Grothendieck topology site

A Grothendieck topology on a category \(C\) consists of a specification, for each object \(X\) of \(C\), of a collection of sieves on \(C\) which we will refer to as covering sieves. The collections of covering sieves are required to satisfy the following properties:

1. For every object \(X\) of \(C\), the identity \(y(X) \to y(X)\) is a covering sieve;

2. For every morphism \(f\colon X \to Y\) in \(C\) and every covering sieve \(U \hookrightarrow y(Y)\) on \(Y\), the pullback sieve \(f^*U \hookrightarrow y(X)\) is a covering sieve on \(X\);

3. Let \(X\) be an object of \(C\), \(U \hookrightarrow y(X)\) a covering sieve on \(X\), and \(V \hookrightarrow y(X)\) an arbitrary sieve on \(X\). Suppose that, for every morphism \(f \colon Y \to X\) belonging to \(U\), the pullback sieve \(f^*V\) is a covering sieve on \(Y\). Then \(V\) is a covering sieve on \(X\).

A Grothendieck site is a category \(C\) equipped with a Grothendieck topology.