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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2012
   • (edited Mar 8th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn the References-section at _[[2-sheaf]]_ I have added three "classical" references: in the 1970s Grothendieck, Giraud and then Bunge usually considered "2-sheaves" -- namely category-valued stacks -- by default. Also there is a good body of work on 2-sheaves realized as _[[internal categories]]_ in the underlying 1-sheaf topos. I have added a pointer to Joyal-Tierney's _[Strong stacks](http://www.pps.jussieu.fr/~mellies/semantics-operads-categories/joyal-tierney-strong-stacks.pdf)_ so far, but I think much more literature exists in this direction. But if one goes this internalization-route at all, what one should _really_ do is, I think, consider [[internal (infinity,1)-category|weak internal categories]] in the [[(2,1)-topos]] over the underlying site. Has this been studied at all? Does anyone know how 2-categories of weak internal categories in $(2,1)$-toposes relate to [[2-toposes]]? At least under nice conditions these should be equivalent, I guess. But I want to understand this better.

   In the References-section at 2-sheaf I have added three “classical” references:

   in the 1970s Grothendieck, Giraud and then Bunge usually considered “2-sheaves” – namely category-valued stacks – by default. Also there is a good body of work on 2-sheaves realized as internal categories in the underlying 1-sheaf topos. I have added a pointer to Joyal-Tierney’s Strong stacks so far, but I think much more literature exists in this direction.

   But if one goes this internalization-route at all, what one should really do is, I think, consider weak internal categories in the (2,1)-topos over the underlying site.

   Has this been studied at all? Does anyone know how 2-categories of weak internal categories in $(2,1)$-toposes relate to 2-toposes? At least under nice conditions these should be equivalent, I guess. But I want to understand this better.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 8th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexPassing to internal categories in a (2,1)-topos to get a 2-topos is analogous to passing to internal groupoids in a 1-topos to get a (2,1)-topos. And just as not every (2,1)-topos is 1-localic, not every 2-topos is (2,1)-localic. In particular, the 2-topos of 2-presheaves on a small 2-category that is not a (2,1)-category cannot be obtained from internal categories in any (2,1)-topos.

   Passing to internal categories in a (2,1)-topos to get a 2-topos is analogous to passing to internal groupoids in a 1-topos to get a (2,1)-topos. And just as not every (2,1)-topos is 1-localic, not every 2-topos is (2,1)-localic. In particular, the 2-topos of 2-presheaves on a small 2-category that is not a (2,1)-category cannot be obtained from internal categories in any (2,1)-topos.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2012
   • (edited Mar 8th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, thanks, sure. I want to concentrate on the suitable "localic" case here. What can one say? What is known? What do you know?

   Okay, thanks, sure. I want to concentrate on the suitable “localic” case here. What can one say? What is known? What do you know?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex[Here](http://ncatlab.org/michaelshulman/show/truncated+2-topos) is what I know about 2-toposes vs (2,1)-toposes, and [here](http://ncatlab.org/michaelshulman/show/exact+completion+of+a+2-category) is an attempt at constructing "exact completions" of 2-categories by way of internal categories. The infinite iteration at the end is ugly, though, and I'm not sure that it is necessary; I'm hoping to come back to this sometime (maybe with David R's help) now that I understand the [1-dimensional case](http://www.math.ucsd.edu/~mshulman/papers/exsitetalk.pdf) better (the preprint version of that talk is almost ready for posting). In any case, the universal property of exact completion, together with the universal properties of 2-sheaves and (2,1)-sheaves, ought to imply that the 2-exact completion of a (2,1)-topos of (2,1)-sheaves on a (2,1)-site agrees with the 2-topos of 2-sheaves on the same site.

   Here is what I know about 2-toposes vs (2,1)-toposes, and here is an attempt at constructing “exact completions” of 2-categories by way of internal categories. The infinite iteration at the end is ugly, though, and I’m not sure that it is necessary; I’m hoping to come back to this sometime (maybe with David R’s help) now that I understand the 1-dimensional case better (the preprint version of that talk is almost ready for posting). In any case, the universal property of exact completion, together with the universal properties of 2-sheaves and (2,1)-sheaves, ought to imply that the 2-exact completion of a (2,1)-topos of (2,1)-sheaves on a (2,1)-site agrees with the 2-topos of 2-sheaves on the same site.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2012
   • (edited Mar 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Mike! That's very useful. While I am reading, let me state my question more in detail: I would like to know if * given a 1-site $C$; * writing $Sh_{(2,1)}(C)$ for the $(2,1)$-topos over it; * writing $Cat(Sh_{(2,1)}(C))$ for the 2-category of category objects in the $(2,1)$-topos; then: how far is $Cat(Sh_{(2,1)})(C)$ from the 2-topos $Sh_2(C)$ of category-valued sheaves on $C$? And by "category objects" I mean the fully weak form. If you wish: complete Segal objects in $Sh_{(2,1)}(C)$. That's what I am trying to understand better (eventually for $(2,1)$ generalized to $(\infty,1)$).

   Thanks, Mike! That’s very useful.

   While I am reading, let me state my question more in detail:

   I would like to know if

   • given a 1-site $C$;

   • writing $Sh_{(2,1)}(C)$ for the $(2,1)$-topos over it;

   • writing $Cat(Sh_{(2,1)}(C))$ for the 2-category of category objects in the $(2,1)$-topos;

   then: how far is $Cat(Sh_{(2,1)})(C)$ from the 2-topos $Sh_2(C)$ of category-valued sheaves on $C$?

   And by “category objects” I mean the fully weak form. If you wish: complete Segal objects in $Sh_{(2,1)}(C)$.

   That’s what I am trying to understand better (eventually for $(2,1)$ generalized to $(\infty,1)$).

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2012
   • (edited Mar 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMike, where do you define "$n$-exactness"? I was lost for a while, until I happened upon your page _[2-congruence](http://ncatlab.org/michaelshulman/show/2-congruence)_, where I find the very useful sentence > One way to express the idea is that in an n-category, every object is internally a (n−1)-category; exactness says that conversely every “internal (n−1)-category” is represented by an object. So now I am back to the page _[exact completion of a 2-category](http://ncatlab.org/michaelshulman/show/exact+completion+of+a+2-category)_. I feel like this is written with a reader in mind who already got more information from elsewhere, but I get the idea now that for instance theorem 5 there is part of the answer to my question. Let me know if I am reading it correctly: in words your are saying that, assuming "enough groupoids" a 2-category is "2-exact" if it is the 2-category of category objects in its underlying $(2,1)$-category. Is that right? I gather, then, that 2-categories of 2-sheaves are 2-exact. Is that by definition of "2-exact"? What's the definition of "2-exact"?

   Mike,

   where do you define “$n$-exactness”?

   I was lost for a while, until I happened upon your page 2-congruence, where I find the very useful sentence

   One way to express the idea is that in an n-category, every object is internally a (n−1)-category; exactness says that conversely every “internal (n−1)-category” is represented by an object.

   So now I am back to the page exact completion of a 2-category. I feel like this is written with a reader in mind who already got more information from elsewhere, but I get the idea now that for instance theorem 5 there is part of the answer to my question.

   Let me know if I am reading it correctly: in words your are saying that, assuming “enough groupoids” a 2-category is “2-exact” if it is the 2-category of category objects in its underlying $(2,1)$-category. Is that right?

   I gather, then, that 2-categories of 2-sheaves are 2-exact. Is that by definition of “2-exact”? What’s the definition of “2-exact”?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, I get it now. So 2-categories of 2-sheaves are going to be 2-exact by a 2-Giraud theorem, I suppose. All right, good. I'll really quit now, will look at this further tomorrow. Thanks again.

   Okay, I get it now. So 2-categories of 2-sheaves are going to be 2-exact by a 2-Giraud theorem, I suppose.

   All right, good. I’ll really quit now, will look at this further tomorrow. Thanks again.

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 9th 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex>maybe with David R's help I'm ready when you are...

   maybe with David R’s help

   I’m ready when you are…

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMar 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexSorry, I linked you to the middle of the web, which isn't inter-linked as well as it could be; the TOC at [[michaelshulman:2-categorical logic]] has links to all the pages, including [[michaelshulman:exact 2-category]]. I feel like the answer to whether internal categories (= "1-truncated" complete Segal objects) in a (2,1)-topos yield the corresponding 2-topos must surely be yes, but I don't think I got around to working out the details. @David R: Great! I'll be in touch when I've got the 1-categorical case done; it should be only a week or two.

   Sorry, I linked you to the middle of the web, which isn’t inter-linked as well as it could be; the TOC at 2-categorical logic (michaelshulman) has links to all the pages, including exact 2-category (michaelshulman).

   I feel like the answer to whether internal categories (= “1-truncated” complete Segal objects) in a (2,1)-topos yield the corresponding 2-topos must surely be yes, but I don’t think I got around to working out the details.

   @David R: Great! I’ll be in touch when I’ve got the 1-categorical case done; it should be only a week or two.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks, that's already very useful to know. One might think that the answer to this question should be somewhere in the work of Marta Bunge. I have tried to scan parts of it, but didn't find a version of this statement so far. But in the course of this I tried to improve the entry _[[2-sheaf]]_ a bit more by adding more classical references, adding remarks on the various different incarnations over lower categorical sites (fibered categories, indexed categories, toposes over a base topos) and added statements of two propositions from Bunge-Pare. Much more could be done, of course.

    Thanks, that’s already very useful to know.

    One might think that the answer to this question should be somewhere in the work of Marta Bunge. I have tried to scan parts of it, but didn’t find a version of this statement so far.

    But in the course of this I tried to improve the entry 2-sheaf a bit more by adding more classical references, adding remarks on the various different incarnations over lower categorical sites (fibered categories, indexed categories, toposes over a base topos) and added statements of two propositions from Bunge-Pare.

    Much more could be done, of course.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexMike, let me come back to that theorem 5 on the page _[[michaelshulman:exact completion of a 2-category]]_: unless I am missing something, the theorem says in particular that if a 2-sheaf 2-topos is 2-exact and has enough groupoids, then the answer to my question in #5 is as hoped for: the 2-topos is equivalent to the 2-category of internal categories (2-congruences) of its underlying $(2,1)$-topos. Right? So what is missing? While I don't see you state it anywhere, I do suppose that every 2-sheaf 2-topos is 2-exact? So when does a 2-topos have "enough groupoids". (I see at _[[michaelshulman:core]]_ that this is the case if "every object admits an eso from a groupoidal one", but I still need to think about what this means precisely). Is it sufficient that it be $(2,1)$-localic?

    Mike,

    let me come back to that theorem 5 on the page exact completion of a 2-category (michaelshulman):

    unless I am missing something, the theorem says in particular that if a 2-sheaf 2-topos is 2-exact and has enough groupoids, then the answer to my question in #5 is as hoped for:

    the 2-topos is equivalent to the 2-category of internal categories (2-congruences) of its underlying $(2,1)$-topos.

    Right?

    So what is missing? While I don’t see you state it anywhere, I do suppose that every 2-sheaf 2-topos is 2-exact?

    So when does a 2-topos have “enough groupoids”. (I see at core (michaelshulman) that this is the case if “every object admits an eso from a groupoidal one”, but I still need to think about what this means precisely). Is it sufficient that it be $(2,1)$-localic?

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAh, yes, you are right. It's been quite a while since I thought about all of this. The [[michaelshulman:2-Giraud theorem]] says, in particular, that Grothendieck 2-toposes are 2-exact. The page [[michaelshulman:truncated 2-topos]] states that a 2-topos has enough groupoids just when it is equivalent to the 2-sheaves on some (2,1)-site (i.e. what we are now calling (2,1)-localic).

    Ah, yes, you are right. It’s been quite a while since I thought about all of this.

    The 2-Giraud theorem (michaelshulman) says, in particular, that Grothendieck 2-toposes are 2-exact. The page truncated 2-topos (michaelshulman) states that a 2-topos has enough groupoids just when it is equivalent to the 2-sheaves on some (2,1)-site (i.e. what we are now calling (2,1)-localic).

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexAh, fantastic. I'll try to collect these bits in an explitcit theorem-statement at _[[2-topos]]_ as soon as I get a chance. That's awesome.

    Ah, fantastic.

    I’ll try to collect these bits in an explitcit theorem-statement at 2-topos as soon as I get a chance. That’s awesome.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2013
    • (edited Oct 23rd 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexIn reaction to [this MO question](http://mathoverflow.net/q/145563/381) about the nLab entry _[[2-sheaf]]_ I have added to the latter a remark, right at the beginning, to clarify the terminology a bit and highlight its purpose better.

    In reaction to this MO question about the nLab entry 2-sheaf I have added to the latter a remark, right at the beginning, to clarify the terminology a bit and highlight its purpose better.