One way of understanding the gluing axiom is to notice that is the colimit of the following diagram: The gluing axiom says that turns colimits of such diagrams into limits. 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 be a topological space with basis . We can define a category to be the full subcategory of whose objects are the . A B-sheaf on with values in is a contravariant functor which satisfies the gluing axiom for sets in . That is, on a selection of open sets of , 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). Clearly a sheaf on can be restricted to a B-sheaf. In the other direction, given a B-sheaf we must determine the sections of on the other objects of . To do this, note that for each open set , we can find a collection whose union is . Categorically speaking, this choice makes the colimit of the full subcategory of whose objects are . Since is contravariant, we define to be the limit of the with respect to the restriction maps. (Here we must assume that this limit exists in .) If is a basic open set, then is a terminal object of the above subcategory of , and hence . Therefore, extends to a presheaf on . It can be verified that is a sheaf, essentially because every element of every open cover of is a union of basis elements (by the definition of a basis), and every pairwise intersection of elements in an open cover of is a union of basis elements (again by the definition of a basis). The logic of C The first needs of sheaf theory were for sheaves of abelian groups; so taking the category as the category of abelian groups was only natural. In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central. This, however, is not quite the same thing; one speaks instead of a locally ringed space, because it is not true, except in trite cases, that such a sheaf is a functor into a category of