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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 7th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexshouldn't [[0-site]] be named [[(0,1)-site]]?

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

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 7th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIMHO, yes. Although 0-site should probably redirect to it, since there isn't really any useful notion of a (0,0)-site.

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

3. • CommentRowNumber3.
   • CommentAuthorEric
   • CommentTimeJul 7th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItex>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? :)

   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? :)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 7th 2010
   • (edited Jul 7th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexby 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]]

   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

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeJul 8th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI don't know whether ‘$n$-site’ should follow ‘$n$-sheaf’ or ‘$n$-topos’. The former (as it is used on the Lab) has an implied $(-,1)$ 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 $(2,2)$-site? If so, then we should call this notion ‘$2$-site’; if not, then we may call a $(2,1)$-site (which notion should certainly exist and be useful) a ‘$2$-site’. Then ‘$0$-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 $(0,1)$-site, just as we have ‘poset’ for a $(0,1)$-category, ‘Heyting algebra’ for a $(0,1)$-topos (with logical morphisms), ‘locale’ for a Grothendieck $(0,1)$-topos (with geometric morphisms), etc. But my source material, Johnston, just says ‘site’!

   I don’t know whether ‘$n$-site’ should follow ‘$n$-sheaf’ or ‘$n$-topos’. The former (as it is used on the Lab) has an implied $(-,1)$ 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 $(2,2)$-site? If so, then we should call this notion ‘$2$-site’; if not, then we may call a $(2,1)$-site (which notion should certainly exist and be useful) a ‘$2$-site’. Then ‘$0$-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 $(0,1)$-site, just as we have ‘poset’ for a $(0,1)$-category, ‘Heyting algebra’ for a $(0,1)$-topos (with logical morphisms), ‘locale’ for a Grothendieck $(0,1)$-topos (with geometric morphisms), etc. But my source material, Johnston, just says ‘site’!

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 8th 2010
   • (edited Jul 8th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex(...)

   (…)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 8th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexSecond 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 $(-,1)$-explicit. As opposed to [[infinity-stack]] (which is debateable, but not the issue here.) So I think the convention is that an $n$-thing is an $(n,n)$ thing and with that and the above, the category-numbering of higher sheaves on the $n$Lab should be pretty consistent. In that vein, I again opt for $(0,1)$-site!

   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 $(-,1)$-explicit. As opposed to infinity-stack (which is debateable, but not the issue here.)

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

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

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeJul 8th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexSorry, I mixed up your usage of ‘$n$-sheaf’ with your usage of ‘$n$-stack’. Do you expect there to be such a thing as a $(2,2)$-site?

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

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

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJul 8th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThere is definitely an important notion of [[michaelshulman:2-site|(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.

   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.

10. • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJul 8th 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>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.)

    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.)

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2010
    • (edited Jul 8th 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItexBy 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.

    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.

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexYes, 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).

    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).

13. • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI have moved [[0-site]] to [[(0,1)-site]].

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

14. • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex>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 [[Fundamental Bigroupoids and 2-Covering Spaces|2-covering spaces]]? They should be the (2,1)-version. (I hope he has at least heard of my thesis :)

    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 :)

15. • CommentRowNumber15.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItexI disagree with the use of "stack" to mean "stack of groupoids", since this is actually incorrect. Therefore, the (oo,1) should be made explicit.

    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.

16. • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex> 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.

    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.

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexMike 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.

    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.

18. • CommentRowNumber18.
    • CommentAuthorHarry Gindi
    • CommentTimeJul 9th 2010
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItex> 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.

    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.

19. • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 12th 2010
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexWe 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.

    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.

20. • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeJul 13th 2010
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex@ David I thought that you were just giving another silly joke, but you're absolutely right!

    @ David

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