Example of a Grothendieck pretopology satisfying a weak saturation condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:40:26Z http://mathoverflow.net/feeds/question/42437 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42437/example-of-a-grothendieck-pretopology-satisfying-a-weak-saturation-condition Example of a Grothendieck pretopology satisfying a weak saturation condition David Roberts 2010-10-16T22:55:07Z 2010-10-17T21:38:37Z <p>Recall that a <em>singleton Grothendieck pretopology</em> (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.</p> <p>Consider the following two conditions:</p> <ol> <li><p>$J$ is saturated: If $U \to V\to X$ is in $J$ then $V \to X$ is in $J$.</p></li> <li><p>$J$ 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 $J$.</p></li> </ol> <p>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 <em>$K$-epimorphisms</em> 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:</p> <blockquote> <p>What is an example of an admissible singleton pretopology which is <em>not</em> of the form ($K$-epi) for some other pretopology $K$?</p> </blockquote> <p>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?</p>