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\begin{document}

\title{Sites and Sheaves}

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\section{Introduction}
\label{section-introduction}

\noindent
The notion of a site was introduced by Grothendieck to be able to study
sheaves in the \'etale topology of schemes. The basic reference for this
notion is perhaps \cite{SGA4}. 

\subsection{Topologies}

\noindent
Let $C$ be a category, see 
Conventions, Section \ref{conventions-section-categories}. 
In the following the notation $\{U_i \to U\}_{i\in I}$ means that 
$U \in \text{Ob}(C)$, that $I$ is a set and that for each $i\in I$ 
we are given a morphism $U_i \to U$ of $C$ with target $U$. The collection
of all $\{U_i \to U\}_{i\in I}$ forms a category. Namely, a morphism
$\{U_i \to U\}_{i\in I} \to \{V_j \to V\}_{j\in J}$ is given by
a morphism $U \to V$, a map of sets $\alpha : I \to J$ and
for each $i\in I$ a morphism $U_i \to V_{\alpha(i)}$ such that
the diagram 
$$
\begin{CD}
U_i	@>>>	V_{\alpha(I)}		\\
@VVV		@VVV			\\
U	@>>>	V			\\
\end{CD}
$$
is commutative. 

\smallskip\noindent
Later we will actually use 
Conventions, Lemma \ref{conventions-lemma-categories-small}, 
but here we refer to it just to test cross-referencing.

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