Editing Gluing axiom (section)

==The logic of ''C''==
The first needs of sheaf theory were for sheaves of [[abelian group]]s; so taking the category <math>C</math> as the [[category of abelian groups]] was only natural. In applications to geometry, for example [[complex manifold]]s and [[algebraic geometry]], the idea of a ''sheaf of [[local ring]]s'' 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 [[ring (mathematics)|rings]], but in general are not close to being ''local''). We can think of a locally ringed space <math>X</math> as a parametrised family of local rings, depending on <math>x</math> in <math>X</math>.

A more careful discussion dispels any mystery here. One can speak freely of a sheaf of abelian groups, or rings, because those are [[algebraic structure]]s (defined, if one insists, by an explicit [[signature (logic)|signature]]). Any category <math>C</math> having [[product (category theory)|finite product]]s supports the idea of a [[group object]], which some prefer just to call a group ''in'' <math>C</math>. 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 <math>r</math> in the ring, one of <math>r</math> and <math>1-r</math> 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.