Gluing Axiom - Sheaves On A Basis of Open Sets

Sheaves On A Basis of Open Sets

In some categories, it is possible to construct a sheaf by specifying only some of its sections. Specifically, let X be a topological space with basis {B_i}_i∈I. We can define a category O ′(X) to be the full subcategory of O(X) whose objects are the {B_i}. A B-sheaf on X with values in C is a contravariant functor

F: O ′(X) → C

which satisfies the gluing axiom for sets in O ′(X). We would like to recover the values of F on the other objects of O(X).

To do this, note that for each open set U, we can find a collection {B_j}_j∈J whose union is U. Categorically speaking, U is the colimit of the {B_j}_j∈J. Since F is contravariant, we define F(U) to be the limit of the {F(B)}_j∈J. (Here we must assume that this limit exists in C.) It can be shown that this new object agrees with the old F on each basic open set, and that it is a sheaf.