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nLab > Latest Changes: site-dependence of model structure on simplicial presheaves

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2009

    I started a section

    dependence on the underlying site at model structure on simplicial presheaves.

    So far this quotes a result from Jardine's lectures and then looks a bit at an example.

    At that example I would really like to conclude that the Quillen adjunction discussed there is actually a Quillen equivalence. But I have to interrupt now to make a telephone call... :-)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2009

    a supposed proof that indeed we have a Quillen equivalence

    SPSh(Diff)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPsh(CartSp)_{inj}^{loc}

    is now

    here in that entry

    of course Diff and CartSp is just one specific example. In as far as the proof is correct, it will work for all such pairs, for instance Schemes vs AffineSchemes .

    But check if it is indeed correct.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2009

    now also added a section good covers with a definition that I think is good and with a proposition that I think is true and useful.

    However, it seems at this point of the night I seem to be unable to write down precisely what in my head seems to be the obvious proof. I wrote down something there, but need to get back to it. Maybe I am wrong, but I am not convinced of that as yet ;-)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2009

    I think I got it right now.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2009

    Danny Stevenson kindly wrote in to say that the fact in question

    (that every simplicial presheaf that is degreewise a coproduct of representables is cofibrant in alll these model structures)

    should be true and should be stated somewhere in Dan Dugger's work.

    He indicates a proof which sounds very much along the lines of the proof that I did give.

    So I regard this as settled for the time being, removed the green query boxes and just left in an indented remark that for the time being the proof is one I dreamed up which still deserves checking.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2009

    renamed the section in question fibrant and cofibrant objects and expanded further.

    meanwhile Danny Stevenson writes in and points out various even stroger statements from the literature. Will try to include them after lunch...

1 to 6 of 6

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