{et}}$ of a smooth scheme $X$ over a field $k$, the presheaf of Kahler differentials assigns

$\Omega^1(U \to X) = \Gamma(U, \Omega^1_{U/k})$

to each etale $U \to X$. Since etale morphisms are smooth (in fact unramified), and $\Omega^1$ commutes with etale base change, this is actually a sheaf.

More precisely, if $f: U \to X$ is etale, then $f^*\Omega^1_{X/k} \cong \Omega^1_{U/k}$, so we have a canonical identification of the differentials.

For a fixed object $X \in \mathcal{C}$, the slice category (or over category) $\mathcal{C}/X$ has objects $(U, f: U \to X)$ and morphisms commuting triangles. A presheaf on $\mathcal{C}/X$ is then a contravariant functor from $\mathcal{C}/X$ to $\mathbf{Set}$.

If $\mathcal{C} = \mathbf{Sch}/S$ and $X$ is an $S$-scheme, then $\mathcal{C}/X = \mathbf{Sch}/X$. The presheaves on the big site of $X$ are precisely presheaves on this slice category with the induced topology.

By Yoneda, there is an equivalence between morphisms $Y \to X$ and natural transformations $h_Y \to h_X$ of presheaves on $\mathcal{C}$.

A fiber functor on a site $(\mathcal{C}, J)$ is an exact functor $\omega: \operatorname{Sh}(\mathcal{C}, J) \to \mathbf{Set}$ that commutes with finite limits and arbitrary colimits. For the small etale site of $\operatorname{Spec}(k)$, a fiber functor is given by evaluation at a separable closure:

$\omega(\mathcal{F}) = \varinjlim_{k'/k \text{ finite separable}} \mathcal{F}(\operatorname{Spec}(k'))$

This is the stalk at the geometric point $\overline{k} \to k$. The automorphism group of this fiber functor is the absolute Galois group $G_k = \operatorname{Gal}(k^{\text{sep}}/k)$, and Grothendieck's Galois theory identifies the category of locally constant sheaves of finite sets on $(\operatorname{Spec}(k))_{\text{et}}$ with the category of finite continuous $G_k$-sets.

In $\operatorname{PSh}(\mathcal{C})$, the coproduct $h_U \sqcup h_V$ sends $W \mapsto \operatorname{Hom}(W, U) \sqcup \operatorname{Hom}(W, V)$. This is typically not representable: in $\mathcal{C} = \mathbf{Sch}/k$, the coproduct $h_U \sqcup h_V$ is the presheaf of "choosing a map to $U$ or a map to $V$," which is represented by the disjoint union $U \sqcup V$ only when the disjoint union exists in $\mathcal{C}$.

This illustrates why presheaf categories are much larger than the original category: they freely adjoin colimits.

By Yoneda, a scheme $X$ over $S$ is determined by the functor $h_X: (\mathbf{Sch}/S)^{\mathrm{op}} \to \mathbf{Set}$. In practice, we often define $X$ by specifying $h_X$ on affine schemes:

For $\mathbb{P}^n_S$, the functor of points is

$\mathbb{P}^n(T) = \{\text{line bundle quotients } \mathcal{O}_T^{n+1} \twoheadrightarrow \mathcal{L}\} / \cong$

For $GL_n$, the functor is $GL_n(T) = GL_n(\Gamma(T, \mathcal{O}_T))$, the group of invertible $n \times n$ matrices with entries in the global sections of $T$.

For the Hilbert scheme $\operatorname{Hilb}^p_{X/S}$:

$\operatorname{Hilb}^p_{X/S}(T) = \{Z \hookrightarrow X \times_S T \mid Z \text{ flat over } T \text{ with Hilbert polynomial } p\}$

The functor of points perspective makes moduli problems precise and motivates the definition of algebraic stacks for non-representable functors.

A formal scheme $\hat{X}$ (e.g., the formal completion of $\operatorname{Spec}(k[[x]])$) represents a presheaf on the category of schemes, but it is not itself a scheme. On affine schemes:

$\hat{\mathbb{A}}^1(T) = \Gamma(T, \mathcal{O}_T)_{\text{nilp}} = \{f \in \Gamma(T, \mathcal{O}_T) \mid f \text{ locally nilpotent}\}$

This presheaf is an ind-scheme (a colimit of schemes), and its functor of points perspective allows us to treat formal objects on equal footing with ordinary schemes.