Post Made Community Wiki by Harry Gindi
5 "locally cover" doesn't mean anything. I just meant cover.

Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (Schemes) on $CRing^{op}$. That is, taking the full subcategory of $Psh(CRing^{op})$ which consists of objects $S$ such that $S$ is a sheaf in the global Zariski topology and $S$ has a cover by representables in the induced topology on $Psh(CRing^{op})$. This is the category of schemes. Then, one takes this category and equips it with the etale topology and repeats the construction of locally representable sheaves on this site (Sch with the etale topology) to get the category of algebraic spaces.

Can we "skip" the category of schemes entirely by putting a different topology on $CRing^{op}$?

My intuition is that since every scheme can be locally covered by affines, and every algebraic space can be locally covered by schemes, we can cut out the middle-man and just define algebraic spaces as locally representable sheaves for the global etale topology on $CRing^{op}$. If this ends up being the case, is there any sort of interesting further generalization before stacks, perhaps taking locally representable sheaves in a flat Zariski-friendly topology like fppf or fpqc?

Some motivation: In algebraic geometry, all of our data comes from commutative rings in a functorial way (intentionally vague). All of the grothendieck topologies with nice notions of descent used in Algebraic geometry can be expressed in terms of commutative rings, e.g., the algebraic and geometric forms of Zariski's Main theorem are equivalent, we can describe etale morphisms in terms of etale ring maps, et cetera. What I'm trying to see is whether or not we can really express all of algebraic geometry as "left-handed commutative algebra + sheaves (including higher sheaves like stacks)". The functor of points approach for schemes validates this intuition in the simplest case, but does it actually generalize further?

The main question is italicized, but feel free to tell me if I've incorrectly characterized something in the motivation or the background.

Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (Schemes) on $CRing^{op}$. That is, taking the full subcategory of $Psh(CRing^{op})$ which consists of objects $S$ such that $S$ is a sheaf in the global Zariski topology and $S$ has a cover by representables in the induced topology on $Psh(CRing^{op})$. This is the category of schemes. Then, one takes this category and equips it with the etale topology and repeats the construction of locally representable sheaves on this site (Sch with the etale topology) to get the category of algebraic spaces.

Can we "skip" the category of schemes entirely by putting a different topology on $CRing^{op}$?

My intuition is that since every scheme can be locally covered by affines, and every algebraic space can be locally covered by schemes, we can cut out the middle-man and just define algebraic spaces as locally representable sheaves for the global etale topology on $CRing^{op}$. If this ends up being the case, is there any sort of interesting further generalization before stacks, perhaps taking locally representable sheaves in a flat Zariski-friendly topology like fppf or fpqc?

Some motivation: In algebraic geometry, all of our data comes from commutative rings in a functorial way (intentionally vague). All of the grothendieck topologies with nice notions of descent used in Algebraic geometry can be expressed in terms of commutative rings, e.g., the algebraic and geometric forms of Zariski's Main theorem are equivalent, we can describe etale morphisms in terms of etale ring maps, et cetera. What I'm trying to see is whether or not we can really express all of algebraic geometry as "left-handed commutative algebra + sheaves (including higher sheaves like stacks)". The functor of points approach for schemes validates this intuition in the simplest case, but does it actually generalize further?

The main question is italicized, but feel free to tell me if I've incorrectly characterized something in the motivation or the background.

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