Definition 59.10.1. Let $\mathcal{C}$ be a category. A family of morphisms with fixed target $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$ is the data of
an object $U \in \mathcal{C}$,
a set $I$ (possibly empty), and
for all $i\in I$, a morphism $\varphi _ i : U_ i \to U$ of $\mathcal{C}$ with target $U$.
There is a notion of a morphism of families of morphisms with fixed target. A special case of that is the notion of a refinement. A reference for this material is Sites, Section 7.8.
Definition 59.10.2. A site1 consists of a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ consisting of families of morphisms with fixed target called coverings, such that
(isomorphism) if $\varphi : V \to U$ is an isomorphism in $\mathcal{C}$, then $\{ \varphi : V \to U\} $ is a covering,
(locality) if $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ is a covering and for all $i \in I$ we are given a covering $\{ \psi _{ij} : U_{ij} \to U_ i \} _{j\in I_ i}$, then
is also a covering, and
(base change) if $\{ U_ i \to U\} _{i\in I}$ is a covering and $V \to U$ is a morphism in $\mathcal{C}$, then
for all $i \in I$ the fibre product $U_ i \times _ U V$ exists in $\mathcal{C}$, and
$\{ U_ i \times _ U V \to V\} _{i\in I}$ is a covering.
\[ \{ \varphi _ i \circ \psi _{ij} : U_{ij} \to U \} _{(i, j)\in \prod _{i\in I} \{ i\} \times I_ i} \]
For us the category underlying a site is always “small”, i.e., its collection of objects form a set, and the collection of coverings of a site is a set as well (as in the definition above). We will mostly, in this chapter, leave out the arguments that cut down the collection of objects and coverings to a set. For further discussion, see Sites, Remark 7.6.3.
Example 59.10.3. If $X$ is a topological space, then it has an associated site $X_{Zar}$ defined as follows: the objects of $X_{Zar}$ are the open subsets of $X$, the morphisms between these are the inclusion mappings, and the coverings are the usual topological (surjective) coverings. Observe that if $U, V \subset W \subset X$ are open subsets then $U \times _ W V = U \cap V$ exists: this category has fiber products. All the verifications are trivial and everything works as expected.
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