While reading Demazure-Gabriel's construction of $\mathcal{S}ch$ as a full subcategory of $\mathcal{P}sh(CRing^{op})$, I've been trying to translate their exposition into the language of covering sieves and Grothendieck topologies. The requirement that a scheme be a sheaf in the Zariski topology on $CRing^{op}$ is essentially already in the language of sieves, but the requirement that there exists a cover of affines is giving me some trouble.


Question, then:


In general, is there a natural induced Grothendieck topology on $\mathcal{P}sh(\mathcal{C})$ or $\mathcal{S}h(\mathcal{C})$, where $\mathcal{C}$ is a site (not a pre-site!)?


Edit: I removed the other parts of the question regarding t-Schemes and algebraic spaces as functors of points to ask them at some other time.



Induced Grothendieck topology on a presheaf or sheaf category of a site?
The answer to your first question is yes. Suppose $C$ a site.


The category $C^{\sim}$ of sheaves on $C$ simply has the canonical topology (SGA 4, Vol. 1, Exp. II, 2.5). The sheaves here are precisely the representable sheaves. (I'm ignoring questions about universes.)


The category $C^{\wedge}$ of presheaves on $C$ inherits a topology in the following manner (SGA 4, Vol. 1, Exp. II, §5). Declare a morphism $F\to G$ of presheaves on $C$ to be a *covering morphism* if the induced morphism $aF\to aG$ of associated sheaves is an epimorphism. Now say that a collection of morphisms $F\_i\to G$ is a *covering family* if the morphism $\amalg F\_i\to G$ is a covering morphism. This gives $C^{\wedge}$ the finest subcanonical topology such that covering families in $C$ give rise to covering families in $C^{\wedge}$. This topology on $C^{\wedge}$ can be seen as the lift of the topology on $C^{\sim}$ described above along the sheafification functor $a$.


