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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.
Thanks. I added a bit more, and changed the “proof” to be more like a proof, with the description of the locale coming afterwards.
What’s the relationship between the pages posite and (0,1)-site?
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.
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 $1$-sheaf, while (0,1)-site (following Johnstone) takes the standard notion of sheaf on a $(0,1)$-site to be a $(0,1)$-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.
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.
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.
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).
I agree that we should not use “sheaf” for a $(0,1)$-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.
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.
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.
@ Urs: Yes, you’re right.
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.
Oh, and yes, I think “cartesian coverage” is reasonable.
@ 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 ‘$Sh(S)$’ to ‘$Id(S)$’.
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.
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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