Gluing Axiom - The Logic of C

The Logic of C

The first needs of sheaf theory were for sheaves of abelian groups; so taking the category C 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 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 X as a parametrised family of local rings, depending on x in X.

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 C having finite products supports the idea of a group object, which some prefer just to call a group in C. 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 r in the ring, one of r and 1 − r 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.