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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2010

    shouldn’t 0-site be named (0,1)-site?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2010

    IMHO, yes. Although 0-site should probably redirect to it, since there isn’t really any useful notion of a (0,0)-site.

    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeJul 7th 2010

    Although 0-site should probably redirect to it, since there isn’t really any useful notion of a (0,0)-site.

    Some people find these trivialities to be interesting. I have no clue at all about this, but I wonder if Toby (the king of nothingness) could say something interesting about that nothing? :)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2010
    • (edited Jul 7th 2010)

    by the way, we also have (0,1)-topos

    @Eric: there is some discussion of the (0,1)-category-logic at (n,r)-category and at poset

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeJul 8th 2010

    I don’t know whether ‘nn-site’ should follow ‘nn-sheaf’ or ‘nn-topos’. The former (as it is used on the Lab) has an implied (,1)(-,1) in it which the latter lacks. Perhaps it shouldn’t.

    I think that this mathematical question gets at the real issue: is there likely to be a useful notion of (2,2)(2,2)-site? If so, then we should call this notion ‘22-site’; if not, then we may call a (2,1)(2,1)-site (which notion should certainly exist and be useful) a ‘22-site’. Then ‘00-site’ can follow.

    By the way, sorry that I forgot to announce this new page. (I had to go to dinner, and when I got back, I forgot that I hadn’t announced it.)

    It would also be nice if there were a special term for a (0,1)(0,1)-site, just as we have ‘poset’ for a (0,1)(0,1)-category, ‘Heyting algebra’ for a (0,1)(0,1)-topos (with logical morphisms), ‘locale’ for a Grothendieck (0,1)(0,1)-topos (with geometric morphisms), etc. But my source material, Johnston, just says ‘site’!

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2010
    • (edited Jul 8th 2010)

    (…)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2010

    Second attempt at a reply, now that I have actually read your message correctly :-):

    I went through some pain trying to consistently write (infinity,1)-sheaf everywhere, with the (,1)(-,1)-explicit. As opposed to infinity-stack (which is debateable, but not the issue here.)

    So I think the convention is that an nn-thing is an (n,n)(n,n) thing and with that and the above, the category-numbering of higher sheaves on the nnLab should be pretty consistent.

    In that vein, I again opt for (0,1)(0,1)-site!

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeJul 8th 2010

    Sorry, I mixed up your usage of ‘nn-sheaf’ with your usage of ‘nn-stack’.

    Do you expect there to be such a thing as a (2,2)(2,2)-site?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 8th 2010

    There is definitely an important notion of (2,2)-site. Why do you say that n-sheaf has an implied (-,1)? I’d be very unhappy about that; I like (2,2)-sheaves a lot.

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJul 8th 2010

    There is definitely an important notion of (2,2)-site.

    Right, OK, then I’m convinced.

    Why do you say that n-sheaf has an implied (-,1)?

    Confusion. See may last comment (and Urs’s comment that it replies to.)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2010
    • (edited Jul 8th 2010)

    By the way, Igor Bakovic is currently thinking about (2,2)(2,2)-sheaves, too In Oberwolfach he briefly presented a (2,2)(2,2)-analog of the equivalence sheaves \simeq Etale spaces.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 9th 2010

    Yes, I remember that Igor’s work came up on the Cafe somewhere. Most of what’s on that page I linked to is from Street’s papers on 2-dimensional sheaves from a while ago (and could & probably should be on the main nLab, actually).

    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeJul 9th 2010

    I have moved 0-site to (0,1)-site.

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 9th 2010

    By the way, Igor Bakovic is currently thinking about (2,2)-sheaves, too In Oberwolfach he briefly presented a (2,2)-analog of the equivalence sheaves \simeq Etale spaces.

    Any relation to 2-covering spaces? They should be the (2,1)-version. (I hope he has at least heard of my thesis :)

    • CommentRowNumber15.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 9th 2010

    I disagree with the use of “stack” to mean “stack of groupoids”, since this is actually incorrect. Therefore, the (oo,1) should be made explicit.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 9th 2010

    It would also be nice if there were a special term for a (0,1)-site

    I’m sure I’ve heard Johnstone use “posite,” but I forget whether it was in print or not.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2010

    Mike says:

    (and could & probably should be on the main nLab, actually).

    I’d think so, too.

    David says:

    (I hope he has at least heard of my thesis :)

    I don’t know if he is aware of it. You 2-guys should get in touch with him.

    • CommentRowNumber18.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 9th 2010

    I don’t know if he is aware of it. You 2-guys should get in touch with him.

    That was criminal…

    Criminally funny, that is.

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2010

    We need to go to a 2-conference, where we not only have talks, but 2-talks, where the ways in which the talks are related are explained.

    • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeJul 13th 2010

    @ David

    I thought that you were just giving another silly joke, but you’re absolutely right!