Toposophical thoughts

For many years the phrase Ringed Topos has been bandied about. I do not much like it. It puts the cart before the horse and gives a false emphasis. I would rather talk of a ring, or anything else, living in a topos. Call it an extended ring or a smeared-out ring, if you must. Let us use the word Ring for a ring in this general sense. If a Ring A lives in a topos E and a Ring B lives in a topos F then a homorphism of Rings A → B is defined by the data:Composition of homomorphisms of Rings is given by the obvious formula. We get a category of Rings and homomorphisms extending the usual category of rings and homomorphisms. I am not proud of this temporary kludge of a terminology. Instead of rings we could do the same with models of any geometric theory, and capitalize the name of practically every beast in the mathematical zoo; how can we manage without grievously abusing language? What is the point of it? The point is that in Rings there exist universal constructions that do not exist within the more trammelled confines of mere rings. The first example of this phenomenon goes back to the notion of the prime spectrum of a ring, and its Zariski topology.

Invertibility

In the context of commutative rings we define 
inv (x) ⇔ ∃ y . xy = 1
and we say that a homomorphism f reflects inverses if 
inv (f(x)) ⇒ inv(x)
We say that a homomorphism f : A → B is fractional if it satisfiesA commutative ring is local if satisfies A homomorphism to a local ring factors uniquely into a fractional homorphism followed by one that reflects inverses.
Note that the notions above are all geometric - that is to say, they define constructions defined by finite limits and colimits, and which are therefore preserved by the inverse image functors of geometric morphisms. In the topos Sets a local ring is a nontrivial ring with precisely one maximal ideal. But this formulation is not geometric. We have a convenient metatheorem that any implication between positive statements that can be proved in Sets also holds in any topos over Sets.

Classifying toposes

Recall that a geometric theory is given by implications between positive statements: these are first order statements that do not contain implications, negations or universal quantifiers. For any such theory T there is a universal T-model U T living in a topos S(T) dubbed the classifying topos of T. For example, the theory of rings has for its classifying topos the category of Sets-valued functors on the category of finitely presented rings and homomorphisms between them, the universal ring (Ring, rather) being the forgetful functor to Sets.
The category of T-models, and maps of T-models in a topos E is equivalent to the category Top(E,S(T)) of geometric morphisms E → S(T) and natural transformations between their inverse image functors. A geometric morphism f : E → S(T) corresponds to a T-model f * (U T ) in E. The theory of local rings is obtained from that of rings by adjoining the two extra axioms given above. The classifying topos S(locring) has a geometric morphism to S(ring)
e: S(locring) → S(ring)
classifying the universal local ring U[ locring ]. If f : E → S(ring) classifies a Ring A in E then the pullback of f along e yields the map
Spec(A) → S(locring)
which classifies the canonical sheaf of local rings A ~ on Spec(A). Pulling the e back along f yields a geometric morphism Spec(A) → E, which is part of the homomorphism of Rings A → A ~ which can be seen as the universal localization of A through which all homomorphisms of Rings from A to a local Ring factor. In other words U locring is the universal localization of U ring and S(locring) is Spec of the universal ring. The fact that A ~ is not necessarily living in the same topos as A is what drives the idea of extending rings to Rings. It is my belief that most of the topos constructions that have appeared in algebraic geometry can be more simply described in terms of appropriate geometric theories.
This is a game which can clearly be played in a great number of fields. Anyone for rigged toposes?
A picture here showing not toposes but distant galaxies. No geometric morphisms or dark matter shown.