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 {Bi}i∈I. We can define a category O ′(X) to be the full subcategory of O(X) whose objects are the {Bi}. 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 {Bj}j∈J whose union is U. Categorically speaking, U is the colimit of the {Bj}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.
Read more about this topic: Gluing Axiom
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