Definition 7.6.1. Let $\mathcal{C}$ be a category, see Conventions, Section 2.3. A family of morphisms with fixed target in $\mathcal{C}$ is given by an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, a set $I$ and for each $i\in I$ a morphism $U_ i \to U$ of $\mathcal{C}$ with target $U$. We use the notation $\{ U_ i \to U\} _{i\in I}$ to indicate this.
7.6 Sites
Our notion of a site uses the following type of structures.
It can happen that the set $I$ is empty! This notation is meant to suggest an open covering as in topology.
Definition 7.6.2. A site1 is given by a category $\mathcal{C}$ and a set $\text{Cov}(\mathcal{C})$ of families of morphisms with fixed target $\{ U_ i \to U\} _{i \in I}$, called coverings of $\mathcal{C}$, satisfying the following axioms
If $V \to U$ is an isomorphism then $\{ V \to U\} \in \text{Cov}(\mathcal{C})$.
If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})$ and for each $i$ we have $\{ V_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})$, then $\{ V_{ij} \to U\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{C})$.
If $\{ U_ i \to U\} _{i\in I}\in \text{Cov}(\mathcal{C})$ and $V \to U$ is a morphism of $\mathcal{C}$ then $U_ i \times _ U V$ exists for all $i$ and $\{ U_ i \times _ U V \to V \} _{i\in I} \in \text{Cov}(\mathcal{C})$.
Clarifications. In axiom (1) we require there should be a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that $I = \{ i\} $ is a singleton set and such that the morphism $U_ i \to U$ is equal to the morphism $V \to U$ given in (1). In the following we often denote $\{ V \to U\} $ a family of morphisms with fixed target whose index set is a singleton. In axiom (3) we require the existence of the covering for some choice of the fibre products $U_ i \times _ U V$ for $i \in I$.
Remark 7.6.3. (On set theoretic issues – skip on a first reading.) The main reason for introducing sites is to study the category of sheaves on a site, because it is the generalization of the category of sheaves on a topological space that has been so important in algebraic geometry. In order to avoid thinking about things like “classes of classes” and so on, we will not allow sites to be “big” categories, in contrast to what we do for categories and $2$-categories. Suppose that $\mathcal{C}$ is a category and that $\text{Cov}(\mathcal{C})$ is a proper class of coverings satisfying (1), (2) and (3) above. We will not allow this as a site either, mainly because we are going to take limits over coverings. However, there are several natural ways to replace $\text{Cov}(\mathcal{C})$ by a set of coverings or a slightly different structure that give rise to the same category of sheaves. For example:
In Sets, Section 3.11 we show how to pick a suitable set of coverings that gives the same category of sheaves.
Another thing we can do is to take the associated topology (see Definition 7.48.2). The resulting topology on $\mathcal{C}$ has the same category of sheaves. Two topologies have the same categories of sheaves if and only if they are equal, see Theorem 7.50.2. A topology on a category is given by a choice of sieves on objects. The collection of all possible sieves and even all possible topologies on $\mathcal{C}$ is a set.
We could also slightly modify the notion of a site, see Remark 7.48.4 below, and end up with a canonical set of coverings.
Each of these solutions has some minor drawback. For the first, one has to check that constructions later on do not depend on the choice of the set of coverings. For the second, one has to learn about topologies and redo many of the arguments for sites. For the third, see the last sentence of Remark 7.48.4.
Our approach will be to work with sites as in Definition 7.6.2 above. Given a category $\mathcal{C}$ with a proper class of coverings as above, we will replace this by a set of coverings producing a site using Sets, Lemma 3.11.1. It is shown in Lemma 7.8.8 below that the resulting category of sheaves (the topos) is independent of this choice. We leave it to the reader to use one of the other two strategies to deal with these issues if he/she so desires.
Example 7.6.4. Let $X$ be a topological space. Let $X_{Zar}$ be the category whose objects consist of all the open sets $U$ in $X$ and whose morphisms are just the inclusion maps. That is, there is at most one morphism between any two objects in $X_{Zar}$. Now define $\{ U_ i \to U\} _{i \in I}\in \text{Cov}(X_{Zar})$ if and only if $\bigcup U_ i = U$. Conditions (1) and (2) above are clear, and (3) is also clear once we realize that in $X_{Zar}$ we have $U \times V = U \cap V$. Note that in particular the empty set has to be an element of $X_{Zar}$ since otherwise this would not work in general. Furthermore, it is equally important, as we will see later, to allow the empty covering of the empty set as a covering! We turn $X_{Zar}$ into a site by choosing a suitable set of coverings $\text{Cov}(X_{Zar})_{\kappa , \alpha }$ as in Sets, Lemma 3.11.1. Presheaves and sheaves (as defined below) on the site $X_{Zar}$ agree exactly with the usual notion of a presheaves and sheaves on a topological space, as defined in Sheaves, Section 6.1.
Example 7.6.5. Let $G$ be a group. Consider the category $G\textit{-Sets}$ whose objects are sets $X$ with a left $G$-action, with $G$-equivariant maps as the morphisms. An important example is ${}_ GG$ which is the $G$-set whose underlying set is $G$ and action given by left multiplication. This category has fiber products, see Categories, Section 4.7. We declare $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ to be a covering if $\bigcup _{i\in I} \varphi _ i(U_ i) = U$. This gives a class of coverings on $G\textit{-Sets}$ which is easily seen to satisfy conditions (1), (2), and (3) of Definition 7.6.2. The result is not a site since both the collection of objects of the underlying category and the collection of coverings form a proper class. We first replace by $G\textit{-Sets}$ by a full subcategory $G\textit{-Sets}_\alpha $ as in Sets, Lemma 3.10.1. After this the site $(G\textit{-Sets}_\alpha , \text{Cov}_{\kappa , \alpha '}(G\textit{-Sets}_\alpha ))$ gotten by suitably restricting the collection of coverings as in Sets, Lemma 3.11.1 will be denoted $\mathcal{T}_ G$. As a special case, if the group $G$ is countable, then we can let $\mathcal{T}_ G$ be the category of countable $G$-sets and coverings those jointly surjective families of morphisms $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ such that $I$ is countable.
Example 7.6.6. Let $\mathcal{C}$ be a category. There is a canonical way to turn this into a site where $\{ f : V \to U \mid f\text{ is an isomorphism}\} $ are the coverings of $U$. Sheaves on this site are the presheaves on $\mathcal{C}$. This corresponding topology is called the chaotic or indiscrete topology.
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