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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.
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.
Okay, thanks, sure. I want to concentrate on the suitable “localic” case here. What can one say? What is known? What do you know?
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.
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 $(\infty,1)$).
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”?
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.
maybe with David R’s help
I’m ready when you are…
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.
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.
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?
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).
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.
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.
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