73.2 The general procedure
In this section we explain a general procedure for producing the sites we will be working with. This discussion will make little or no sense unless the reader has read Topologies, Section 34.2.
Let S be a base scheme. Take any category \mathit{Sch}_\alpha constructed as in Sets, Lemma 3.9.2 starting with S and any set of schemes over S you want to be included. Choose any set of coverings \text{Cov}_{fppf} on \mathit{Sch}_\alpha as in Sets, Lemma 3.11.1 starting with the category \mathit{Sch}_\alpha and the class of fppf coverings. Let \mathit{Sch}_{fppf} denote the big fppf site so obtained, and let (\mathit{Sch}/S)_{fppf} denote the corresponding big fppf site of S. (The above is entirely as prescribed in Topologies, Section 34.7.)
Given choices as above the category of algebraic spaces over S has a set of isomorphism classes. One way to see this is to use the fact that any algebraic space over S is of the form U/R for some étale equivalence relation j : R \to U \times _ S U with U, R \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}), see Spaces, Lemma 65.9.1. Hence we can find a full subcategory \textit{Spaces}/S of the category of algebraic spaces over S which has a set of objects such that each algebraic space is isomorphic to an object of \textit{Spaces}/S. We fix a choice of such a category.
In the sections below, given a topology \tau , the big site (\textit{Spaces}/S)_\tau (resp. the big site (\textit{Spaces}/X)_\tau of an algebraic space X over S) has as underlying category the category \textit{Spaces}/S (resp. the subcategory \textit{Spaces}/X of \textit{Spaces}/S, see Categories, Example 4.2.13). The procedure for turning this into a site is as usual by defining a class of \tau -coverings and using Sets, Lemma 3.11.1 to choose a sufficiently large set of coverings which defines the topology.
We point out that the small étale site X_{\acute{e}tale} of an algebraic space X has already been defined in Properties of Spaces, Definition 66.18.1. Its objects are schemes étale over X, of which there are plenty by definition of an algebraic spaces. However, a more natural site, from the perspective of this chapter (compare Topologies, Definition 34.4.8) is the site X_{spaces, {\acute{e}tale}} of Properties of Spaces, Definition 66.18.2. These two sites define the same topos, see Properties of Spaces, Lemma 66.18.3. We will not redefine these in this chapter; instead we will simply use them.
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