local rings. It is the stalks of the sheaf that are local rings, not the collections of sections (which are rings, but in general are not close to being local). We can think of a locally ringed space as a parametrised family of local rings, depending on in . A more careful discussion dispels any mystery here. One can speak freely of a sheaf of abelian groups, or rings, because those are algebraic structures (defined, if one insists, by an explicit signature). Any category having finite products supports the idea of a group object, which some prefer just to call a group in . In the case of this kind of purely algebraic structure, we can talk either of a sheaf having values in the category of abelian groups, or an abelian group in the category of sheaves of sets; it really doesn't matter. In the local ring case, it does matter. At a foundational level we must use the second style of definition, to describe what a local ring means in a category. This is a logical matter: axioms for a local ring require use of existential quantification, in the form that for any in the ring, one of and is invertible. This allows one to specify what a 'local ring in a category' should be, in the case that the category supports enough structure. Sheafification To turn a given presheaf into a sheaf , there is a standard device called sheafification or sheaving. The rough intuition of what one should do, at least for a presheaf of sets, is to introduce an equivalence relation, which makes equivalent data given by different covers on the overlaps by refining the covers. One approach is therefore to go to the stalks and recover the sheaf space of the best possible sheaf produced from . This use of language strongly suggests that we are dealing here with adjoint functors. Therefore, it makes sense to observe that the sheaves on form a full subcategory of the presheaves on . Implicit in that is the statement that a morphism of sheaves is nothing more than a natural transformation of the sheaves, considered as functors. Therefore, we get an abstract characterisation of sheafification as left adjoint to the inclusion. In some applications, naturally, one does need a description. In more abstract language, the sheaves on form a reflective subcategory of the presheaves (Mac Lane–Moerdijk Sheaves