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February 25, 2008

A Question or Two

Posted by David Corfield

Points of a set, XX, correspond to certain maps from the Boolean algebra of subsets, P(X)P(X), to 22, namely those corresponding to prime ideals of the algebra.

Points of a space, YY, correspond to certain functors from the topos of locally constant sheaves to Set, via evaluation at a point again. Is there a way to construe this by analogy to the prime ideal story? Is there a ‘spectrum’ around?

How does one characterise the fibre functors to Set which correspond to points? Is there something ‘ideal’ going on?

Cartier’s Mad Day’s Work paper seems to suggest there is such a story going on here. In the same paragraph (p. 404) as the description of the fundamental group as the automorphism group of a fibre functor, he speaks of the Galois group of a field extension in terms of the fields’ spectra.

Posted at February 25, 2008 11:43 AM UTC

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Re: A Question or Two

Here are some quick thoughts on this.

First, a nitpick: if you start with a set XX, then points xXx \in X do of course give prime ideals of PXP X (kernels of Boolean ring maps eval x:PX2eval_x: P X \to \mathbf{2}). But if XX is infinite, then by the axiom of choice, there are other prime ideals too: namely, maximal ideals containing the prime ideal consisting of finite subsets of XX. These are in bijection with nonprincipal ultrafilters on XX, which can be considered “ideal points at infinity” that are adjoined to XX to form its Stone-Cech compactification.

But we can get XX precisely if we change the language, to include arbitrary joins instead of just finite ones. Then an “ideal” II of PXP X would be defined to be closed under arbitrary joins (and would be a submodule of PXP X under the meet operation); the prime ideals in that case do correspond to points of XX. In other words, work in the category of (Boolean) locales, not just (Boolean) rings.

Actually, this is not just a nitpick; I think the change in point of view is important. The idea is that the concept of Grothendieck topos is a categorification of the concept of locale. The analogue of locale map (one which preserves arbitrary joins and finite meets) would be a functor which preserves arbitrary colimits and finite limits, i.e., the left adjoint part of a geometric morphism.

So, for example, suppose we start with a topological space XX. Under a fairly mild assumption (that XX is sober), those functors

Sheaves(X)SetSheaves(X) \to Set

which preserve colimits and finite limits correspond exactly to points xx of XX; the corresponding “evaluation” functor eval xeval_x sends a sheaf to its stalk at xx (or its fiber, restricting to locally constant sheaves as you mention).

So a first answer to how to characterize these fiber functors is to say they’re the cocontinuous finitely continuous ones to SetSet.

The analogue to ‘ideal’ is less clear-cut. In ring theory, the notion of ideal is a simplification of an appropriate notion of congruence relation, a simplification which is possible because of the presence of additive inverses (i.e., we say aa is congruent to bb modulo II if abIa - b \in I). Since such congruence relations are kernel pairs of ring maps f:ABf: A \to B, one could categorify that by taking the kernel pair = weak pullback of a left exact left adjoint f:EFf: E \to F against itself:

E× FEE.E \times_F E \stackrel{\to}{\to} E.

This kind of thing is studied by the topos theorists, especially in conjunction with descent theory, starting with the Joyal-Tierney monograph An Extension of the Galois Theory of Grothendieck. (I would have to reread that monograph to remind myself of the precise way in which all this is connected with Grothendieck’s Galois theory, and anyway the explanation wouldn’t fit in the margin here.)

In a rather different direction, Steve Awodey in his thesis pursued the following analogy: just as a commutative ring can be reconstituted as the ring of global sections of a sheaf of local rings on its space of prime ideals, so in the 2-category of elementary toposes and logical morphisms, a small topos EE could be reconstituted by taking global sections of a sheaf [or actually stack] of [“hyper”]local toposes over a certain site. The first such site is relatively tautological: it is the category EE itself equipped with the finite covering topology. The second is more sophisticated: the site is a certain topological space! Awodey goes on to give some interesting applications to higher-order logic, but as I say this direction would seem to be a little different to the one you want to go in.

Posted by: Todd Trimble on February 25, 2008 4:03 PM | Permalink | Reply to this

Re: A Question or Two

Thanks!

…this direction would seem to be a little different to the one you want to go in.

The thing is you may well have a better idea of this direction than I do.

Perhaps the Spec idea isn’t worth pursuing to the extent that I seem to be looking in a ‘rig’-like direction, though Bertrand Toën et Michel Vaquié seem to be able to do things with Spec \mathbb{N}, see p. 4 of Au-dessous de Spec \mathbb{Z}.

[If AA is an associative, unitary, commutative monoid object in a certain kind of symmetric monoidal category CC, so an object in Comm(C)Comm(C), then Spec(A)Spec(A) is the corresponding object in Comm(C) opComm(C)^{op}. \mathbb{N} is a commutative monoid object in Comm(Mod)Comm(\mathbb{N}-Mod), the category of commutative monoids.]

Posted by: David Corfield on February 25, 2008 5:42 PM | Permalink | Reply to this

Re: A Question or Two

The general question is a bit too abstract for my level of competence, but there are much more restrictive situations in which the issues are very deep. The one I’m familiar with is when X is a compact curve of higher genus defined over Q with a rational point b. On the category of finite covering spaces of X, b determines a fiber functor F_b that’s `defined over Q’ in a suitable sense. There is a conjecture that proposes that *all* fiber functors that are defined over Q should come from rational points. This is a version of Grothendieck’s section conjecture.

Some details are contained in my colloquium talk at Leeds

MK

Posted by: Minhyong Kim on February 26, 2008 3:11 PM | Permalink | Reply to this

Re: A Question or Two

I get the odd glimpse of what’s going on from your exposition, but would need it to be Baezified a little more.

The remarkable upshot of this formulation is that the study of solutions to equations is subsumed into the study of maps whose very nature compels us to consider as the most basic in all of in mathematics. (p. 3)

Which word(s) are missing here?

Does anything π 2\pi_2-like go on in Diophantine geometry?

Posted by: David Corfield on February 28, 2008 12:22 PM | Permalink | Reply to this

Re: A Question or Two

I know my limitations well-enough! Baezification is infinitely harder than categorification.

Higher homotopy is interesting and has been used in various aspects of arithmetic geometry. Higher *rational* homotopy has even a motivic nature. But curiously enough, there are no applications so far to Diophantine problems. That is, at the moment \pi_1 seems to be much more powerful.

In the paragraph you cite, no words are missing. The end of the sentence should be `…in all of mathematics.’

The meaning of it has to do with the intuition that algebras that are finitely generated over \Z or \Q are much more basic than manifolds over the real or complex numbers, while maps between the corresponding schemes is the usual preoccupation of Diophantine geometry. In their more fanciful moods, arithmetic geometers like to speculate that the real or complex numbers need to be understood much better in a `bottom up’ fashion, and that a proper such understanding might eventually express spaces modeled on complete fields also as rather constructive limits of Diophantine spaces. In that sense, Diophantine geometry should be a foundation to the study of *all* spaces.

Posted by: Minhyong Kim on February 28, 2008 11:51 PM | Permalink | Reply to this

Re: A Question or Two

Upon re-reading the paragraph, I realize that limitations in my English are probably showing up as well. Oh well. I hope the meaning was clear enough.

MK

Posted by: Minhyong Kim on February 29, 2008 12:42 AM | Permalink | Reply to this

Re: A Question or Two

The inclusion of ‘them’ would do the trick:

…maps whose very nature compels us to consider them as the most basic in all of mathematics.

That’s interesting about Diophantine geometry as a ‘foundation’, in presumably the conceptual sense.

It is striking how philosophers seem wedded to the idea that numbers are some of the simplest mathematical entities around, when we know how number theory has generated such a wealth of key ideas, as Barry Mazur pointed out in Number Theory as Gadfly.

I remember also a couple of essays by André van Es on the generative power of number theory. This is the only one I can find online:

…mathematically speaking, arithmetic and geometry may be heavily intertwined. Nonetheless, the spirit of this essay is that arithmetic has very different ‘parameters of evolution’. These are not the ones described above as the isolation of aspects by abstraction and generalization. One could describe them alternatively as the isolation of aspects by analogy.

And later,

There exists a natural fusion of aspects in the objects of arithmetic. Analogy serves to isolate these aspects in separate analogous contexts.

Posted by: David Corfield on February 29, 2008 9:04 AM | Permalink | Reply to this

Re: A Question or Two

Ugh. I do have occasional blind spots in English, not having been born into the language. But is the construction really *incorrect*? I considered the following sequence of transformations:

…sentences that David compels us to consider as incorrect.

…sentences David compels us to consider as incorrect.

…maps Grothendieck compels us to consider as basic.

…maps whose nature compels us to consider as basic.

and so on. Sorry for cluttering up the discussion with silly questions on grammar.

Now back to mathematics.

In fact, my own tendency is to think of Diophantine geometry as foundational in a very *physical* sense, rather than a conceptual one. In particular, generative power at a conceptual level and analogies, while undoubtedly important, seem to receive too much emphasis.

Some feeling for this I explained in a talk for graduate students long ago. What I wrote there appears painfully naive, but it’s probably self-deception to think I could express matters any better now. I should point out however that the second half of that talk dealing with Matiyasevich is actually much less relevant to the present discussion than the first bit.

Posted by: Minhyong Kim on February 29, 2008 12:05 PM | Permalink | Reply to this

Re: A Question or Two

Your way is possible. It’s just that there’s something about the singular ‘nature’ requiring the ‘s’ in ‘compels’ and the plural ‘maps’ which interferes with my parsing.

I’ll have to look at it afresh later.

Posted by: David Corfield on February 29, 2008 2:01 PM | Permalink | Reply to this

Re: A Question or Two

The last of those transformations (to “maps whose nature…”) is illegitimate. Putting the “that” back makes this obvious (“maps that whose nature…”). I’m not certain of the precise nature of the problem, but it’s to do with restrictions on what relative pronouns are allowed to do. You can’t have the relative possessive “whose” referring either to the relative pronoun “that” that heads its clause (particularly if the latter isn’t actually there…), or to the antecedent of the relative clause in which it is embedded (that is, “maps”). So I suspect this is due to a general restriction on relatives in embedded clauses in English whose precise details I can’t remember just now. “Maps that their nature compels us to consider as…” would be OK.

Posted by: Tim Silverman on March 1, 2008 12:06 AM | Permalink | Reply to this

Re: A Question or Two

Thanks! Your explanation definitely demystifies the strange transformation whose very nature compelled me to overlook the error of.

MK

Posted by: Minhyong Kim on March 1, 2008 1:11 AM | Permalink | Reply to this

Re: A Question or Two

“Diophantine geometry should be a foundation”

I wonder if this would not imply that one finds arithmetic or function field analogies to all constructions in geometry, e.g. the mathematics behind the geometrisation theorem or Connes’ noncommutative geometry?

Posted by: Thomas Riepe on February 29, 2008 2:25 PM | Permalink | Reply to this

Re: A Question or Two

In fact, this is the usual direction of research. The Weil conjectures form a well-known example and Connes believes in arithmetic non-commutative geometry with characteristic passion. In his anabelian letter to Faltings, Grothendieck alludes to the rigidity of hyperbolic geometry as inspiration for anabelian geometry.

But own fanciful view is that such macroscopic phenomena should, in some sense, *originate* in Diophantine structures.

Posted by: Minhyong Kim on February 29, 2008 4:19 PM | Permalink | Reply to this

Re: A Question or Two


Thanks for pointing to that conference! I looked the past days a bit into NC-Geometry, because of the apparent links with modular forms (else I doubt that Dixmier traces etc. would ever have interested me). In case you know articles connecting NC-Geometry with F1, would you please post the bibliographic infos?

Posted by: Thomas Riepe on February 29, 2008 7:48 PM | Permalink | Reply to this

Re: A Question or Two

Minhyong, if you’d be willing to put your PR hat on and explain in your opinion the best case for an interesting connection between Connes-style non-commutative geometry and arithmetic algebraic geometry, I’d be very interested.

(To put all my cards on the table, I’m tad skeptical. But I’m much more ignorant than skeptical, so I’m keeping an open mind. In any event, I’m truly interested and not setting anyone up for an ambush.)

Posted by: James on February 29, 2008 11:17 PM | Permalink | Reply to this

Re: A Question or Two

That PR hat is much too big for me, since both my ignorance and skepticism may well run deeper than your own. I think I mentioned once that neither reference nor quotation signify endorsement.

However, I have acquired over the years a great deal of respect for Connes the person on account of the absolute devotion and seriousness with which he pursues his scientific convictions. They certainly make me want to sympathize with him, even when my understanding is rather dim. There is a similar situation with another F_1-theorist, Shinichi Mochizuki.

For this reason, I may make an attempt to fulfill your request in some version of the future, regardless of my own beliefs (or lack of them).

Posted by: Minhyong Kim on March 1, 2008 1:32 AM | Permalink | Reply to this
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