Functor of points definition of a space modeled on a site I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples.
Let $(C,J)$ be a grothendieck site and $Sh(C,J)$ the corresponding sheaf topos. Let $P$ be a property of morphisms in $C$ which containes all isomorphisms and is stable under composition and pullbacks. The aim is to define a space as a $J$-sheaf with a cover by representable sheaves that glue by $P$ morphisms.

Definition 1: A "nice" $P$-local space is a sheaf $F \in Sh(C,J)$ s.t. there exists a representable sheaf $h_U$ and an epimorphism $\varphi: h_U \to F$ satisfying the following:

*

*$\varphi: h_U \to F$ is representable and has property $P$ (meaning all pullbacks to representable sheaves have property $P$).


Example 1: If $C= Aff$, $J=Zariski$ and $P=\text{open immersion}$, we get a a scheme.
Example 2: If $C=Aff$, $J= \text{etale}$ and $P= \text{etale}$ we get an algebraic space (I hope. Separability conditions for algebraic spaces vary so I'm not so sure about the conventions here).
Example 3: If $C= \text{CartSp}$, $J = \text{open immersion}$ and $P =C^{k}$ we get a $C^k$-manifold.
Here is the wierd part. The definition above (although rather natural) only gives us some subcategory of "nice" schemes/algebraic spaces.

Question 1: What subcategory of schemes does Example 1 correpond to? If I require that the covers be finite coproducts of affine scheme do I get qcqs schemes?

To get the category of schemes we need an extra step:

Definition 2: A general $P$-local space is a sheaf $F \in Sh(C,J)$ s.t. there exists a representable sheaf $h_U$ and an
epimorphism $\varphi: h_U \to F$ satisfying:

*

*$h_U \times_F h_U$ is a "nice" $P$-local space.

*Both projections $\pi_i : h_U \times_F h_U \to h_U$ are representable by a $P$ morphism.

Question 2: Is the above condition equivalent to requiring that the
diagonal $h_U \to h_U \times h_U$ be representable by a separated
$P$-local space?

Now wer'e fine since any intersection of affines is a quasi-affine scheme which satisfies definition 1.

Question 3: Why is this step reasonable from this point of view? Does this have something to do with closure under pullbacks? Does "definition 1" generate "definition 2" by taking arbitrary pullbacks?.

 A: First things first: we have to define what open immersions are. (!) We take for granted some class of basic open immersions – in the case of schemes, the ones corresponding to principal localisations of rings – and then close under unions and descent. (Making this precise is rather complicated if one wants to be general.)
Once we know what open immersions are, we can define local isomorphisms: these are the morphisms $X \to Y$ for which there is a cover of $X$ by open immersions $U \to X$ such that the composite $U \to X \to Y$ is also an open immersion. This is a completely straightforward generalisation of the definition of local homeomorphism.
Finally, we can define "spaces": these are the sheaves $Y$ for which there is an epimorphic local isomorphism $X \twoheadrightarrow Y$ where $X$ is a disjoint union of representable sheaves. This actually gives all schemes – to cut down to qcqs schemes, simply restrict to finite unions everywhere.
As far as I know, there is no shortcut that completely avoids the notion of local isomorphism. Demazure and Gabriel simply unfold the definition of local isomorphism into the definition of scheme.
