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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeAug 6th 2013
• (edited Aug 6th 2013)

In the examples section of extensive category, it is stated that the category of affine schemes is infinitary extensive.

For all I know, I was the one who stuck in that example. But is that statement actually true? I’m having trouble seeing it.

If $S$ is a commutative ring over $R$ (by which I mean under $R$ (-:), does the functor $S \otimes_R -: CAlg_R \to CAlg_R$ preserve arbitrary cartesian products? Because it seems that’s what we basically need for the statement to be true.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeAug 7th 2013

Just for the record: of course $S \otimes_R - : CAlg_R \to CAlg_R$ cannot be expected to preserve arbitrary products in general, i.e., products in $CRing$ are not stable under pushout. In the opposite category of affine schemes, this means coproducts are not stable under pullback, so affine schemes are not infinitary (l)extensive. Which leads me to say: I’m going to fix the nLab entry.

The lowly details of the proof of the first sentence in the preceding paragraph are perhaps best left to the reader’s discretion, but if anyone cares, my own demonstration can be found here.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeAug 7th 2013

Interesting! Looks like the error was probably a result of confusion at the time we were settling on the terminology “finitary extensive” and “infinitary extensive”.

I presume that the category of not-necessarily-affine schemes is infinitary extensive. If so, that means that infinite coproducts of affine schemes are no longer affine. Is that right?

• CommentRowNumber4.
• CommentAuthorZhen Lin
• CommentTimeAug 7th 2013

That’s right: the disjoint union of infinitely many copies of $\operatorname{Spec} k$ ($k$ a field) is not affine because, well, its underlying space is not quasicompact.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeAug 7th 2013

First of all, Mike: congratulations on the birth of Arthur! And hope mom (Megan, right?) is doing well.

I suppose that the category of schemes is infinitary extensive, but I don’t know where the details might have been written down. Unless someone knows, I might take a crack at it – although I’m a little sketchy on limits and colimits of schemes and how they behave. It would obviously be good to know and have recorded.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeAug 7th 2013

Come to think of it, infinitary extensivity of the category of schemes should be conceptually obvious, given the observations that this is a full subcategory of the category of functors $CRing \to Set$ that is closed under finite limits and small coproducts (finite limits is a well-known result, and small coproducts follows very quickly from the Demazure-Gabriel functorial definition of scheme). I don’t see any set-theoretic difficulties getting in the way of that.

• CommentRowNumber7.
• CommentAuthorZhen Lin
• CommentTimeAug 7th 2013

I don’t think it is literally closed under small coproducts: you have to apply Zariski sheafification, if I remember correctly. Basically this is because not all affine schemes are connected.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeAug 7th 2013

Hm.. need to think on this. But may need some time…

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeAug 7th 2013

Thanks Todd! Yes, Megan is doing well, thanks for asking.

Could it be a similar full subcategory of the category of presheaves on connected affine schemes? Relatedly, is the category of affine schemes the free finite-coproduct completion of the subcategory of connected ones?

• CommentRowNumber10.
• CommentAuthorZhen Lin
• CommentTimeAug 7th 2013

The two questions are closely related, I think. The answer to the second question is no: there are affine schemes that are not the disjoint union of their connected components. For example, $\operatorname{Spec} \mathbb{F}_2^{\mathbb{N}}$ is (homeomorphic to) the ultrafilter space $\beta \mathbb{N}$, which is totally disconnected.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeAug 7th 2013

Okay, I’m back. Thanks for pointing out #7, Zhen: that makes sense. So I should say instead that schemes are closed under finite limits and small coproducts of Zariski sheaves, and then the argument I had tried before should apply (since Zariski sheaves form an infinitary lextensive category, so must any full subcategory closed under finite limits and small coproducts).

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeAug 8th 2013

Interesting! I had no idea there were affine schemes that looked like that topologically.

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeAug 8th 2013

But there’s stuff here that I’ve never been too clear on, and that I’d love to get sorted out.

In #6, I blithely referred to the category of functors $CRing \to Set$. Implicitly I’m taking the functorial approach to schemes advanced by Demazure-Gabriel; if we follow the nLab’s description of this in the article scheme, then we consider $CRing^{op}$ as a large site, and form the category of sheaves with respect to the big Zariski topology, which is subcanonical. The category of schemes is by definition the full subcategory of sheaves that admit (small?) coverings by open affine subsheaves. (The nLab article doesn’t say “small” with regard to coverings; it’s just “family”, whatever that is supposed to mean.)

It looks as if the category of Zariski sheaves wants to be what some people call the (big) Zariski topos, but due to size considerations, it’s hard for me to see it as a Grothendieck topos. Actually, the nLab seems just a bit waffly on this: in the article Zariski site, the idea section refers to sheaves on a class-sized site $CRing^{op}$, and the definition section starts with this site, but then reverts to the small site of finitely presented rings with the inherited topology. I’ve seen people use one or the other, and I’ve never been too clear on the relationship between the sheaf categories or why people routinely invoke the class-sized site. (I see John Baez and Mike Shulman had similar concerns in 2009, and I didn’t see the issues truly clarified in that discussion.)

Demazure-Gabriel (i.e., their Introduction to Algebraic Geometry and Algebraic Groups), to which the nLab refers, do this: they assume two fixed universes $V$ and $U$, with $\mathbb{N} \in U \in V$, calling members of $U$ “small sets”. (One might suppose they mean that $V$ (with its membership relation) is assumed to satisfy ZFC, as does $U$ with the inherited membership relation, but they don’t actually say.) Then they call a small commutative ring with identity a “model”, and denote the category of models by $M$, which will be the supporting site for things like the Zariski topology. In other words, they will consider presheaves and sheaves on $M^{op}$, where presheaf means a functor $M \to Set$ to the category of sets (that live as elements in $V$).

They wave their hands a bit, essentially saying they don’t really need universes. Since they want $[M, Set]$ to be locally small, they don’t want $M$ to be “too large”. On the other hand, they want to be able to freely apply a whole bunch of commutative algebra constructions to models: residue class fields, rings of fractions, completions… They say it would be enough to assume that for any model $R$, every model with cardinality less than or equal to $Card(R)^\mathbb{N}$ is also a model (notice that finitely presented rings will not suit that purpose!), but justify the approach they do take by saying that many mathematicians “are accustomed to universes by now”, and also that they would like to use freely direct limits in their category of models.

Anyway, it looks like some of this could use some sorting out.

• CommentRowNumber14.
• CommentAuthorZhen Lin
• CommentTimeAug 8th 2013
• (edited Aug 8th 2013)

Yes, there are some set-theoretic issues at play.

1. The functor of points of a scheme is a sheaf on the large Zariski site. (In fact, it is even a sheaf on the large fpqc site.) I think it is small (i.e. a colimit of a small diagram of representables) but I could be wrong.
2. The large Zariski site is not essentially small in the sense of Johnstone, i.e. it is not Morita-equivalent to a small site… or at least it is not obviously so. Certainly what is true is that the cardinality of the non-trivial principal localisations of an integral domain $A$ is bounded below by the cardinality of $A$ itself, and there are integral domains of arbitrarily large cardinality.
3. One can certainly look at gros Zariski sites that are not large. For example, for $M = {CRing}_{fp}$, the category of locally finitely presentable schemes embeds as a full subcategory of the topos of Zariski sheaves on $M$. The general principle is that one can always get the schemes that are “locally $M$”.
• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeAug 8th 2013

Let me try asking a concrete question: is there a small Zariski site such that schemes are recognized as a full subcategory of locally affine Zariski sheaves?

Looking at what Demazure-Gabriel say, the answer seems to be a tentative ’yes’, but it’s somewhat hard for me to be sure.

• CommentRowNumber16.
• CommentAuthorZhen Lin
• CommentTimeAug 8th 2013
• (edited Aug 8th 2013)

Edit. The argument below is not quite correct…

No, there is no such a site. Suppose $M$ is a small full subcategory of $CRing$ such that the Yoneda representation ${CRing}^{op} \to [M, Set]$ is fully faithful. Since $M$ is small and $CRing$ is complete, we get a left adjoint, making ${CRing}^{op}$ a reflective subcategory of $[M, Set]$. But then ${CRing}^{op}$ would be an l.f.p. category, and we know that is impossible because $CRing$ itself is l.f.p. but not a preorder category.

• CommentRowNumber17.
• CommentAuthorTodd_Trimble
• CommentTimeAug 8th 2013
• (edited Aug 8th 2013)

Thanks very much, Zhen: that’s of course a very convincing argument.

In particular, sheaves on the large Zariski site cannot possibly form a Grothendieck topos. So what sort of category do they form? Cartesian closure, for instance, looks highly doubtful to me. How about exactness properties? Do they form an infinitary extensive pretopos? (Perhaps this is easy; I haven’t thought hard about it.)

Edit: A more general question is this. Suppose that $E$ is a infinitary extensive pretopos, and that $F \to E$ is a full embedding with a left exact left adjoint. Is it true that $F$ is an infinitary extensive pretopos?

• CommentRowNumber18.
• CommentAuthorZhen Lin
• CommentTimeAug 8th 2013
• (edited Aug 9th 2013)

Oops, what I said there was not quite correct: ${CRing}^{op}$ has to be accessibly embedded in order to get a contradiction. But in any case the point is that to get a small Zariski site we would have to find a small codense subcategory of $CRing$, which seems like an odd thing to have. I still think that the category of all schemes does not embed into the category of sheaves on any small Zariski site.

As for reflective subcategories of infinitary-pretoposes: while it is true that $[CRing, Set]$ is always an infinitary-pretopos, it’s not obvious whether or not Zariski sheaves form a reflective subcategory! (See, for instance, [Waterhouse, 1975].) But in fact they do, because of the following fact: for each commutative ring $A$, the class of Zariski-covering sieves on $Spec A$ (partially ordered by inclusion) has a small coinitial cofiltered family. So the colimit (which is indexed over the opposite of the class of Zariski-covering sieves) in the Grothendieck plus construction exists and preserves finite limits.

So we are in the situation you describe. Such a full subcategory is indeed an infinitary-pretopos: small colimits and finite limits in $F$ can be computed by applying the reflector to the corresponding construction in $E$, so any good properties those have in $E$ are inherited by $F$.

• CommentRowNumber19.
• CommentAuthorTodd_Trimble
• CommentTimeAug 8th 2013

Okay, great – thanks again, Zhen. I’ve now amended the relevant example at extensive category, referring back to your last comment.

• CommentRowNumber20.
• CommentAuthorDavidRoberts
• CommentTimeAug 9th 2013

@Zhen Lin - so $(CRing^{op},Zar)$ satisfies WISC?

• CommentRowNumber21.
• CommentAuthorZhen Lin
• CommentTimeAug 9th 2013

Yes, that looks right. Every Zariski-covering sieve contains one that is generated by a finite set of principal affine open subschemes.

• CommentRowNumber22.
• CommentAuthorDavidRoberts
• CommentTimeAug 9th 2013

Conversely, the fpqc site doesn’t. The question is, where is the transition?

• CommentRowNumber23.
• CommentAuthorZhen Lin
• CommentTimeAug 9th 2013
• (edited Aug 9th 2013)

The étale and fppf topologies satisfy WISC, at least over a locally noetherian base scheme. This is basically because they are defined by adjoining to the Zariski topology certain (finite) families of the form $\{ \operatorname{Spec} B \to \operatorname{Spec} A \}$ where $B$ is finitely presented over $A$. The fpqc topology has no such finiteness requirement.

• CommentRowNumber24.
• CommentAuthorHurkyl
• CommentTimeJul 18th 2020

I’m looking at the equivalent characterizations of extensive. Am I missing something or is condition #3 not actually equivalent to the others? In particular, given condition #3, it’s not clear how you would show that a span $a \to a + b \leftarrow z$ has a pullback.

• CommentRowNumber25.
• CommentAuthorSam Staton
• CommentTimeJul 20th 2020

I agree – corrected point 3 having checked Carboni-Lack-Walters

• CommentRowNumber26.
• CommentAuthorDavidRoberts
• CommentTimeJul 20th 2020

This thread needs to be merged with the other discussion thread.

• CommentRowNumber27.
• CommentAuthorHurkyl
• CommentTimeJul 20th 2020

I’ve made the analogous change to the infinitary version, on the assumption it’s required there too.

• CommentRowNumber28.
• CommentAuthorHurkyl
• CommentTimeJul 20th 2020

For the sake of minimizing confusion, add a parenthetical mentioning the finite case implies the infinite case.

• CommentRowNumber29.
• CommentAuthorHurkyl
• CommentTimeJul 20th 2020

For the sake of minimizing confusion, add a parenthetical mentioning the finite case implies the infinite case.

• CommentRowNumber30.
• CommentAuthorThomas Holder
• CommentTimeDec 11th 2020

Added some links and references. I would also suggest to move the terminological discussion at the end into the forum thread in line with the current approach to query boxes.

• CommentRowNumber31.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 11th 2020

Here is the discussion:

## Discussions

While creating this page, we had the following discussion regarding “finitely” versus “infinitary.”

+–{.query} Can we say ’small-extensive’? Or even redefine ’extensive’ to have this meaning, using ’finitely extensive’ for the first version? —Toby

I think “extensive” is pretty well established for the finite version, and I would be reluctant to try to change it. I wouldn’t object too much to “small-extensive” for the infinitary version in principle, but $\infty$-positive is used in the Elephant and possibly elsewhere. I think the topos theorists think by analogy with $\infty$-pretopos, which I don’t think we have much hope of changing, despite the unfortunate clash with “$\infty$-topos.” But you can use “finitary disjunctive” and “disjunctive” in the lex case, which most examples are. -Mike

Mike: Okay, I just ran across one paper that uses “(infinitary) extensive” for the infinitary version the first time it was introduced, and then dropped the parenthetical for the rest of the paper. I also recall seeing “extensive fibration” used for a fibration having disjoint and stable indexed coproducts, which is certainly a (potentially) infinitary notion. So perhaps there is no real consensus on whether “extensive” definitely implies the finite version or the infinitary one.

Toby: It would be nice to not overload the prefix ’$\infty$-’ so much. It's like ’continuous’; default to small.

Mike: I agree that it would be nice to avoid $\infty$-. What if we do what we did for omega-category? That is, if you want to be unambiguous, say either “finitary extensive” or “infinitary extensive,” and in any particular context you are allowed to define “extensive” at the beginning to be one of the two and use it without prefix in what follows.

Toby: Sure. Of course, the general concept is $\kappa$-extensive, where $\kappa$ is any cardinal (which we may assume to be regular).

=–

• CommentRowNumber32.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 11th 2020

Removed the discussion.

1. Re #26: I overlooked this at the time, but have merged the two threads now.

• CommentRowNumber34.
• CommentAuthorDmitri Pavlov
• CommentTimeDec 14th 2020

Corrected a mistake in the definition of an infinitary extensive category.

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeJan 22nd 2021

Thomas has alerted me that this page had claimed that smooth manifolds “lack all pullbacks”. I have changed it to: “… does not have all pullbacks (only those along transversal maps)”.

• CommentRowNumber36.
• CommentAuthorDavidRoberts
• CommentTimeJan 22nd 2021

Edited it to

The category Diff of smooth manifolds is infinitary extensive, though it does not have all pullbacks (only those involving a cospan of transversal maps).

to make the case match. The link to transversal maps (plural) before was referring to a single map along which one is pulling back.

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeJan 22nd 2021

It’s not important, but it was okay the way it was:

Pullback is along a map. Fiber product is of a cospan. Pullback along a transversal map, transversal to the map being pulled back, is what the text naturally needed here.

Just saying. But let’s please not discuss this further. :-)

• CommentRowNumber38.
• CommentAuthorDavidRoberts
• CommentTimeJan 23rd 2021

Urs, please, I trust you can make the reasonable assumption I know the mathematics. I’m just editing the phrasing to make it read more smoothly. My reason above was not phrased the best, maybe.

A smooth map has pullbacks along maps transversal to it. But “$Diff$ has pullbacks along transversal maps” makes it sound to my ears like “transversal maps” is a class of morphisms of $Diff$ along which an arbitrary map can be pulled back, like the class of submersions.

That’s all. :-)

• CommentRowNumber39.
• CommentAuthorMike Shulman
• CommentTimeJun 30th 2021

I wonder if it would be too presumptuous to appropriate the name “disjunctive category” for what the page currently calls “pre-lextensive categories”. It seems that some people have used “disjunctive category” to mean “extensive category”, but it seems a waste of good terminology to have two names for the same thing, and since pre-lextensivity suffices to interpret disjunctive logic it seems a natural back-formation.

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