Site

Let $\mathcal{C}$ be a small category.

Definition. A Grothendieck topology $\tau$ on $\mathcal{C}$ assigns to each object $U \in \mathcal{C}$ a collection $\tau(U)$ of covering sieves — subfunctors $S \hookrightarrow \mathrm{Hom}(-, U)$ — satisfying:

1. (Maximal sieve.) The maximal sieve $\mathrm{Hom}(-, U)$ is in $\tau(U)$.
2. (Stability.) If $S \in \tau(U)$ and $f: V \to U$ is any morphism, then the pullback sieve $f^*S = \{g: W \to V \mid f \circ g \in S\}$ is in $\tau(V)$.
3. (Transitivity.) If $S \in \tau(U)$ and $R$ is a sieve on $U$ such that $f^*R \in \tau(V)$ for every $f: V \to U$ in $S$, then $R \in \tau(U)$.

A site is a pair $(\mathcal{C}, \tau)$ of a small category equipped with a Grothendieck topology.

Equivalently (and more concretely), a Grothendieck pretopology on $\mathcal{C}$ specifies covering families $\{U_i \to U\}_{i \in I}$ satisfying: (i) isomorphisms cover; (ii) covering families are stable under pullback: if $\{U_i \to U\}$ covers $U$ and $V \to U$ is any morphism, then $\{U_i \times_U V \to V\}$ covers $V$; (iii) coverings compose: if $\{U_i \to U\}$ covers $U$ and each $\{V_{ij} \to U_i\}_j$ covers $U_i$, then $\{V_{ij} \to U\}_{i,j}$ covers $U$. Every pretopology generates a unique Grothendieck topology.

Proposition. A presheaf $F: \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ is a sheaf for $(\mathcal{C}, \tau)$ if and only if for every covering sieve $S \in \tau(U)$, the canonical map $\mathrm{Hom}(\mathrm{Hom}(-, U), F) \to \mathrm{Hom}(S, F)$ is a bijection. The full subcategory $\mathrm{Sh}(\mathcal{C}, \tau) \hookrightarrow \mathrm{PSh}(\mathcal{C})$ of sheaves is a Grothendieck topos.

Proposition. A site $(\mathcal{C}, \tau)$ is called subcanonical if every representable presheaf $\mathrm{Hom}(-, C)$ is a sheaf for $\tau$. This holds if and only if every covering sieve is an effective-epimorphic family.

Proposition. Two sites $(\mathcal{C}, \tau)$ and $(\mathcal{D}, \sigma)$ can give rise to equivalent categories of sheaves: $\mathrm{Sh}(\mathcal{C}, \tau) \simeq \mathrm{Sh}(\mathcal{D}, \sigma)$. A Grothendieck topos determines a unique locale of subterminal objects, while distinct sites produce equivalent sheaf categories when they share the same sheafification.

Examples.

• Topological spaces: For a topological space $X$, the site $\mathrm{Open}(X)$ has open subsets as objects, inclusions as morphisms, and open covers as covering families.
• Zariski site: For a scheme $X$, objects are Zariski-open subschemes with covering families given by Zariski-open covers.
• Etale site: For a scheme $X$, objects are etale morphisms $U \to X$, and covering families are jointly surjective families of etale maps. This topology is finer than the Zariski topology and gives rise to etale cohomology.
• Canonical topology: On any category $\mathcal{C}$, the finest subcanonical topology. Its covering sieves are exactly the effective-epimorphic sieves.