                                              Source file: src/sites.tex

SITES AND SHEAVES

1. Introduction

The notion of a site was introduced by Grothendieck to be able to sudy sheaves in the étale topology of schemes. The basic reference for this notion is perhaps [MA71].

1.1. Topologies. Let C be a category, see Conventions, Section 3. In the following the notation {U_{i} → U}_{i∈I} means that U ∈ Ob(C), that I is a set and that for each i ∈ I we are given a morphism U_{i} → U of C with target U. The collection of all {U_{i} → U}_{i∈I} forms a category. Namely, a morphism {U_{i} → U}_{i∈I} → {V_{j} → V }_{j∈J} is given by a morphism U → V, a map of sets α : I → J and for each i ∈ I a morphism U_{i} → V_{α(i)} such that the diagram

U_{i} ⟶ V_{α(I)}
↓  ↓
U ⟶ V

is commutative. Later we will actually use Conventions, Lemma 3.1.1, but here we refer to it just to test cross-referencing.

References

[MA71] J.L. Verdier M. Artin, A. Grothendieck. Theorie de Topos et Cohomologie Etale des Schemas I, II, III, volume 269, 270, 305 of Lecture Notes in Mathematics. Springer, 1971.