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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 1st 2011

    I have edited posite

    In particular I tried to work the query box into the text. Mike and David R. please check if you agree with the result.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 1st 2011

    Thanks. I added a bit more, and changed the “proof” to be more like a proof, with the description of the locale coming afterwards.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJan 2nd 2011

    What’s the relationship between the pages posite and (0,1)-site?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2011

    It seems to me that these pages should be merged and the hierarchy of general and special cases should be sorted out. At (0,1)-site there are some additional conditions being imposed (the poset is required to be a meet-semilattices, the sheaves are required to take values just in truth values) which need not be imposed in the general case.

    Also; I think that the decision at (0,1)-site to say just “site” for this concept is a bit dangerous.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeJan 2nd 2011

    See the References at the bottom of (0,1)-site; that page follows Johnstone (who says “site”, etc throughout).

    The really big difference is that posite takes the standard notion of a sheaf on a posite to be a 11-sheaf, while (0,1)-site (following Johnstone) takes the standard notion of sheaf on a (0,1)(0,1)-site to be a (0,1)(0,1)-sheaf. More broadly, posite is about topos theory, while (0,1)-site is about locale theory.

    Since one of my projects with the nLab is to use it record my understanding of locale theory as I learn about it, I would like to have a page about sites in locale theory. I don’t know that this needs to be separate from the page about posites in topos theory, however. But merging them won’t be as easy as it appears at first glance.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeJan 2nd 2011

    I should have written comment #3 as

    What’s the intended relationship between the pages posite and (0,1)-site?

    That is, intended by the authors of posite (which came afterwards). One possible answer being that they didn’t know or remember the existence of (0,1)-site.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2011

    One possible answer being that they didn’t know or remember the existence of (0,1)-site.

    That’s likely.

    Whatever we do with the two entries, we should interlink them so that future readers don’t fall into the same trap again and instead be aware that both pages/notions exist.

    The comparison remarks you just made to me, and more details along these lines, should go into these nLab pages.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 2nd 2011

    My bad – I created posite without knowledge of (0,1)-site. I believe that they should be merged. And I suppose the consistent thing to do in terms of naming would be to merge the contents of posite into (0,1)-site.

    I don’t think there needs to be a “standard” “notion of sheaf” on a (0,1)-site. In fact one can consider n-sheaves on an m-site for any values of n and m, although if m>n then some information will be lost. (e.g. the (0,1)-sheaves on a 1-site form the localic reflection of the corresponding 1-topos.) We can disambiguate by saying “1-sheaf” and “(0,1)-sheaf” (or “0-sheaf” or “ideal”) if necessary.

    I do think that it’s good, when possible, to adhere to the general convention that an unqualified “foo” is the same as a “1-foo”, except when using the implicit ∞-category theory convention. This applies both to sheaves and to sites (and to topoi, for that matter).

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeJan 3rd 2011

    I agree that we should not use “sheaf” for a (0,1)(0,1)-sheaf. I will edit (0,1)-site to include a discussion of higher sheaves on posites, including the material from posite. Then I’ll rename it, since “posite” is probably the better term. I don’t think that we’ll actually need separate articles.

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJan 11th 2011

    OK, I’ve done this. See posite (the former (0,1)-site) and posite > history (the former posite) to check that I got everything correctly.

    • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeJan 11th 2011

    By the way, is it reasonable to call a coverage cartesian when it satisfies the stronger condition often required when a category has pullbacks (as here)? I used that term on posite (and also added it to the previous link).

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    Thanks, Toby. Looks good.

    A very minor comment: I have changed “A posite is a decategorification of a site” to “The notion of posite is a decategorificaiton of that of site”,

    Maybe my English rendering of the point here is not optimal, but there is a difference, and I think it matters.

    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeJan 11th 2011

    @ Urs: Yes, you’re right.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2011

    Thanks! I changed the definition of the “canonical coverage” since I think we have to explicitly require “meet-stable joins,” i.e. universally effective-epimorphic families.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2011

    Oh, and yes, I think “cartesian coverage” is reasonable.

    • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeJan 12th 2011

    @ Mike #14:

    I think that there was a typo there (missing prime), which I’ve fixed. I also put back the original definition as a simplification in the case of locally cartesian sites.

    Finally, I completed the change from ‘sheaf’ to ‘ideal’, since I forgot to change ‘Sh(S)Sh(S)’ to ‘Id(S)Id(S)’.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJan 12th 2011

    Thanks for the typo fix. Are you really sure that the original definition is right when bounded meets exist? I think you would still need to consider only joins that are distributed over by the bounded meets.

    • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeJan 12th 2011

    Yeah, I was just thinking about this. I’m going to put that the original version is correct when the poset is a frame, since that’s when it’s really useful, even though that’s probably not the most general situation in which it’s correct.

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