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On the page Grothendieck topos, Toby Bartels raised the question whether local smallness should be part of the Giraud axioms, referring to a math overflow discussion on a related issue. A while ago I added some comments to the math overflow discussion saying that local smallness has to be part of the Giraud axioms, giving an argument with universes, but nobody seems to have noticed. I raise the issue here again since it’s beside the point of the original math overflow discussion, but clearly related to the nlab.
Concretely, if U, V are universes with U in V, then V is not a Grothendieck topos relative to U, but satisfies all the conditions of the Giraud axioms.
The confusion on math overflow seems to come from C2.2.8(vii) in the Elephant, where Grothendieck toposes are characterized as infty-pretoposes with a generating set of objects.
The crucial point is that the assumption of local smallness is implicit in the definition of infty-pretopos in the Elephant, but not in the nlab.
To actually see that one has to look fairly closely at the Elephant:
Before Lemma A.1.4.19, Johnstone writes ” … C is an infty-pretopos if it is an infty-positive geometric category which is effective as a regular category.
a geometric category is defined as one “satisfying the conditions of Lemma 1.4.18”, which can be reformulated as “regular well-powered with pullback-stable small joins of subobjects”.
So geometric categories are well-powered, but any regular well powered category is also locally small, since Hom(A,B) can be embedded into sub(AxB) via graphs.
On the other hand, well poweredness is not a requirement for geometric categories on the nlab.
I also prefer the convention on the nlab, since there are certain interesting settings where we don’t have well-poweredness, and then it is possible to have small joins of subobjects w/o having small meets.
But this means that the Giraud theorem on the nlab has to contain the assumption of local smallness explicitely.
I will now go ahead and remove the comments of Toby Bartels and add the condition. Feel free and rollback or modify if you don’t agree.
(By the way, my favourite definition of infty-pretopos is “exact, infty-extensive category”. I find this easier to digest and memorize)
Thanks! I agree with the change. I added a remark about this issue to the page as well.
Incidentally, regarding the opinions of Urs and Toby at the top of the page (which I’d not noticed before!), I agree with Toby. Not primarily because it’s a nontrivial theorem, but because to me “category of sheaves” would imply a particular presentation given by a site. It would be analogous to the distinction between a group and a group presentation.
That’s a good point, I agree with that too.
Partly with a view to making that analogy more precise, I added Street’s characterization to Grothendieck topos (that a Grothendieck topos is the same as a lex-total category having the same size as $Set$). This is sort of saying that a Grothendieck topos is a category of sheaves with respect to the canonical topology on itself, which might be viewed as analogous to the idea that each group has a canonical group presentation (as a quotient of the free group on itself). The analogy isn’t perfect since this doesn’t give a 2-monad on $Cat$ whose algebras are Grothendieck toposes, but lex-totality is nevertheless sort of in that vein; I might come back to that point later.
@Todd can we say anything more if we take instead Street’s view of an isolated category, the 2-category of categories, absorb some foundational/size issues into that and then make some sort of size comparison in the 2-category, rather than talking of a bijection of sets $Mor(E) \simeq \kappa$? Something like a (essentially?)surjective functor from $Set^\mathbf{2}$?
re #3: Sure, I’ve removed the editorial comment at the beginning.
@David: I’m not sure what you mean by Street’s view of an isolated category, but maybe you’re suggesting something along the lines of formulating the notion of lex total object w.r.t. a yoneda structure on a 2-category? A yoneda structure has a built-in notion of smallness which with luck might be adapted to speak of “moderate” objects which could replace the ${|Mor(E)|} = \kappa$ business I wrote down in the article. Aside from that, I have a strong suspicion that Street and Walters would have played around with lex total objects in that context, without any extra size assumptions. Maybe Mike or Mark Weber would know about lex total objects in a 2-topos and how much one could say about them (sadly, the link to Mark’s paper at 2-topos no longer works).
I think the paper is still available from here.
Bob Walters had a series of posts on toposes as ’glorified locales’ last year which give an interesting account of his views.
Thanks, Thomas – I’ve updated the link to Mark Weber’s paper at 2-topos.
I wanted to understand these things about “lex total” for a long time, maybe this is an occasion to look into it closer.
@Todd: why do you phrase the size condition as one on the class of morphisms? Street (in ‘Notions of topos’) states it as a condition on the objects, saying that there has to be a moderate collection of objects that can cover any other objects (via extremal epis). I think the condition that the category is locally small is implicit since otherwise the Yoneda embedding is not well defined. Under suitable conditions it might be possible to deduce from a moderate covering collection of objects that the collection of all objects is essentially small, and if the category is skeletal and locally small this probably induces that the collection of maps is moderate as well. But for this we have at least to assume that the category is skeletal or the size of isomorphism classes is somehow bounded. (With a normal set theoretic understanding it might seem pointless a bit to consider categories which are essentially moderate but not moderate in size, but if we want invariance under equivalence, then we have to allow that I think)
Finally, I think the size condition in the characterization has to be $\leq\kappa$ and not $=\kappa$ (“at most the size of $\mathrm{Set}$”, not “the same size as $\mathrm{Set}$”), since otherwise we exclude the initial Grothendieck topos.
Oh, you’re right, Jonas. It’s the size of the class of isomorphism classes of objects, not the size of the class of objects, that counts. Of course. (I was thinking the class of objects, which under local smallness would be the same as the size of the class of morphisms, which normally I would consider a more meaningful way to speak of “size” of a category (not considering more refined notions due variously to Baez-Dolan, Leinster, and others). So that explains why I had ${|Mor(E)|}$.) Anyway, I’ll fix it – thanks.
Finally, I think the size condition in the characterization has to be ≤κ\leq\kappa and not =κ=\kappa (“at most the size of Set\mathrm{Set}”, not “the same size as Set\mathrm{Set}”), since otherwise we exclude the initial Grothendieck topos.
Ha ha ha ha! Okay, that’s true, the terminal category, or something equivalent to it. Touché. Wouldn’t that be the only case?
Okay, that’s true, the terminal category, or something equivalent to it. Touché. Wouldn’t that be the only case?
Yes, I think so. If $\mathcal{E}$ is a Grothendieck topos which is not equivalent to $1$, then it has an object $A$ which is not initial, and then for each set $I$, all the coproduct injections into the $I$-fold coproduct of $A$ are different (by disjointness, since the intersection of two coproduct injections is $A$ if they are equal, and $0$ if they are different). Therefore $\mathrm{mor}(\mathcal{G})$ has already the size of the universe.
This depends on classical logic, but so does Street’s characterization of Grothendieck toposes (the proof assumes the existence of a well-ordering on a possibly large set of objects).
I’ve never known quite what to make of this theorem, that with lex-totality and a moderate generating set you somehow magically also get a small generating set. The fact that it uses nonconstructive arguments like large well-orderings makes it seem to me more like an accident than something we ought to take seriously as a definition of topos.
@Todd - yes, that’s roughly what I was thinking of. If Vopenka held, then we cannot talk of the large discrete category $V$ inside $Cat$, so it would be better to have a proof that didn’t use the class of sets.
@Jonas #13 - is a well-quasi-ordering (i.e. a well-founded preorder) enough?
Mike, if nothing else, the concept that lex-totality provides a viable ’notion of topos’ ought to be taken seriously and recorded somewhere in the nLab, IMO. (And obviously it’s more a ’notion of Grothendieck topos’ than just ’notion of (elementary) topos’.)
Other than that, I can understand your discomfort. Freyd has never been someone to shy away from a nonconstructive argument, it seems to me.
Oh, certainly, I don’t object to recording lex-totality elsewhere on the lab.
Regarding my question to Todd:
In fact Street raises the idea himself (page 201 in internal numbering, third page of the pdf) in Notions of topos: he mentions ’size structures’ on bicategories, for instance the bicategory of fibrations over an elementary topos. He says that in the absence of AC in B, it’s unlikely all lex-total objects in Fib(B) will arise as an internal elementary topos with generators in Fib(B).
Is “size structure” synonymous to “Yoneda structure”? Or is it meant to be more general and informal?
Regarding David’s question, I’m not sure. The main point seems to be to well-order a “moderate” covering set of objects in a way such that all its initial segments are small, and then to derive a contradiction from the assumption that none of the initial segments are generating. So in any case it’s by contradiction, I’m not sure at the moment if there is a sensible way to replace the well ordering by a quasi-well ordering.
One way to look at Street’s result that for me makes it less ‘scary’ is to observe that it’s parametric in an arbitrary strongly inaccessible cardinal. So in particular one can choose moderate=countable and small=finite. I’m trying to understand now what the result means in this case …
It may be worth noting that unlike Yoneda structures, other contexts for formal category such as virtual equipments can make sense without any “locally large” categories. In particular, there is a virtual equipment whose objects are locally small and moderate (or small) categories. If “lex-totality” could be interpreted in that context, therefore, the size assumption would be automatic.
Jonas
as far as I can tell, from Street’s “Cosmoi…” paper, a size structure on a (1-)category is an internal full subcategory, hence I guess a universe in the case of a topos. The 2-category $Cat(C)$, for $C$ cartesian closed, then gets a Yoneda structure. I guess the comment about $Fib(B)$ must be referring to a Yoneda structure. He then writes
Indeed, this structure arises from a fibrational cosmos (= “cosmos” in the sense of Street [28]).
where [28] is “Elementary cosmoi” in Springer LNM, which I’d have to dig out. I think you’re right about it being informal, but we can probably pin it down in this case.
@Mike
I guess you mean for arrows to be profunctors? Then instead of an adjoint to the Yoneda embedding, we want [something] relating to $Hom:C\times C^{op} \to Set$?
A (virtual) equipment has two kinds of morphism, the functors and the profunctors. As remarked at total category, totality is equivalent to saying that the colimit of $Id_C$ weighted by any $W:C^{op}\to Set$ exists, and this can be phrased as soon as you have functors and profunctors without needing the presheaf category. I’m not as sure about how to add lexness, though.
Yes, I meant the ’weaker’ arrows. As to lexness, hm….
@David: Thanks, the “Cosmoi” paper is the reference that I was looking for!
I missed this discussion the first time around; I want to address the original topic, but it's drifted a bit, so I'll put my remarks at Grothendieck toposes in weak foundations.
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