The notion of a site was introduced by Grothendieck to be able to study sheaves in the étale topology of schemes. The basic reference for this notion is perhaps [SGA4]. Our notion of a site differs from that in [SGA4]; what we call a site is called a category endowed with a pretopology in [Exposé II, Définition 1.3, SGA4]. The reason we do this is that in algebraic geometry it is often convenient to work with a given class of coverings, for example when defining when a property of schemes is local in a given topology, see Descent, Section 35.15. Our exposition will closely follow [ArtinTopologies]. We will not use universes.
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