Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable under pullback (i.e. pullbacks of them exist, and they are stable). Each map is to be considered a covering family with a single element.

Consider the following two conditions:

  1. $J$ is saturated: If $U \to V\to X$ is in $J$ then $V \to X$ is in $J$.

  2. $J$ is admissible: $J$ contains the split epimorphisms, and if $U \to V\to X$ is in $J$ and $U \to V$ is a split epimorphism, then $V\to X$ is in $J$.

Now clearly saturated singleton pretopologies are admissible (notice saturated implies $J$ contains the split epis). An example of a saturated pretopology is the class ($K$-epi) of $K$-epimorphisms in a fintely complete category: maps $p:Q\to X$ such that there is a $K$-cover $k:U\to X$ and a map $s:U\to Q$ with $p\circ s = k$ (or more generally local sections for any pretopology, not necc. singleton). Now my question is this:

What is an example of an admissible singleton pretopology which is not of the form ($K$-epi) for some other pretopology $K$?

I know the general situation in categories like $Top$, $Diff$, but not in algebraic settings. As far as I know, pretopologies like ($K$-epi) don't turn up much (I may be wrong, and happy to be corrected), but do admissible singleton pretopologies otherwise arise in algebraic geometry?

  • $\begingroup$ there's a typo in the title $\endgroup$ – Yemon Choi Oct 17 '10 at 2:36
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    $\begingroup$ I've found in the past here that people basically never answer questions about grothendieck topologies. In fact, I asked a question recently concerning them, and I got a bunch of votes up after I removed the topos-theory tag. The trick is to make your question look like it's about algebraic geometry. Also, the definition you're using of a singleton topology is confusing (you should note that the morphisms are supposed to be single-morphism covering-families). $\endgroup$ – Harry Gindi Oct 17 '10 at 12:36
  • $\begingroup$ But I gave you a +1 anyway because I don't want to sound like a complainer =p. $\endgroup$ – Harry Gindi Oct 17 '10 at 12:37
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    $\begingroup$ I think other people actually filter MO by tags, as opposed to scan all the questions for interesting things like some of us. $\endgroup$ – theHigherGeometer Oct 17 '10 at 21:39

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