Several Topos theory questions
About 1: Yes!
About 2: (Internal logic of Zariski topos) I don't think it has been done systematically. A glimpse of it is in Anders Kock, Universal projective geometry via topos theory, if I remember well, and certainly in some other places. But one point is that it is not at all easy to find formulas in the internal language which express what you have in mind. See my answer at "synthetic" reasoning applied to algebraic geometry
About 3:You can indeed glue all sorts of things:
Things fitting into the axiomatic framework of "geometric contexts": Look at the "master course on Algebraic stacks" here: http://perso.math.univ-toulouse.fr/btoen/videos-lecture-notes-etc/ This one is great reading to understand the functorial point of view on schemes and manifolds!
Commutative Monoid objects in good monoidal (model) categories: http://arxiv.org/abs/math/0509684
Commutative monads (here you can glue monoids, semirings and other algebraic structures mixing them all): http://arxiv.org/abs/0704.2030
In Shai Haran's "Non-Additive Geometry" you can even glue the monoids and semirings etc. with relations (although I wouldn't know why)
You can also glue things "up to homotopy instead" of strictly - this is roughly what Lurie's infinity-topoi are about, and also the model catgeory part of the 2nd point, or any oter approaches to derived algebraic geometry
(Edit in 2017) The PhD thesis by Zhen Lin Low is relevant. "The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces."
One of several good points of view on what a Grothendieck topology does, is to say it determines which colimits existing in your site should be preserved under the Yoneda embedding, i.e. what glueing takes already place among the affine objects. So, if you insist on glueing groups it could be a good idea to look e.g. for a topology which takes amalgamated products (for me this means glueing groups, you may want only selected such products, e.g. along injective maps) to pushouts of sheaves... Then feel free to develop a theory on this and send me a copy!
About 4: (Why don't people work with sheaves instead of schemes) They do. One situation where they do is when taking the quotient of a scheme by a group action. The coequalizer in the category of schemes is often too degenerate. One answer is taking the coequalizer in the category of sheaves, the "sheaf quotient" (but sometimes better answers are GIT quotients and stack quotients).
Just to be clear: the category of sheaves on the big Zarsiki site is a topos only if "big Zariski site" refers to the category of finitely presented commutative rings, or else some other small subcategory of CommRingop. Or else you allow your sheaves to take values in large sets. The category of small-set-valued sheaves on a large site is not in general a topos.
Regarding the internal logic of the Zariski topos, you may be interested in VIII.6 of "Sheaves in geometry and logic" which shows that it is the classifying topos for local rings. And regarding generalization to other algebraic contexts, you may be interested in this paper.
(1) Yes, I think that's one of the ways to define schemes. Look for representable functors and you'll get lots of literature.
It was a crazy idea about 50 years go, part of establishment nowadays.
I'm not an expert, but I think in (3) it's crucial that rings can be localized. I think there's some notion of localizability in category theory and it boils down to something any localizable thing is a (subthing) of sheaves on a site (the formal statement is "any presentable category can be obtained as a localization of some category of sheaves of sets").
For (4) I think the situation is quite simple. Schemes are easy to imagine for most people, so people work in scheme language unless there's a need for more general topoi.
Here are also my earlier questions:
- What is a topos?
- How to think about model categories?