Editing Gluing axiom (section)

==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 <math>X</math> be a topological space with [[basis of a topological space|basis]] <math>\{ B_i \}_{i \in I}</math>.  We can define a category <math>\mathcal{O}'(X)</math> to be the full subcategory of <math>{\mathcal O}(X)</math> whose objects are the <math>\{ B_i \}</math>.  A '''B-sheaf''' on <math>X</math> with values in <math>C</math> is a contravariant functor

:<math>{\mathcal F}:{\mathcal O}'(X) \rightarrow C</math>

which satisfies the gluing axiom for sets in <math>{\mathcal O}'(X)</math>.  That is, on a selection of open sets of <math>X</math>, <math>\mathcal F</math> specifies all of the sections of a sheaf, and on the other open sets, it is undetermined.

B-sheaves are equivalent to sheaves (that is, the category of sheaves is equivalent to the category of B-sheaves).<ref>Vakil, [http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf Math 216: Foundations of algebraic geometry], 2.7.</ref>  Clearly a sheaf on <math>X</math> can be restricted to a B-sheaf.  In the other direction, given a B-sheaf <math>\mathcal F</math> we must determine the sections of <math>\mathcal F</math> on the other objects of <math>{\mathcal O}(X)</math>.  To do this, note that for each open set <math>U</math>, we can find a collection <math>\{ B_j \}_{j \in J}</math> whose union is <math>U</math>.  Categorically speaking, this choice makes <math>U</math> the colimit of the full subcategory of <math>{\mathcal O}'(X)</math> whose objects are <math>\{ B_j \}_{j \in J}</math>.  Since <math>\mathcal F</math> is contravariant, we define <math>{\mathcal F}'(U)</math> to be the [[projective limit|limit]] of the <math>\{ {\mathcal F}(B_j) \}_{j \in J}</math> with respect to the restriction maps.  (Here we must assume that this limit exists in <math>C</math>.)  If <math>U</math> is a basic open set, then <math>U</math> is a terminal object of the above subcategory of <math>{\mathcal O}'(X)</math>, and hence <math>{\mathcal F}'(U) = {\mathcal F}(U)</math>.  Therefore, <math>{\mathcal F}'</math> extends <math>\mathcal F</math> to a presheaf on <math>X</math>.  It can be verified that <math>{\mathcal F}'</math> is a sheaf, essentially because every element of every open cover of <math>X</math> is a union of basis elements (by the definition of a basis), and every pairwise intersection of elements in an open cover of <math>X</math> is a union of basis elements (again by the definition of a basis).