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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 29th 2010

    See this MO question. I got the answer wrong, because I forgot the definition of sieves generated by a Grothendieck pretopology. However, the sort of thing in the question I use in my thesis (albeit the singleton version), so now I curious as to how this works. The following is my thought process, please tell me if it is wrong.

    If J is a pretopology on a category CC with pullbacks, then let JJ-epi denote the singleton pretopology with covers maps that admit local sections over a JJ-cover. Now if CC is extensive, JJ-epi is the same as ⨿J\amalg J-epi, where ⨿J\amalg J is a class of maps ⨿U iX\amalg U_i \to X for {U iX}\{ U_i \to X\} a JJ-covering family. If JJ isn’t a superextensive pretopology, then JJ-sheaves are not necessarily the same as ⨿J\amalg J-sheaves, but sheaves for JJ-epi can’t be sheaves for both JJ and ⨿J\amalg J, so we have a pretopology KK on CC countering the claim that KK-sheaves are the same as KK-epi sheaves.

    However, the question relates not to JJ-epi, but to a non-singleton version: the new covering families are such that old covering families factor through them. We can think about just the sheaf condition for a single object XX. I could take a new covering family of XX to be {U i⨿YX}\{ U_i \amalg Y \to X\} for any old map YXY \to X, and hardly expect the original sheaves to be sheaves for this covering family.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 29th 2010

    Incidentally, in the reference - Goodwillie and Lichtenbaum’s paper on the h-topology - the OP (what on earth does that mean??) gave, it mentions the f-topology on schemes as being the smallest topology generated by the extensive topology and the topology of finite surjections. Another good example for a superextensive (pre)topology. G&L cite SGA4 for their definition of ’topology’ and that the question has an obvious answer ’yes’. I don’t have access to SGA4 at the moment, so I can’t check the precise definition. It may be hidden in there.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 29th 2010

    Also G&L mention a condition of a topology having enough points: that a sheaf if trivial iff all its stalks are trivial. Do we have something like this on the Lab? I really need to get back to work, else I’d dig around myself..

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeOct 29th 2010

    OP = Original Poster. I added an answer to the MO question; despite what one might “hardly expect,” it is actually true. (-:

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 29th 2010

    Cool, thanks!

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 29th 2010

    I suppose the obvious thing to ask is whether this is true for sites without all pullbacks, and replacing local-section-admitting maps (or the non-singleton version as in the question) with only those of which all pullbacks exist.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 29th 2010

    The version in the Elephant is stated for categories without pullbacks. The right thing to do is still talk about local-section-admitting maps, just phrase it without using pullbacks.