Several Topos theory questions Hey. I have a few off the wall questions about topos theory and algebraic geometry.


*

*Do the following few sentences make sense? 


Every scheme X is pinned down by its Hom functor Hom(-,X) by the yoneda lemma, but since schemes are locally affine varieties, it is actually just enough to look at the case where "-" is an affine scheme.  So you could define schemes as particular functors from CommRing^op to Sets.  In this setting schemes are thought of as sheaves on the "big zariski site".
If that doesn't make sense my next questions probably do not either.
2 The category of sheaves on the big zariski site forms a topos T, the category of schemes being a subcategory. It is convenient to reason about toposes in their own "internal logic". Has there been much thought done about the internal logic of T, or would the logic of T require too much commutative algebra to feel like logic?  Along these lines, have there been attempts to write down an elementary list of axioms which capture the essense of this topos?  I am thinking of how Anders Kock has some really nice ways to think of differential geometry with his SDG.
3 What is it about the category of commutative rings which makes it possible to put such a nice site structure on it, but not other algebraic categories? Gluing rings together lead to huge advancements in algebraic geometry. What about gluing groups?  Is there a nice Grothendieck topology you could put on Groups^op, and then you could start studying sheaves on this site? If not, why not - what about rings makes them so special?
4 Why do people work with the category of schemes instead of the topos of sheaves on CommRing^op - toposes have every nice categorical property you could possibly ask for.
About me:  I am a 1st year grad student who is taking a first course on schemes, and I just have a lot of crazy ideas floating around.  I don't feel comfortable engaging in such wild speculation with my professors.  Could you offer any insight into these ideas?
 A: 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). 
A: (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?
A: One short answer to #4:  If you're trying to do geometry with sheaves on a category of affine objects, you want to work with sheaves that are "geometric" in the sense that they can be covered by affine objects. 
A: 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.
A: PROBLEM 1 
people do think scheme like that. Category of associative (not necessarily commutative) k-algebra is equivalent to category of affine schemes. We have Yoneda embedding to category of presheaves of sets. It is well known that considering presheaves of sets is equivalent to consider presheaves of some category. Then all the representable functor are affine schemes. In fact, it is not neccessary to consider category of sheaves but category of presheaves. There is huge advantage to consider category of presheaves instead of category of sheaves because we have very good functoriality in presheave category. Once one need sheaves, he just need to take sheafification functor according to subcanonical topology to get sheaves. 
