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$\DeclareMathOperator{\Spec}{Spec}$ Consider the following procedure for defining a class of morphisms of schemes:

  1. Take a suitable class of homomorphisms of rings (e.g., canonical maps to localization at an element, standard étale homomorphisms, faithfully flat homomorphisms).
  2. Identify the category of rings with the opposite category of affine schemes. This identifies our chosen ring homomorphisms with certain morphisms of affine schemes.
  3. Consider the morphisms $f\colon X\to Y$ of schemes with the following property: whenever $\Spec A$ is an affine scheme with a morphism $\Spec A\to Y$, the fibre product $\Spec A\times_Y X$ is affine and the natural map $\Spec A\times_Y X\to\Spec A$ corresponds to one of our chosen ring morphisms from step (1).

I'm really interested in this process because I think it might give a way to define the Zariski/étale/flat topologies by defining a topology on the opposite category of commutative rings and then extending it as follows:

  1. Identify the category of schemes with the full subcategory of the topos of sheaves on $\mathbf{CRing}^{\text{op}}$ given by the functor of points (assuming that these functors of points are indeed sheaves).
  2. Take the canonical topology on the topos of sheaves, and restrict to the category of schemes.

Some pages on the nLab seem to suggest that this should yield certain important topologies on the category of schemes.

  1. If we start with the Grothendieck pretopology on $\mathbf{Aff}$ given by jointly surjective families of morphisms corresponding to localization-at-a-single-element maps, do we get the Zariski topology?
  2. If we start with the pretopology given by jointly surjective families of formal duals of standard étale homomorphisms, do we get the étale topology?
  3. If we start with the pretopology given by jointly surjective families of formal duals of faithfully flat homomorphisms, do we get the flat topology?

At the ring level, jointly surjective should correspond to jointly injective, since localization maps, standard étale homomorphisms etc. behave nicely topologically (they give closed maps).

Pretopologies for the Zariski, étale and flat topologies are given by jointly surjective families of Zariski embeddings, étale morphisms and fpqc/fppf morphisms respectively.

In this case, the procedure defined at the start of the question does the following (replacing [...]/[***] with localization map/Zariski embedding, standard étale/étale and faithfully flat/fpqc):

  1. Take the topology on $\mathbf{Aff}$, given by jointly surjective families of [...] homomorphisms.
  2. Form a pretopology generating the canonical topology on the topos of presheaves on $\mathbf{Aff}$ in a standard way, and restrict it to the category of schemes.
  3. This pretopology should correspond to jointly surjective families of certain morphisms of schemes. Then the [***] morphisms are those morphisms that appear in a covering family.

Is this correct? For example, are the étale morphism of schemes precisely those morphisms $f\colon X\to Y$ that appear in some covering family in the canonical extension to the category of schemes of the 'standard étale topology' on the opposite category of commutative rings? More concretely, are the étale morphisms precisely those morphisms $f\colon X\to Y$ such that

whenever $\Spec A$ is an affine scheme with a morphism $\Spec A\to Y$, the fibre product $\Spec A\times_Y X$ is affine (say $\Spec A\times_Y X\cong\Spec B$) and the natural map $\Spec A\times_Y X\to\Spec A$ corresponds to a standard étale morphism $B\to A$

?

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    $\begingroup$ Like @c_c_chaos said, you are requiring your étale morphisms to be affine (in the last paragraph). However I quite like your idea. I think it could work if one relaxes the condition a little and onyl require it (Zariski-)locally. For instance, an étale morphism is Zariski-locally a standard étale morphism. $\endgroup$ – Ingo Blechschmidt Apr 11 '17 at 7:53
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Unless I missed something, this cannot work because you are requiring way too many schemes to be affine. Indeed, just consider the case that $Y = \operatorname{Spec} A$ is itself affine. You are then asking whether for any open immersion/étale morphism/"fpqc on its image" morphism $X \to Y$ the source $X$ is necessarily affine which is wrong in all three cases. For example, in the Zarisiki case, there clearly are affine schemes $Y$ which have non-affine open subschemes.

However, I do not really understand what you are trying to do, so it might be possible that your idea works after making suitable modifications.

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