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?