Grothendieck Topology

Idea

A Grothendieck topology tells you, for each object of a category, which families of morphisms should count as “covers.” It replaces open covers in topology with a purely categorical notion of covering.

Definition

Let $\mathcal{C}$ be a category. A Grothendieck topology $J$ assigns to each object $U$ a collection $J(U)$ of covering sieves on $U$, subject to axioms given below.

Sieve

A sieve $S$ on $U$ is a collection of morphisms $f:V\to U$ that are closed under precomposition.

Covering Sieve

A covering sieve is a collection of sieves $J(X)$ for each object $X$ such that:

• Stability: The pullback of a covering sieve along an arbitrary morphisms is itself a ccovering sieve.
• Local character: If a sieve $S$ containers a covering sieve $J^{\prime }$ after pulling back then $S$ is itself a covering sieve.

In a Grothendieck topology the following must also hold:

• Maximality: The maximal sieve is always a covering sieve.