Page "Grothendieck topology" Paragraph 5

The classical definition of a sheaf begins with a topological space X.

A sheaf associates information to the open sets of X.

This information can be phrased abstractly by letting O ( X ) be the category whose objects are the open subsets U of X and whose morphisms are the inclusion maps V → U of open sets U and V of X.

We will call such maps open immersions, just as in the context of schemes.

Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

The gluing axiom is phrased in terms of pointwise covering, i. e.,