[Math] Categorical construction of the category of schemes

Question

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" question may be (rather then stating that the answer is 42;))

Is there a purely categorical procedure that takes the category of commutative rings as input and produces the category of schemes (over $\mathbf{Z}$) as output?

A possible place to start would be to consider a scheme $X$ as a functor from the category $CommRing$ of commutative rings to the category of sets: $A\mapsto Hom_{Sch}(Spec(A),X)$ where $A$ a commutative ring. If we instead of $Spec(A)$'s we consider all schemes, then we simply get the Yoneda embedding. But some questions arise.

1. Does this give a fully faithful functor from schemes to functors from commutative rings to sets? Or loosely speaking, do $Spec(A)$-valued points ($A$ a commutative ring) suffice to determine a scheme? (My guess is that the answer is yes and this is classical.)

2. Is there a way to characterize those functors that actually come from schemes? For example one can introduce a Grothendieck topology on $CommRing$ (or its opposite) and require that the functor should be a sheaf in that topology. But in that case, can one describe the topology without referring to the fact that the objects of $CommRing$ are commutative rings? (Here my guess is the first question is probably too complicated but there are some necessary conditions.)

3. Regardless whether the answer to 2. is positive or negative, is there a way to describe algebraic spaces or stacks as presheaves on $CommRing^{op}$ that satisfy some conditions?

Answer 1

1. The highbrow way of reformulating your question is as follows. Consider the category $Sch$ of all schemes endowed with the Zariski topology. There is a fully faithful embedding of the category of affine schemes $Aff = CommRing^{op}$ into $Sch$; the topology induced on $Aff$ by that on $Sch$ is also the Zariski topology. The comparison lemma ([SGA4] III, 4.1) then says that, because any object in $Sch$ can be covered by objects in $Aff$, the categories of sheaves on both sites are equivalent. In particular, representable sheaves in $Sch$ (i.e., schemes) are determined by their values on affine schemes.
2. For a sheaf $F$ on $Aff$ to be represented by a scheme it is enough that it be covered by affine schemes, i.e., that there exist affine schemes $U_i$ together with open immersions $U_i \to F$ (you have to define what this means, of course) such that $\coprod_i h_{U_i} \to F$ is an epimorphism of sheaves. Actually, you can take this as a definition of schemes. The compatibility of the gluings in the classical definition is taken care of here by the sheaf condition.
3. Algebraic spaces can be similarly defined. While I was writing this, Harry beat me to giving the reference to the excellent notes of Bertrand Toën from a course of his on algebraic stacks.

In 2, you also ask if you can construct schemes from $Aff$ without actually using the fact that you are dealing with commutative rings. I think not. The categorical nonsense can get you only so far: at some point you have to introduce the geometry itself, and that is given by the $Aff$ with its topology. If you replace $Aff$ by the category of open sets in some $\mathbb{R}^n$ with open immersions you would end up defining manifolds. This is what Toën calls geometric contexts.