Speculation and background
--------------------------


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


According to [nLab](https://ncatlab.org/nlab/show/scheme#translation_between_the_two_approaches), 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 $\operatorname{Sh}(\mathcal{C})$ 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 $\coprod U\_i \to \coprod U\_i \times\_X \coprod U\_i$ 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.



Precise definition of a scheme (Key question: How to define an open subfunctor without resorting to classical scheme theory)
Check out the paper of Kontsevich-Rosenberg [noncommutative space.](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), they defined formally open immersion and open immersion completely functorial way. This definition is nothing to do with "noncommutative" 


Definition:
Formally open immersion is formally smooth monomorphism.


But one thing need to point out, they are working on Q-category which is a generalization of Grothendieck topologies(destination is dealing with topology without base change property).But just disregard this notion and work in usual grothendieck topology as you want


