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Dear Theo, I think that you're oversimplifying things a bit too much here. The notion of a submersion depends very much on an "admissibility structure" in the sense of Lurie, or a "geometric context" in the sense of Toën-Vezzosi. That is, in addition to a Grothendieck topology, you also need a "geometry" satisfying certain properties to give further structure to your category.

I was confused a while ago about a similar point, and after learning more about the subject, I realized that it's not nearly so simple as I had hoped.

For instance, the proper notion of a submersion in the algebro-geometric context is a smooth map, but there is no notion of a smooth map between sheaves on the affine étale site before first discussing what it means for a morphism to be "relatively representable". You may want to check out Toën-Vezzosi's paper "Homotopical algebraic geometry II", where they give an inductive definition of an n-geometric algebraic stack (where stack here means simplicial sheaf) (and this inductive definition holds true for sheaves of sets as well). For a map between schemes to be smooth (resp. a submersion) you need for the map to be "relatively representable" by a "scheme" (resp. a "manifold") (when you restrict to the case of sheaves of sets, $n$ really only varies between $-1$, $0$, and $1$).

This may all sound like gibberish, but if you take a look at Toën-Vezzosi's HAG II chapter 2 and ignore the homotopical stuff, the basic idea should be clear. The moral of the story is that a Grothendieck topology alone cannot characterize the geometry of the sheaves on that site.

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Dear Theo, I think that you're oversimplifying things a bit too much here. The notion of a submersion depends very much on an "admissibility structure" in the sense of Lurie, or a "geometric context" in the sense of Toën-Vezzosi. That is, in addition to a Grothendieck topology, you also need a "geometry" satisfying certain properties to give further structure to your category.

I was confused a while ago about a similar point, and after learning more about the subject, I realized that it's not nearly so simple as I had hoped.

For instance, the proper notion of a submersion in the algebro-geometric context is a smooth map, but there is no notion of a smooth map on the affine étale site before first discussing what it means for a morphism to be "relatively representable". You may want to check out Toën-Vezzosi's paper "Homotopical algebraic geometry II", where they give an inductive definition of an n-geometric algebraic stack (where stack here means simplicial sheaf) (and this inductive definition holds true for sheaves of sets as well). For a map between schemes to be smooth (resp. a submersion) you need for the map to be "relatively representable" by a "scheme" (resp. a "manifold") (when you restrict to the case of sheaves of sets, $n$ really only varies between $-1$, $0$, and $1$).

This may all sound like gibberish, but if you take a look at Toën-Vezzosi's HAG II chapter 2 and ignore the homotopical stuff, the basic idea should be clear. The moral of the story is that a Grothendieck topology alone cannot characterize the geometry of the sheaves on that site.