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Speculation and background

Let $\mathcal{C}:=CRing^{op}_{Zariski}$, the affine Zariski site. Consider the category of sheaves, $Sh(\mathcal{C})$.

According to nLab, schemes are those sheaves that "have a cover by Zariski-open immersions of affine schemes in the category of presheaves over Aff."

In SGA 4.1.ii.5 Grothendieck defines a further topology on $Sh(\mathcal{C})$ using a "familles couvrantes", which are families of morphisms $\{U_i \to X\}$ such that the induced map $\coprod U_i \to X$ is an epimorphism. Further, he gives another definition. A family of morphisms $\{U_i \to X\}$ is called "bicouvrante" if it is a "famille couvrante" and the map $\coprod U_i \times_X to \coprod U_i \to times_X \coprod U_i$ is an epimorphism. [Note: This is given for a general category of sheaves on a site, not sheaves on our affine Zariski site.]

Speculation: I assume that the nLab definition means that we have a (bi)covering family of open immersions of representables, but as it stands, we do not have a sufficiently good definition of an open immersion, or equivalently, open subfunctor.

It seems like the notion of a bicovering family is very important, because this is precisely the condition we require on algebraic spaces (if we replace our covering morphisms with etale surjective morphisms in a smart way and require that our cover be comprised of representables).

Questions

What does "open immersion" mean precisely in categorical langauge? How do we define a scheme precisely in our language of sheaves and grothendieck topologies? Preferably, this answer should not depend on our base site. The notion of an open immersion should be a notion that we have in any category of sheaves on any site.

Eisenbud and Harris fail to answer this question for the following reason: they rely on classical scheme theory for their definition of an open subfunctor (same thing as an open immersion). If we wish to construct our theory of schemes with no logical prerequisites, this is circular.

Once we have this definition, do we require our covering family of open immersions to be a "covering family" or a "bicovering family"?

Further, how can we exhibit, in precise functor of points language, the definition of an algebraic space?

This last question should be a natural consequence of the previous questions provided they are answered in sufficient generality.

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Precise definition of a scheme (FunctorofpointsKeyquestion:Howtodefineanopensubfunctorwithoutresortingtoclassicalschemetheory)

Speculationandbackground

In

Speculation: I assume that the usual nLab definition of a scheme, it seems like means that we require the sheaves to have a "famille couvrante" given by (bi)covering family of open immersions of representables, and we require that a separated scheme have a "famille bicouvrante" by representables (but as it might be the case here that we actually require "familles bicouvrantes" regardlessstands, and for separated schemes we require that we do not have a "famille bicouvrante" over $Spec(\mathbb{Z})$). Howeversufficiently good definition of an open immersion, this still doesn't seem totally rightor equivalently, open subfunctor.

It seems like the notion of a bicovering family is very important, because this is precisely the condition we don't just want morphisms from affines, require on algebraic spaces (if we specifically want "open immersions". I suspect replace our covering morphisms with etale surjective morphisms in a smart way and require that we further want our "familles (bi)couvrantes" to cover be families of monomorphisms of representables, but I'm not sure comprised of thisrepresentables).

Then the questions:

Questions

What does "open immersion" mean precisely in categorical langauge(with respect to our familles (bi)couvrantes)? ? How do we define a scheme precisely in our language of sheaves and grothendieck topologies?

Is the category of Schemes itself a topos Preferably, this answer should not depend on the category our base site. The notion of sheaves equipped with an open immersion should be a topology defined by "familles (bi)couvrantes" with appropriate restrictions (affine monomorphisms maybe)? [Yes, I know notion that usually the we have in any category of sheaves on a any site"extends" .

Eisenbud and Harris fail to answer this question for the categoryfollowing reason: they rely on classical scheme theory for their definition of an open subfunctor (same thing as an open immersion). If we wish to construct our theory of schemes with no logical prerequisites, but this topology is not at all subcanonical (with respect circular.

Once we have this definition, do we require our covering family of open immersions to $Sh(\mathcal{C})$).]be a "covering family" or a "bicovering family"?

Further, how can we exhibit, in precise functor of points language, the definition of an algebraic space? (Wikipedia notes that we can just as well require that the equivalence relation is given by affines rather than general schemes. I don't see how this can be true unless we require that algebraic spaces have an equivalence relation given by separated schemes, but please, enlighten me.)

My guess on this last one: It seems like taking sheaves on the sitey $CRing^{op}_{\acute{e}tale}$, applying the same criteria we used for schemes, and further requiring that all covering families have the property that they are "familles bicouvrantes" over $Spec(\mathbb{Z})$ is enough. (

This is based on the assumption from Wikipedia.)

I hope I did enough research and haven't made too many mistakes.

Edit: It appears that having affine diagonal is weaker than being separated (oh well?). Then this leads to a further last question : If we require that algebraic stacks have affine diagonal, do we still have should be a respectable theory natural consequence of algebraic stacks?

Edit2: According to a footnote in SGA, being "couvrant" et "bicouvrant" are stable under base change, so the requirements over $Spec(\mathbb{Z})$ previous questions provided they are unnecessaryanswered in sufficient generality.

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