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


1. 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?



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](https://mathoverflow.net/questions/606/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](https://www.repository.cam.ac.uk/handle/1810/256998) 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). 


