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December 10, 2007

This Week’s Finds in Mathematical Physics (Week 259)

Posted by John Baez

In week259 of This Week’s Finds, hear what may be hiding in the Egg Nebula:



Then, learn how a mathematical phantom called the ‘field with one element’ is gradually becoming real. It may explain the deep inner meaning of q-deformation, and the 3-dimensional aspect of the integers!

Ever wonder what a ‘scheme’ is? You’ll learn that too.

Posted at December 10, 2007 12:17 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1531

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Nebulae and Meadows; Re: This Week’s Finds in Mathematical Physics (Week 259)

That was great!

The astrophotography, vision of the hot dusty electron-atmospheric future (don’t bother Al Gore, though, as he’s busy today in Oslo), and Field ponderings.

Anything to this?:

arXiv:0712.0917
Title: Some properties of finite meadows
Authors: Inge Bethke, Piet Rodenburg
Comments: 8 pages, 1 table
Subjects: Symbolic Computation (cs.SC)

Posted by: Jonathan Vos Post on December 10, 2007 1:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

You say

Durov’s framework generalizes Deitmar’s!

even though they disagree about vector spaces over F 1F_1.

For Deitmar

A “vector space over F 1F_1” is just a … plain old set,

while for Durov it’s a pointed set. This must surely be an indication of a big difference which will show when they come to do algebraic geometry over F 1F_1.

Posted by: David Corfield on December 10, 2007 2:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Durov’s framework has a generalized ring whose modules are sets, and one whose modules are pointed sets. So, his framework generalizes Deitmar’s.

You make it sound like they disagree over the definition of ‘vector spaces over F 1F_1’, but I don’t think Deitmar even talks about ‘vector spaces over F 1F_1’. I wrote:

A ‘vector space over F 1F_1’ is just a set on which this monoid acts via multiplication… but that amounts to just a plain old set. The ‘dimension of such a ‘vector space’ is just its cardinality.

but you shouldn’t really blame Deitmar for this. I was just trying to stretch his idea

ringsmonoids rings \mapsto monoids

to include

modulesofringsactionsofmonoids modules of rings \mapsto actions of monoids

and noting that a module over the 1-element monoid (which is what he calls F 1F_1) is nothing but a set.

But you see, he’s an algebraic geometer in a hurry. He’s not interested in vector spaces over F 1F_1 — he’s interested in going straight to schemes defined over F 1F_1. And, I think his example of the ‘projective line over F 1F_1’ matches what Durov says that should be: a 2-point set.

To be blunt, I think Durov’s approach is more systematic and general. So, he talks a bit about both generalized rings: the one whose modules are pointed sets, and the one whose modules are sets. It’s probably good for algebraic geometers to deploy both, since the obvious ‘throw out the basepoint’ functor

[pointedsets][sets][pointed sets] \to [sets]

is just what algebraic geometers would call ‘projectivization’ — turning a ‘vector space’ for Durov’s F 1F_1 into a ‘projective space’. It just so happens that a ‘projective space’ for Durov’s F 1F_1 is a module of some other generalized ring… the one whose modules are just sets.

There’s also the obvious ‘keep the basepoint’ functor

[pointedsets][sets][pointed sets] \to [sets]

and the obvious ‘throw in a basepoint’ functor

[sets][pointedsets] [sets] \to [pointed sets]

but I think it’s good to have a total of 3 functors around. After all, whenever we have a homomorphism of (generalized) rings f:RSf: R \to S we get the obvious pullback

f *:SModRModf^* : S Mod \to R Mod

but also, one might hope, both a left and right adjoint to this — what Lawvere would call ‘free’ and ‘fascist’ functors.

In short, there’s room for plenty of fun and confusion here, but it should all make nice sense in the end. If I seem a bit casual about pointed sets versus sets, it’s because of my faith in that.

Posted by: John Baez on December 10, 2007 9:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Is there any reason to think more about my observation concerning the subobject classifier of the 2-topos of categories as scalar restriction of F 1F_1-modules to F F_{\empty}-modules, i.e., pointed sets to sets?

Perhaps it’s more truthfully described as a discrete opfibration classifier, as Weber does. He also notes Lawvere’s idea: the category of sets is a generalised object of truth values.

Posted by: David Corfield on December 11, 2007 9:24 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

but something else, something more mysterious…

I am wondering: rings have a nice neat categorical description as categories enriched over abelian groups.

One problem with fields seems to be that they lack an equally neat characterization. This is a nuisance not only when looking for the field with one element, but also, for instance, when one tries to categorify the notion of a field.

Maybe somebody should think harder about what fields really are.

(And maybe somebody is already doing that. But then maybe somebody should think harder about what the first somebody is really doing. If you know what I mean.)

Posted by: Urs Schreiber on December 10, 2007 5:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

What if one tried to categorify a generalized field? Perhaps after doing the same for a generalized ring.

What do we know about algebraic 2-theories?

No doubt TWF won’t let us down. Ah yes, Week 170. So Noson Yanofsky gives us a definition of algebraic 2-theories in Coherence, Homotopy and 2-Theories.

What kind of 2-theory corresponds to a commutative theory? And what to the terminal theory?

Posted by: David Corfield on December 10, 2007 6:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

David wrote:

What if one tried to categorify a generalized field?

One shouldn’t — ones brain might explode!

Seriously, it’s an interesting idea, and it might serve to unify a bunch of work on ‘categories’ and ‘2-vector spaces’, just as generalized fields one notch down unify a bunch of work on ‘sets’ and ‘vector spaces’.

But, one notch down there’s a lot more evidence staring us in the face, making it incredibly obvious that we should unify a bunch of work about ‘sets’ and ‘vector spaces’. I’m not talking about the overall family resemblance of SetSet and Vect FVect_F for any field FF, which existing category theory understands fairly well, and which we already know how to categorify to some extent. I’m talking about the much more detailed, exciting and mysterious sense in which Vect F qVect_{F_q} converges to SetSet as q1q \to 1.

If one found something like this one notch up, there’d really be a good reason to get excited about categorified generalized fields. Right now the idea seems a bit too ethereal to grab my interest.

(Right now I’m having fun trying to learn some basic algebraic geometry while simultaneously learning the far-out generalizations proposed by Durov, especially algebraic geometry over F 1F_1 and algebra geometry over \mathbb{Z}_\infty, which is all about ‘the real prime’ and ‘Arakelov geometry’.)

Posted by: John Baez on December 12, 2007 7:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

If one found something like this one notch up…

Hmmm. Shouldn’t we expect that the q1q \to 1 limit of the 2-category of Vect F qVect_{F_q}-modules is the 2-category of (pointed sets)-modules?

I know Urs has talked about things like Vect-mod. But what is known? So we have Kapranov-Voevodsky 2-vector spaces as finitely generated free Vect-modules. And then further entities like Bim(Vect).

Is it even known what Set-mod looks like?

Is there an (abelian) algebraic 2-theory resembling Set \to Pointed set, which sends categories to pointed categories?

Hmmm, once more. Weren’t your lectures teeming with pointed categories?

Posted by: David Corfield on December 12, 2007 9:36 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Lots of good questions, David! Please help me out and answer some.

Posted by: John Baez on December 13, 2007 12:14 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

The odd thing about participating in this enterprise as an amateur is that one has much less sense of what ought to be already known. (Now which mathematician said something to the effect that what he could bring to the study of a field was a reliable sense of what ought to be already known? Oh yes, Ulam quoted on p. 107 of my book.)

If John Gray wrote about algebraic 2-theories in 1974, then surely there’s a whole heap of collected wisdom on them out there somewhere. Not that working things out for yourself should be avoided.

There ought to be a 2-adjunction between categories and pointed categories where the left adjoint merely adds a pointed object. So perhaps pointed categories are the 2-vector spaces for some generalized 2-ring.

On the other hand there are (Pointed Set)-modules, including those such as (PointedSet) n(Pointed Set)^n.

With FinSet-modules, we want a map FinSetEnd(C)FinSet \to End(C), for some category CC. Is it that you couldn’t expect an equivalent of \mathbb{Z}’s module /2\mathbb{Z}/2\mathbb{Z}?

Maybe the groupoid FinSet 0FinSet_0 could have torsion modules.

Posted by: David Corfield on December 13, 2007 10:44 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

David wrote:

If John Gray wrote about algebraic 2-theories in 1974, then surely there’s a whole heap of collected wisdom on them out there somewhere.

Probably not a ‘whole heap’ — not in published form, anyway. I’m sure if you asked Martin Hyland he could tell you everything you wanted to know about algebraic 2-theories… but it’d mostly be in his head, not in papers.

You have to remember, John Gray was way ahead of his time when it came to nn-categories. Also, category theory is a strongly deprecated field, especially in the USA — so not many people work on it, and not every idea gets followed up.

I bet you’ll find more published work on ‘2-monads’ or ‘pseudomonads’ than ‘2-theories’. That’s another way to skin the same cat. But again, the number of people actually working on these is darn small: Street, Hyland, Lack, Powers, a few more… mainly the usual suspects.

There may be a few people who have thought about ‘2-operads’, too.

Anyway, I should answer some of your questions, but I gotta go give a final exam!

Posted by: John Baez on December 13, 2007 6:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

I see the point that we need enough pressure to build up to prompt us to do the work.

one notch down there’s a lot more evidence staring us in the face, making it incredibly obvious that we should unify a bunch of work about ‘sets’ and ‘vector spaces’.

Then again perhaps this particularly way of pushing for a unification of ‘categories’ and ‘2-vector spaces’ might promote the generation of a similar bunch of work.

One of the things we lack is a parallel to the nice formula for the size of GL(n,F q)GL(n, F_q) tending to that of S nS_n, as q1q \to 1.

This is all very close to our 2-Klein geometry work.

As Urs has told us, your and Alissa’s 2-vector spaces are categories internal to Vect KVect_K. So we might expect that BC 2-vector spaces over F qF_q will tend to categories internal to PointedSet as q1q \to 1.

Now what’s one of those? A pointed category (C,*)(C, *) with id *id_* a pointed arrow?

How about this for a profound fact – a category is a BC 2-vector space over the field without elements.

Posted by: David Corfield on December 14, 2007 9:16 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Haven’t thought about 2-operads;, but what sort of thoughts are worth pursuing? There’s a whole bunch of `exercises’ one could perform - just using our experience with other 2-things.

Posted by: jim stasheff on December 14, 2007 1:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Jim writes:

Haven’t thought about 2-operads; but what sort of thoughts are worth pursuing?

It’s not instantly clear, so it might be worth noting the work that’s been done so far:

Batanin has a whole theory of ω\omega-operads, such that his weak ω\omega-categories are algebras of a certain ‘initial contractible’ ω\omega-operad. There’s a marvelous relation to associahedra and their generalizations. I see David had some questions about this that never got answered… I know some answers.

Posted by: John Baez on December 14, 2007 8:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Maybe somebody should think harder about what fields really are.

(And maybe somebody is already doing that. But then maybe somebody should think harder about what the first somebody is really doing.)

Lol, this is brilliant!

Someone’s thinking my Lord, come to him,

Someone’s thinking my Lord, come to him,

Someone’s thinking my Lord, come to him,

Oh lord, kumbaya.

Posted by: Bruce Bartlett on December 10, 2007 9:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Bruce wrote:

Someone’s thinking my Lord, come to him,

Oh lord, kumbaya.

Interesting, a quote from Joan Baez here! (according to Wikip.) ;-)

Posted by: Urs Schreiber on December 11, 2007 6:08 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Indeed. You can’t keep a good Baez down.

Posted by: Bruce Bartlett on December 12, 2007 12:46 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Urs wrote:

I am wondering: rings have a nice neat categorical description as categories enriched over abelian groups.

One problem with fields seems to be that they lack an equally neat characterization. This is a nuisance not only when looking for the field with one element, but also, for instance, when one tries to categorify the notion of a field.

Maybe somebody should think harder about what fields really are.

Indeed!

In theory, algebraic geometers are the ones who should do this. Early on in algebraic geometry, the idea arose that the same equation, say

x n+y n=z n, x^n + y^n = z^n,

could have drastically different solutions depending on what field the variables (here x,y,zx,y,z) lived in. This eventually led to the idea of a ‘scheme’ as something that could have ‘kk-points’ for any field kk.

But in the process, people realized that fields are in many ways less convenient to work with than commutative rings. Grothendieck can probably take a lot of credit for this. It’s a great example of a principle I attribue to him: ‘it’s better to work with a nice category containing some nasty objects, than a nasty category containing only nice objects’. While commutative rings can be ‘nasty’ compared to fields (e.g. they can contain nilpotents, which were traditionally considered ‘nasty’), the category of commutative rings is very ‘nice’ compared to the category of fields (basically because commutative rings involve only everywhere defined operations satisfying equational laws, so they’re models of an algebraic theory).

So, Grothendieck’s later approach to schemes (not the textbook one) is based on the idea that a scheme is nothing but a gadget that has ‘RR-points’ for any commutative ring RR. In this later definition, a scheme is just a functor

F:CommRingSetF: CommRing \to Set

assigning to each commutative ring RR a set of ‘RR-points’, F(R)F(R). In particular, any commutative ring SS gives a scheme

hom(S,):CommRingSethom(S, -) : CommRing \to Set

and these are what we call ‘affine schemes’.

This is very slick and beautiful, and I explained some of its virtues in week205 — for example, it gives a nice way of understanding topics in number theory like ‘inert primes’.

But, after posting week205 I got a reply from a well-known number theorist saying this stuff was useless.

I wasn’t convinced. But, I get the impression that many algebraic geometers and number theorists are still recovering from Grothendieck. They don’t really want further drastic reformulations of the subject, of the sort that might follow from pondering questions like ‘what are fields, really’.

I was reminded of this by something James wrote:

I think it is fair to say that the weirdness in scheme theory around closed points vs all points and maximal ideals vs prime ideals is really an artifact of scheme theory relying on point-set topology rather than topos theory. And things are this way because Grothendieck invented toposes after he invented schemes, and no one since then has bothered/wanted to rewrite the foundations in terms of topos theory.

and Tom’s reply:

That’s really bizarre! You’d think that algebraic geometry would be important enough that someone would have done it. Maybe it’s one of those things where people are afraid to rewrite foundations because they think people are too set in their ways. Or maybe people have tried to do it, but got the reaction ‘why are you telling me things I already know, just in some obscure language?’

It’s certainly forgivable to want to take a break from revamping the foundations of a subject, but I think eventually enough difficulties will build up to force algebraic geometry to further heights of abstraction. Some problems can only be solved that way.

In particular, Durov’s work shows there are lots of advantages of thinking of ‘commutative rings’ as a special case of ‘commutative algebraic theories’ — theories with a bunch of nn-ary operations that all commute. We need this to understand what sort of thing the ‘field with one element’ is.

But, we may not need to decide which commutative algebraic theories deserve to be called ‘fields’! You can see Durov struggling with this question, but I’m not sure how important the answer is.

Posted by: John Baez on December 10, 2007 10:41 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

John wrote:
‘it’s better to work with a nice category containing some nasty objects, than a nasty category containing only nice objects’.

Some such generalisations of schemes:
Stacks and Artins algebraic spaces (or Knutsons book)

Best,
Thomas

Posted by: Thomas Riepe on December 14, 2007 8:14 AM | Permalink | Reply to this

Phield Theory: Phanton Fields; Re: This Week’s Finds in Mathematical Physics (Week 259)

How general should generalizations be, beyond the phantom Field with 1 element?

Should the general Phield theory include the phantom field with 0 elements? The phantom field with -1 elements? The phantom field with 1/2 elements? The phantom field with pi elements? The phantom field with i elements?

I have discussed with Michael Aschbacher to what extent the sporadic simple groups are examples of something deeper, which happen to be groups, and whose distribution reflect phantom sporadic groups that don’t quite make it into grouphood.

It seems to me that your periodic table of higher-order Lie algebras may be related.

Posted by: Jonathan Vos Post on December 10, 2007 7:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Typo: it’s Nikolai Durov, not Anton Durov.

Posted by: jon on December 10, 2007 8:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Whoops — thanks for catching that! I’ll fix it.

Posted by: John Baez on December 10, 2007 10:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

“They keys are, like, so close to one another”

;)

Posted by: Mikael Vejdemo Johansson on December 10, 2007 10:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Great, now I’ve gotta keep straight “ghosts” and “phantoms”. The fact that Gavin Wraith wrote up the article doesn’t help.

When we get to liches I quit.

Posted by: John Armstrong on December 10, 2007 10:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

That cracked me up while reading it as well.

Hmmmm… If we do have liches, then I’d like to dub the maximal lich a lich lord. At last! Mathematics that sounds like an AD&D-game!

Posted by: Mikael Vejdemo Johansson on December 10, 2007 10:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

If we do have liches, then I’d like to dub the maximal lich a lich lord. At last! Mathematics that sounds like an AD&D-game!

Would it come with a no-ghost theorem for attacking the Lich lattice?

(Sorry. :-P)

Posted by: Todd Trimble on December 11, 2007 12:20 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

If you like Halloween-themed mathematical concepts, you’ll love Dan Christensen’s paper on phantoms, ghosts, and skeleta. Too bad Gavin Wraith didn’t write it.

Posted by: John Baez on December 11, 2007 12:57 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

I was trying to figure out why Durov likes commutative generalized rings so much better than generalized rings in general. Of course this should be related to why algebraic geometers like commutative rings better than rings in general.

And, this morning while I was waiting for my pupils to dilate at the optometrist’s, I came up with some ideas. I’m hoping Tom Leinster or Todd Trimble or someone could say if these are new or not.

First, let’s stop calling them ‘generalized rings’ and call them what they are: ‘algebraic theories’.

Second, let’s start by seeing how much we can do using operads.

There’s a down-to-earth definition of what it means for an operad to be ‘commutative’. But the main thing this definition accomplishes, I think, is this:

Given an operad OO we can talk about its algebras in any symmetric monoidal category CC. We get a category of OO-algebras in CC, which I’ll call Oalg(C)O alg(C).

If OO is commutative, all its operations act on any algebra AA not just as morphisms

A nA A^{\otimes n} \to A

in CC, but as OO-algebra homomorphisms!

So, for OO commutative, an OO-algebra in CC automatically becomes an OO-algebra in Oalg(C)O alg(C)! We also have a forgetful functor going the other way, so we get an equivalence

Oalg(C)Oalg(Oalg(C))O alg (C) \simeq O alg (O alg(C))

There’s something very Eckmann–Hiltonesque about this. And in fact it seems to be Eckmann–Hiltonesque in two ways — two ways related by a curious ‘level shift’.

First, consider an operad with only unary operations. This is just a monoid in disguise. Further, the operad is commutative iff the monoid is commutative. We don’t usually talk about ‘algebras’ of a monoid, though — we call them ‘actions’. The above story, restricted to this special case, thus says ‘if MM is a commutative monoid, an action of MM in CC is the same as an action of MM in the category of actions of MM in CC’.

(And, if C=SetC = Set, this statement is an ‘if and only if’.)

Second, consider the operad OO whose algebras are commutative monoids. This OO is a commutative operad! So, we get

Oalg(C)Oalg(Oalg(C))O alg (C) \simeq O alg (O alg(C))

or in other words, ‘a commutative monoid in CC is the same as a commutative monoid in the category of commutative monoids in CC’.

Anyway, while all this is cute, I don’t yet see how to use it to give a slick definition of ‘commutative operad’. Or maybe I do: maybe we can say OO is commutative if the forgetful functor

Oalg(Oalg(C))Oalg(C)O alg(O alg(C)) \to O alg (C)

is an equivalence of categories! Does that seem right?

Also, I don’t see why this makes commutative operads better than general operads as a method of generalizing algebraic geometry. For that, I think I want something more like this:

There’s a tensor product of rings, but for commutative rings this tensor product is more interesting, since it makes CommRing opComm Ring^{op} into a cartesian category. This is a fundamental sense in which affine schemes act like spaces.

There’s also a tensor product of operads. Is this more interesting for commutative operads? Does it make the opposite of the category of commutative operads into a cartesian category?

I thought of some other stuff too, but this is already more than anyone wants to read.

Posted by: John Baez on December 10, 2007 11:13 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Jim Dolan called with a few remarks.

I’d written:

So, for OO commutative, an OO-algebra in CC automatically becomes an OO-algebra in Oalg(C)O alg(C)! We also have a forgetful functor going the other way, so we get an equivalence

Oalg(C)Oalg(Oalg(C))O alg (C) \simeq O alg (O alg(C))

This is correct up to the phrase ‘so we get an equivalence’, which is wrong. For any operad OO we have a ‘forgetful’ functor

Oalg(Oalg(C))Oalg(C) O alg (O alg(C)) \to O alg (C)

and when OO is commutative my argument gives a functor going the other way:

Oalg(C)Oalg(Oalg(C))O alg (C) \to O alg (O alg(C))

but these are not weak inverses of each other!

An abstract way to see this uses the tensor product of operads. This is defined by the property that an O 1O 2O_1 \otimes O_2-algebra in CC is the same as an O 1O_1-algebra in O 2alg(C)O_2 alg(C), or more precisely:

(O 1O 2)alg(C)O 1alg(O 2alg(C))(O_1 \otimes O_2) alg (C) \simeq O_1 alg (O_2 alg(C))

Syntactically, we generate the operad O 1O 2O_1 \otimes O_2 by throwing in all the operations of O 1O_1 and all the operations of O 2O_2, and imposing relations saying that they commute.

If we really had

Oalg(Oalg(C))Oalg(C) O alg (O alg(C)) \simeq O alg (C)

we would have

(OO)alg(C)Oalg(C)(O \otimes O)alg(C) \simeq O alg (C)

But, it’s easy to find counterexamples, even when OO is commutative. Why? Simply because OOO \otimes O is generated by two commuting copies of all the operations in OO. And that can be a lot more operations than there are in OO.

In fact, the easiest counterexamples arise when our commutative operad OO has just unary operations, so that it’s a commutative monoid in disguise.

For example, suppose OO is just \mathbb{N} in disguise, and take C=SetC = Set. Then an OO-algebra in CC is a set XX with a function f:XXf: X \to X. An OOO \otimes O-algebra in CC is a set XX with two commuting functions f,g:XXf,g : X \to X. So, they’re very different.

So, the following ‘corollary’ of my false claim is also false:

First, consider an operad with only unary operations. This is just a monoid in disguise. Further, the operad is commutative iff the monoid is commutative. We don’t usually talk about ‘algebras’ of a monoid, though — we call them ‘actions’. The above story, restricted to this special case, thus says ‘if MM is a commutative monoid, an action of MM in CC is the same as an action of MM in the category of actions of MM in CC’.

(And, if C=SetC = Set, this statement is an ‘if and only if’.)

The rest of what I wrote seems true.

In particular, it seems correct that the tensor product of operads, restricted to commutative operads, is the coproduct of commutative operads. So, the answer to this:

Does it make the opposite of the category of commutative operads into a cartesian category?

seems to be ‘yes’.

Posted by: John Baez on December 11, 2007 12:49 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Jim adds:

by the way, about this business of why the “forgetful” functor from (o tensor o)-algebras to o-algebras isn’t an inverse to the (“co-diagonal”) functor going the other way that exists (precisely?) when o is commutative, it sort of has to do with the fact that there are _two_ forgetful functors; you can forget either the first (“inside”) o-algebra structure or the second (“outside”) one. again, this is the way it generally works; neither of the two projections from xXx to x is inverse to the diagonal x->xXx.

Posted by: John Baez on December 11, 2007 2:53 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

In particular, it seems correct that the tensor product of operads, restricted to commutative operads, is the coproduct of commutative operads.

Right, and I think the mistake that Jim pointed out can be understood in this language:

If OO is a commutative operad, then there is a codiagonal map

:OOO\nabla: O \otimes O \to O

in the category of commutative operads since, as you say, the operadic tensor product becomes the coproduct there. By pulling back, this gives one direction

Oalg(OO)algOalg(Oalg)O alg \to (O \otimes O)alg \simeq O alg(O alg)

of that “equivalence” you were conjecturing. The other direction, a forgetful functor

(OO)algOalg(Oalg)Oalg(O \otimes O)alg \simeq O alg(O alg) \to O alg

can be seen as induced from the unique operad map IOI \to O, where II is the initial operad. That is, pulling back along this operad map is the forgetful functor

Oalg(C)Ialg(C)C.O alg(C) \to I alg(C) \simeq C.

In other words, the forgetful functor direction you were alluding to corresponds to the operad map

OIOi1OOO \simeq I \otimes O \stackrel{i \otimes 1}{\to} O \otimes O

which is nothing other than the second coproduct inclusion i 2:OOOi_2: O \to O \otimes O. Now of course i 2\nabla \circ i_2 is the identity by the way coproducts work, but the other composite i 2i_2 \circ \nabla is generally not the identity.

Are you unhappy with the standard way of defining the notion of commutative operad? If OO is a Set-valued operad, we have a composite map

O(m)×O(n)1×δO(m)×O(n) mμO(mn)O(m) \times O(n) \stackrel{1 \times \delta}{\to} O(m) \times O(n)^m \stackrel{\mu}{\to} O(m n)

which we may call α m,n\alpha_{m, n}, and another composite map

O(m)×O(n)O(n)×O(m)α n,mO(mn),O(m) \times O(n) \cong O(n) \times O(m) \stackrel{\alpha_{n, m}}{\to} O(m n),

and commutativity of OO is the condition that these maps agree for all mm, nn.

One thing that I can imagine might be slightly ‘irritating’ is that we are evidently using the presence of a diagonal map δ\delta to define this notion (i.e., that the objects O(n)O(n) are cocommutative comonoids :-) ). Assuming our symmetric monoidal category CC has coproducts which are preserved under tensoring, the change of base

SetCSet \to C

is a strong symmetric monoidal functor which preserves such commutative comonoids, so the notion of commutative operad would cleanly carry over to CC under such base change.

Posted by: Todd Trimble on December 11, 2007 3:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Pipped by Jim. Curses, foiled again…

Posted by: Todd Trimble on December 11, 2007 3:38 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Pipped?
and Jim = me?

Posted by: jim stasheff on December 11, 2007 1:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Jim Stasheff asked:

and Jim = me?

I am pretty sure that the Jim meant was Jim Dolan. Todd was referring , I guess, to this and this forwarded comment from Jim Dolan, which beat his own comment by, let’s see, apparently some 40 minutes or so.

Posted by: Urs Schreiber on December 11, 2007 3:57 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

So what’s a commutative (or do we want some other term, like symmetric?) algebraic 2-theory? Presumably a 2-theory made up of things like nn-ary Vect-linear combinations would count.

Posted by: David Corfield on December 11, 2007 9:18 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Though Todd was pipped by Jim, Todd’s post is probably easier to understand, since it’s more detailed. So, it’s all for the best in this best of all possible worlds.

Todd wrote:

Are you unhappy with the standard way of defining the notion of commutative operad?

I was at the time, mainly because it seemed too ‘syntactical’, as opposed to ‘semantic’. I was trying to figure out why Durov prefers to do algebraic geometry with commutative operads (or really, algebraic theories). And, it seemed to me that one of the the main things he does with an operad OO is play around with their categories of algebras Oalg(C)O alg(C). (Not having read all of his 568-page time, I’m not sure how true this really is, but anyway, this is one of the main things everyone always does with operads, so I was willing to guess it’s true for him too.) So, I was seeking a characterization of commutative operads solely in terms of their categories of algebras!

And so, I was slowly blundering my way towards a realization that you and Jim had more clearly.

Namely: while OpOp (the category of operads) has a tensor product, this becomes the coproduct when restricted to CommOpCommOp (the category of commutative operads). A tensor product is a coproduct when it has a codiagonal

:OOO\nabla: O \otimes O \to O

and a unit

!:1O!: 1 \to O

satisfying some nice properties. In paarticular, they make OO into a commutative monoid in OpOp. So, Jim also hazarded a speculation that perhaps a commutative operad is the same as a commutative monoid in OpOp. Is anyone up to figuring this out?

Anyway, in terms of categories of models, our operad being commutative gives us

Δ *:Oalg(C)(OO)alg(C)Oalg(Oalg(C))\Delta^* : O alg(C) \to (O \otimes O) alg(C) \simeq O alg(O alg (C))

as we have been discussing. It also gives us

! *:1alg(C)Oalg(C) !^*: 1 alg(C) \to O alg(C)

Hmm. Since the algebras of the terminal operad 11 are just commutative monoids, 1alg(C)1 alg (C) is the category of commutative monoids in CC. So, we’re saying any commutative monoid in CC becomes an algebra of OO in CC, which is quite familiar to me — just a turned-around, semantic way of saying 11 is the terminal operad.

Anyway, trying to assemble these observations into something a bit more cogent, I’d like to say that OO being commutative makes it a commmutative monoid in OpOp, so that Oalg(C)O alg(C) gets a kind of ‘cocommutative comonoid’ structure. But, I’m not quite sure how to make that precise, because what symmetric monoidal 2-category is Oalg(C)O alg(C) an object in, exactly?

To be frank, I’m not sure where all this stuff is going. It now seems to me that something else is more important. We’re trying to generalize algebraic geometry. Algebraic geometers are called ‘geometers’ because CommRing opCommRing^{op} (the category of affine schemes) has many of the features typical of a category of ‘spaces’. For starters, it’s cartesian! But CommRing opCommRing^{op} lacks some nice features: namely, we can’t always glue together affine schemes to get other affine schemes. So, algebraic geometers embed CommRing opCommRing^{op} in a larger category, the category of schemes, which has more colimits. (All colimits?)

Now we’re seeing that CommOpCommOp is a bit similar to CommRingCommRing. In particular, CommOp opCommOp^{op} — the punster in me wants to call this the category of commutative ‘erads’ — is cartesian, just like the category of affine schemes.

I’d like to know exactly how far the similarity goes.

And then, I’d like to see Deitmar and Durov’s tricks for generalizing the concept of ‘scheme’ as very general tricks, applicable whenever we have a cartesian category with some extra features, and want to be able to from new objects by ‘gluing together’ objects form this cartesian category.

(We could take presheaves on our cartesian category, to get arbitrary colimits, or we could do something weirder, analogous to the textbook definition of ‘scheme’.)

When I’m done, I’ll probably discover that this stuff is already well-known among experts in topos theory. But that’s okay.

Posted by: John Baez on December 11, 2007 7:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Namely: while OpOp (the category of operads) has a tensor product, this becomes the coproduct when restricted to CommOpCommOp (the category of commutative operads). A tensor product is a coproduct when it has a codiagonal :OOO\nabla: O \otimes O \to O and a unit !:1O!:1 \to O satisfying some nice properties. In paarticular, they make OO into a commutative monoid in OpOp. So, Jim also hazarded a speculation that perhaps a commutative operad is the same as a commutative monoid in OpOp. Is anyone up to figuring this out?

Two things. One, there seems to be a typo, where you have ‘1’ (the terminal operad) instead of ‘II’ (the initial operad). (Later, you do mention 1 again as the operad for commutative monoids; II is the operad for general objects.)

Two, yes to Jim’s surmise, but I think you can also say that a commutative operad is an operad OO equipped with just a monoid structure with respect to \otimes; that is, monoids here are automatically commutative. This for approximately the same reason that a commutative group is a group equipped with a monoid structure with respect to ×\times, and a commutative ring is a ring equipped with a monoid structure with respect to \otimes (commutation of operations being built into the definitions of these monoidal products).

Posted by: Todd Trimble on December 11, 2007 9:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Thanks for catching the mistakes, Todd — and thanks for confirming Jim’s guess!

As for this thing I wrote…

(We could take presheaves on our cartesian category, to get arbitrary colimits, or we could do something weirder, analogous to the textbook definition of ‘scheme’.)

When I’m done, I’ll probably discover that this stuff is already well-known among experts in topos theory.

… maybe this ‘something weirder’ is just called forming a topos of sheaves on a site — in which case it’s very well-known, and not weird at all.

I don’t know. I guess I need some kindly soul to answer the following two questions:

Every scheme XX gives a functor

F:CommRingSetF : CommRing \to Set

where F X(R)F_X(R) is the set of ‘RR-points’ of XX — this being just another name for scheme morphisms Spec(R)XSpec(R) \to X.

Which functors F:CommRingSetF: CommRing \to Set come from a schemes in this way?

A functor F:CommRingSetF: CommRing \to Set is called a ‘presheaf on CommRing opCommRing^{op}’. Could there be some Grothendieck topology on CommRing opCommRing^{op} such that FF comes from a scheme iff FF is a sheaf with respect to this topology?

If this is true, I bet Deitmar’s ‘schemes over F 1F_1’ are really just sheaves on CommMon opCommMon^{op} with respect to some Grothendieck topology on CommMon opCommMon^{op}.

And, maybe this is part of a big pattern…

Posted by: John Baez on December 12, 2007 3:12 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Which functors F:CommRingSetF: CommRing \to Set come from a scheme in this way?

James said something about this back here. The topology he was referring to (“closure under adjoining quotients by equivalence relations which are covers in your topology”) is the Zariski topology on affine schemes, where covers E αEE_{\alpha} \to E are covering families by Zariski open affine subschemes.

I’m sure he (among others) could answer this question definitively:

A functor F:CommRingSetF: CommRing \to Set is called a ‘presheaf on CommRing opCommRing^{op}’. Could there be some Grothendieck topology on CommRing op such that F comes from a scheme iff F is a sheaf with respect to this topology?

That is, do schemes form a topos? I doubt it: I thought I heard somewhere that they don’t even admit all coequalizers. James?

But maybe it doesn’t matter. The sheaf-theoretic characterization of schemes James mentioned is of a general nature which would probably apply to this speculation:

If this is true, I bet Deitmar’s ‘schemes over F 1F_1’ are really just sheaves on CommMon opCommMon^{op} with respect to some Grothendieck topology on CommMon opCommMon^{op}.

So maybe we just tweak that, referring to a Zariski topology on CommMon opCommMon^{op}, and close up coproducts of objects in this category under quotients of equivalence relations which are covers?

Posted by: Todd Trimble on December 12, 2007 7:22 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Todd wrote:

The topology he was referring to… is the Zariski topology on affine schemes, where covers E αEE_\alpha \to E are covering families by Zariski open affine subschemes.

Good. I think I get this idea of using the Zariski topology on affine schemes to define general schemes. I was wondering about a more ‘global’ way to pick out the schemes from among the presheaves on CommRing opCommRing^{op}.

That is, do schemes form a topos? I doubt it: I thought I heard somewhere that they don’t even admit all coequalizers. James?

We’ll have to wait until that masked man returns.

If schemes don’t even admit all coequalizers, no wonder Grothendieck became dissatisfied with them and wanted to switch to general presheaves over CommRingCommRing. I didn’t think they were that bad. (But maybe it’s easy to find two topological spaces that locally look like spectra of rings, and glue them together in a way to get a space that’s not like this?)

Anyway, while I’d like to know the slickest and most up-to-date approach to algebraic geometry before generalizing it, it seems most people are sticking with the usual approach to schemes…

So maybe we just tweak that, referring to a Zariski topology on CommMon opCommMon^op, and close up coproducts of objects in this category under quotients of equivalence relations which are covers?

If I understand what’s going on, that’s what Deitmar is already doing here (from week259):

He starts by defining a “commutative ring over F1” to be simply a commutative monoid. The simplest example is F1 itself.

Now, watch how he gets away with never using addition:

He defines an “ideal” in a commutative monoid R to be a subset I for which the product of something in I with anything in R again lies in I. He says an ideal P is “prime” if whenever a product of two elements in R is in P, at least one of them is in P.

He defines the “spectrum” Spec(R) of a commutative monoid R to be the set of its prime ideals. He gives this the “Zariski topology”. That’s the topology where the closed sets are the whole space, or any set of prime ideals that contain a given ideal.

He then shows how to get a sheaf of commutative monoids on Spec(R). He defines a “scheme” to be a space equipped with a sheaf of commutative monoids that’s locally isomorphic to one of this sort.

Does that sound right?

I spent some time reading Durov’s paper last night, and it seems he does the same sort of thing for commutative algebraic theories. But, he has to work a bit harder. It’s all in section 6, “Localization, Spectra and Schemes”, starting on page 269.

First he tries to copy the cases we’ve just talked about (commutative rings and commutative monoids), defining a ‘prime ideal’ for a commutative algebraic theory in a way that only depends on its unary operations. He uses this to define the ‘prime spectrum’ of a commutative algebraic theory.

But then he seems to decide this is misguided. Which makes sense: there’s obviously something fishy about focusing too much attention on the unary operations in a general algebraic theory!

He then defines a ‘total spectrum’ and works with that.

You can get a vague flavor of what he’s up to in section 6.4.14, on page 301, where he says:

Whenever we want to consider spectra of generalized rings, and generalized schemes, obtained from such spectra by gluing, we have to fix a localization theory T ?T^? and a topology (weak or finite; strong topology won’t do for schemes since quasicoherence is not local for strong topology). We have just seen that these choices do not affect spectra of classical rings and classical schemes, so any choice would be compatible with the classical theory of Grothendieck schemes in this respect.

Posted by: John Baez on December 12, 2007 5:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

We’ll have to wait until that masked man returns.

While we’re waiting, I’ll blab a little about the rumor I’d thought I’d heard that the category of schemes does not admit all coequalizers. It’s just that I got the impression that that to some extent is why algebraic spaces were invented – to repair this difficulty, at least in some cases of interest to algebraic geometers. James would be able to tell us instantly, I’m sure.

Reading the wikipedia article on algebraic spaces, I gather that it’s possible to have a scheme YY and an equivalence relation

EYE \stackrel{\to}{\to} Y

where the maps involved are étale coverings but not coverings in the Zariski topology, and which doesn’t have good descent (i.e., there’s no scheme Y/EY/E which would play the role of coequalizer of the pair of maps from EE to YY above). (But by how algebraic spaces are defined, there’s always an algebraic space which fits the bill.)

Besides James, surely there are other people knowledgeable in algebraic geometry who can help out! (David Ben-Zvi? Someone from the Secret Blogging Seminar?)

Posted by: Todd Trimble on December 12, 2007 7:33 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Yet another James, James Borger, works on this kind of problem. At the Morgan-Phoa workshop he gave a talk entitled Algebraic spaces, what are they categorically? , but it was actually about Lambda rings.

Posted by: Kea on December 12, 2007 8:32 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Todd,

Yes, algebraic stacks add certain colimits - namely quotients by etale equivalence relations - to the world of schemes. But if we want a good theory of quotients or more general colimits - for example quotients by (nonfree) algebraic group actions - then we are naturally led to the world of algebraic stacks and higher stacks.

Put another way, we can replace schemes by sheaves of sets on the category of schemes, (sometimes known as “spaces” in algebraic geometry), thereby adding colimits while retaining some of the ones we already had in our
category (namely those captured by whatever topology we are using, Zariski, etale, flat etc). Algebraic spaces come from closing up schemes in this world under simple colimits.

But if we want to do more complicated operations, like quotients by groups, we know the world of sets is not a great world to work in — we are used to replacing sets
by one of the equivalent worlds: topological spaces/homotopy, simplicial sets, or higher groupoids. e.g. we can take the quotient pt/G in the world of sets, and just stay with a point, but for most applications we might want to retain information about stabilizers and end up with BG, as an object in one of the above equivalent worlds. Likewise in the world of schemes to have a reasonable theory of moduli spaces, quotients and general colimits we should replace functors of points valued in sets by those valued in one of the above equivalent worlds. This is the world of (higher) stacks, which has a good notion of all colimits (e.g. it underlies a model category). This is explained very nicely in Toen’s survey article on higher and derived stacks. (Usually algebraic geometers don’t deal with simplicial sets but only with 1-truncated spaces, or 1-groupoids, leading to usual Artin stacks, but the above is more natural and more in the spirit of this blog…)

Of course next one should be asking about a good theory of LIMITS, which is where derived stacks originate, but I’ll
refer you to Toen’s article for that…

Posted by: David Ben-Zvi on December 13, 2007 6:44 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

David,

Thanks for the confirmation and your succinct explanations.

So from what you say, higher stacks and derived stacks seem to be the right framework for a lot of basic constructions one wants in algebraic geometry (moduli spaces, good descent objects for group actions, …) – could it then be said that the language of higher/derived stacks provide the most up-to-date working foundations for algebraic geometry (supplanting schemes or algebraic spaces as good foundations, as these are not yet rich enough to carry out all the geometric operations one would like)?

Recalling John Baez’s comment here, it sounds like the world of stacks is a “nice category containing some nasty objects, [rather] than a nasty category containing only nice objects”. John’s comment was paying out more of a thread (mainly between Tom Leinster and James) on foundations for algebraic geometry, but it almost sounds like stacks already constitute a de facto technical foundation for much of modern algebraic geometry (as I think Thomas Riepe may have been suggesting as well), thus prompting my (probably naively expressed) question.

Posted by: Todd Trimble on December 14, 2007 11:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

aha! so if we could just face up to the nasty objects and not worry about the cat or the foundations???
;-D

cf manifolds with ‘bad’ group actions

Posted by: jim stasheff on December 15, 2007 1:40 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

I certainly agree that higher and derived stacks are emerging as a good new set of foundations for algebraic geometry..I don’t know an authoritative history, but certainly some of the names to mention are Toen and Vezzosi, Simpson, and Lurie - the latter’s DAG series (0-4 are available so far) appears as a kind of new EGA (combining the foundations of Grothendieck and Quillen in a beautiful package with a ribbon on top. And these foundations are already seeing lots of applications in homotopy theory, moduli spaces, geometric representation theory, etc.

About nasty objects: of course we need a notion of what’s a non-nasty object — these are the Artin (or algebraic) stacks.

Enlarging our world to include (higher and derived) stacks is in some sense orthogonal to allowing nasty objects: we could already have been considering schemes as just sheaves of sets on some topology on rings, and that’s already a world that has nice technical features and nasty objects. I think adding the stacky and derived directions is there to make things NICER – making quotients and intersections that we wanted to carry out already better, by deriving them (or doing their homotopy versions). There are really awful stacks out there, but the real point is that various things that we may have suspected were awful in the usual world of schemes now become nice because they’re treated correctly — this is for example expressed in the “hidden smoothness” philosophy of Kontsevich, Drinfeld, Deligne etc that “any” reasonable moduli space is smooth when considered in the proper (namely derived stack) world – for example this is the source of the “virtual fundamental class” so vital to Gromov-Witten theory.

Posted by: David Ben-Zvi on December 15, 2007 6:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

“I think adding the stacky and derived directions is there to make things NICER – making quotients and
intersections that we wanted to carry out already better, by deriving them (or doing their homotopy versions).”

Curious! mike Schlessinger and I invoke stacks to express what we’ve already done in homotopy theory!

Posted by: jim stasheff on December 15, 2007 8:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

A few more comments…

I think it was mentioned above that the category of schemes isn’t even closed under quotients by free group actions. I never actually read the examples, but it’s possible to write down a scheme defined over C\mathbf{C} with an action of a finite group such that the induced action on the corresponding complex manifold has no fixed points but such that the quotient complex manifold is not a scheme. In particular, the fiber product of C\mathbf{C} with itself over the quotient is not the graph of the original equivalence relation, which is bad.

This is one reason why the category of schemes is ridiculous. The category of algebraic spaces is much better. In particular, it is closed under quotienting by “etale” equivalence relations (Etale means more or less that the equivalence relation has no fixed points. Also algebraic spaces are closed under many more quotients, by a theorem of Artin’s.)

But algebraic spaces do not form a topos either! For instance the quotient of the group C\mathbf{C} (or more properly, the affine line over C\mathbf{C} with its usual group structure) by Z\mathbf{Z} in the category of algebraic space is just the point. But the corresponding equivalence relation is not the one where everything is equivalent to everything else.

The category of schemes probably doesn’t form an anything, or at least an anything which a) internally, has a reasonable definition and b) externally, captures something interesting about geometry. On the other hand, algebraic spaces probably do form a category with extra structure and properties whose definition is worth pining down. I gave a lecture on this at the Morgan—Phoa workshop in Canberra last year, and I gave them the (bad) name “half toposes”. (Kea mentioned this above, but I think she’s confusing two different talks I gave there.) The hope was to inspire some category theorists to work things out, but nothing ever went anywhere.

Regarding derived algebraic geometry, I think stacks etc are all well and good, but I think they will never supplant non-derived spaces. (By “space” I mean algebraic spaces, or even just objects of some topos.) Just like you want to be able to talk about sets as well as n-categories and modules as well as chain complexes, you’ll always want to be able to talk about usual algebraic geometry as well as derived algebraic geometry. In fact, the whole point of talking about the derived things is to understand the non-derived ones better.

Posted by: James on December 18, 2007 5:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Oops. At the end of the first paragraph above, I should have said the “fiber product of *the original scheme* with itself over the quotient” rather than “C\mathbf{C}”.

Posted by: James on December 18, 2007 7:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Would it be possible to say a few words here on what you were thinking about algebraic spaces and half-toposes, or maybe just supply a URL?

Posted by: Todd Trimble on December 20, 2007 2:15 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Unfortunately, I can’t remember things in detail, and I won’t be anywhere near my notes anytime soon, but I’ll say what I remember.

The basic idea is that you can get algebraic spaces using a purely categorical machine if you input two concepts: ring (always commutative) and nilpotent ideal. First you define formally etale maps of rings to be those that satisfy the lifting property with respect to quotients by nilpotent ideals. (I can’t remember whether you call it the left or right lifting property. It’s written out in EGA, Johnstone’s book Topos Theory, and probably many other places.) Then you define etale maps to be formally etale maps that are finitely presentable, which can be expressed using category theory: the (covariant) functor it represents commutes with colimits. This allows you to define the etale topology on the opposite of the category of rings, and hence the topos of sheaves on this site.

Then you define the category of algebraic spaces to be the closure of the representable sheaves (which are usually called “affine schemes”) under adjoining quotients by etale equivalence relations. An equivalence relation on XX is etale if its graph is etale over XX. In fact, this closure procedure stops quickly. After one step you get what most people call “algebraic spaces with affine diagonal”, and after the second step it stabilizes. (Note that I didn’t say what it means for a map of sheaves to be etale. But the definition above refers only to the presheaf the ring represents, so just use the naive generalization of this definition.)

In my talk, I asked what the best way of thinking about the categorical aspect of this is. I thought there should be some kind of category, which I called a “half topos”, and that the category of algebraic spaces should be the half topos generated by affine schemes together with the etale maps between them, or even surjections with nilpotent kernels if you want to push it back even further. I mean that every topos should be a half topos, and that the category of algebraic spaces should be the smallest (in some natural universal sense) half topos containing the opposite of the category of rings in the category of all sheaves on that category in the etale topology.

It’s probably possible to immediately give an uninteresting answer to this question, something along the lines of the construction I just gave: add all quotients of some objects by some kind of equivalence relations in some ambient topos. There are probably many ways in which this is bad, but one of them is that this definition would be of the form “A category C is a half topos if there exists a category R (rings in the example above) and a kind of morphism in R (etale) such that…”. It would be much better to define half toposes without quantifying over all categories, much like how Giraud’s theorem gives a more reasonable definition of (Grothendieck) topos than the one in terms of sheaves on a site. Probably another important property is that the topology in the example above and the type of equivalence relations we can quotient out by come from the same kind of maps (etale ones).

Then you would want analogues of lots of the basic functoriality results you have in topos theory. For instance, if you have a functor between to sites which has some basic properties (Grothendieck called these “continuous” functors, but I think they’ve been renamed by category theorists), then it prolongs to a (geometric) morphism of the toposes of sheaves. You’d want something similar for some kind of functors between half toposes.

That’s all I can think of now. I hope that was somewhat understandable. I should also add that this problem might be very easy to solve. Other than a little chit chat at dinner, I don’t think anyone thought about it at all.

Posted by: James on December 20, 2007 4:08 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

David Ben-Zvi wrote:

… we could already have been considering schemes as just sheaves of sets on some topology on rings, and that’s already a world that has nice technical features and nasty objects.

Hey! That’s what I was asking about! I was asking if there were a way to pick out schemes among the presheaves on CommRing opCommRing^{op}, perhaps by saying they’re the sheaves with respect to some topology! Todd convinced me this was a dumb idea. Are you saying it’s possible?

By the way, I’m not sure stacks alleviate my desire to have a ‘convenient category’ in which to do algebraic geometry — a cartesian closed category with all limits and colimits, for starters.

It’s an interesting and somewhat convincing thought. It certainly fits the overall philosophy of categorification. Why bother with limits and colimits when what matters are weak limits and colimits? So, why bother seeking a convenient category? — what we really need is a convenient \infty-category!

But, homotopy theorists have surely been through this before. What really matters most to them are weak limits and colimits — which they call ‘homotopy’ limits and colimits. But this doesn’t seem to stop them from wanting a convenient category of topological spaces, one with all ordinary limits and colimits.

Why? It basically just sucks having to check some conditions before taking a limit or colimit. It’s like having to ask your mommy’s permission each time you cross the street.

“Can I take a quotient now?” “No! That quotient is bad!” “Aw, mom! Come on! Pleeeease?

Also, one of the big themes in ‘brave new algebra’ is getting stuff to work ‘on the nose’, not just up to homotopy. This is why people are so happy to have symmetric monoidal categories (not just \infty-categories!) of spectra, like symmetric spectra and orthogonal spectra.

So, while I don’t actually know anything about the subject, I would think algebraic geometers should start with some category sort of like “schemes” but which is complete, cocomplete, cartesian closed… as good as possible… before going on to study higher categorical things.

I’m glad to discover that James, who actually knows something, agrees with me:

Regarding derived algebraic geometry, I think stacks etc are all well and good, but I think they will never supplant non-derived spaces. (By “space” I mean algebraic spaces, or even just objects of some topos.) Just like you want to be able to talk about sets as well as nn-categories and modules as well as chain complexes, you’ll always want to be able to talk about usual algebraic geometry as well as derived algebraic geometry.

Yeah!

In fact, the whole point of talking about the derived things is to understand the non-derived ones better.

Well, I wouldn’t say it’s “the whole point” — one can learn to love new worlds for their own sake. But, it’s part of thte point… and that’s why I’d like to see the “old world” of algebraic geometry made as pretty as possible. I’ll need to learn more about algebraic spaces… but I really want a category that has all limits and colimits. Maybe it’s just Set CommRing opSet^{CommRing^{op}}.

Posted by: John Baez on December 18, 2007 7:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

I momentarily thought I was reading a different blog with the nostalgia for sets and categories vs spaces and higher categories! et tu, James? I certainly couldn’t agree less that stacks (or other objects of derived geometry) are there to help us learn about schemes - why are schemes so special?? For two basic examples, the moduli of elliptic curves and the Schubert variety space B\G/B are both fundamental objects which are important for their own sake (yes I know the moduli of elliptic curves parametrizes schemes, but it is undoubtedly an object of independent interest, say as a Shimura “variety”). My interest is from geometric representation theory: if you want to model a given category of representations geometrically, you don’t really have a choice about whether to stick to good ol’ schemes or to pass to richer objects,
it’s imposed on you by the representation theory. Or to put it in other terms: if I give you a topos and I want to think about it as sheaves on something, you don’t really get to insist on it being a scheme, it might very well end up a stack, and if this was a topos you cared about intrinsically you have to take what you get..

Posted by: David Ben-Zvi on December 19, 2007 10:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

To put it more precisely: we can all agree probably that we care about commutative rings (or one’s favorite abstraction thereof), aka affine schemes. Now schemes are obtained by a very special kind of gluing (colimit) construction on affine schemes - ie they’re roughly simplicial affine schemes where the gluing maps are Zariski open immersions.
Why are these objects more fundamental than other classes of gluings of rings, i.e. simplicial affine schemes (aka algebraic stacks, roughly)? Certainly if we want to do geometry we want to put some restrictions on the kinds of gluing maps we allow — say etale, flat or smooth — but insisting that the Zariski opens are of a fundamentally different caliber of importance seems to me unmotivated. (well ok there’s some obvious devil-advocacy going on, but not very much)

(While we’re thinking that way we might want to add cosimplicial directions too, ie do intersections or fiber products of rings – and then we are doing full-fledged derived algebraic geometry).

Posted by: David Ben-Zvi on December 20, 2007 2:05 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

I basically said this below, but just to be clear, I think nothing (or very little) is special about schemes, and I think a lot is special about algebraic spaces, precisely because they are 0-categorical things. In the same way, I think that among n-categories the 0-categories are very special ones and that they deserve to be singled out.

Posted by: James on December 20, 2007 3:09 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Hi David. You said

I certainly couldn’t agree less that stacks (or other objects of derived geometry) are there to help us learn about schemes - why are schemes so special??

Well, schemes themselves are silly, and I’d be one of the last to defend them, but you would probably say the same things about algebraic spaces, which I think are great.

I definitely I overreached when I said that the whole point of talking about derived things is to understand the non-derived ones better, and I bet we don’t really disagree about this. It would have been better to say it’s one of the main points. If you’re really only interested in algebraic geometry for its applications to some other field, then it seems reasonable to say Why is usual algebraic geometry so special? On the other hand, if you’re really only interested in algebraic geometry itself, it seems reasonable to say the reason we care about derived algebraic geometry (as well as food, water, etc) is only for its applications to usual algebraic geometry.

But I still stand by the statement that derived algebraic geometry will never supplant usual algebraic geometry, it will only add to it. I would also say the same thing about sets and n-categories. 100 years after everything is understood about infinity-categories, people will still want to talk about 0-categories. Even the most close-minded infinity-category theorists will at a minimum need 0-categories to define infinity-categories.

But maybe I have become a bit disaffected in the past few years with derived stuff—after we derive everything in sight up to the moon, then what? The Riemann hypothesis is still open, the Weil conjectures still have a ridiculous proof, and so on. My own interests are now mostly in the then-what part. Of course, I’m still interested in the deriving, and I think it would be good for mathematics if lots of people think about how to do the deriving and I would encourage other people to hire them (but for the sake of everyone they shouldn’t forget completely about the then-what part).

I hope that makes sense.

Posted by: James on December 20, 2007 3:00 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Let me be devil’s advocate: seems to me that deriving is essentially adopting a homotopy point of view, but when most people speak in the derived language (which I do not speak fluently), I have to remind myself it’s really homotopy theory (which I do).

Posted by: jim stasheff on December 20, 2007 1:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Jim Stasheff writes:

Let me be devil’s advocate: seems to me that deriving is essentially adopting a homotopy point of view, but when most people speak in the derived language (which I do not speak fluently), I have to remind myself it’s really homotopy theory (which I do).

I think you’re being an angel’s advocate here, reminding us that seemingly different ideas are actually the same!

Let me add yet another viewpoint: “switching to the derived category” and “adopting a homotopy point of view” are different approaches to \infty-categorifying. Instead of treating our mathematical gadgets as forming a mere category, we enhance them (e.g. by introducing simplicial gadgets) so we can treat them as forming an \infty-category.

In both cases, all the higher morphisms we get are (weakly) invertible, so we only get a special sort of \infty-category: namely, an (,1)(\infty,1)-category. There are different ways to formalize the concept of (,1)(\infty,1)-category, but thanks to Lurie and others, the “quasicategory” idea seems to be catching on.

By the way — as a Christmas present, Joyal has finally sent Peter May and me an “almost final” version of his 200-page introduction to quasicategories! This will appear in the proceedings of our IMA conference on nn-categories.

I hope that when this comes out, quasicategories will become even more popular.

Posted by: John Baez on December 20, 2007 9:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

I think you’re being an angel’s advocate here, reminding us that seemingly different ideas are actually the same!

So your angelological beliefs, John, are different from Weyl’s. Where you take the angel to be a revealer of unity, Weyl takes it to be a partisan for one approach:

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.

Posted by: David Corfield on December 21, 2007 8:42 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Weyl’s view would be closer to the origin of the term. For those who don’t know, the Devil’s Advocate is a certain Vatican office charged with finding all the reasons a candidate for canonization might not be in heaven. Everyone in the Catholic Church would bet on Mother Teresa being in heaven, and thus being a saint, but the Church must be absolutely certain (for the Church’s definition of “absolutely certain”) that she is before canonizing her and declaring its certitude. To this end, someone’s job is to pick all the nits and find out any possibility that she, in fact, is not a saint. If all those arguments can be refuted, then the Church considers the matter settled.

In mathematics, we are (or at least should be) all playing Devil’s Advocates all the time. We all know what “should” happen, but we still have to counter all possible misgivings before we’re absolutely certain.

Posted by: John Armstrong on December 21, 2007 9:01 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

How far we’ve come though at times I think we are disunited purely linguistically: viz. weak cokernel vs. homotopy quotient.

Posted by: jim stasheff on December 21, 2007 1:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

On another note, that’s an interesting question about whether topologists want categories or infinity-categories of spectra eg for brave new ring theory, or a world with true (co)limits and on-the-nose operations rather than their homotopy versions.

My impression was that quasicategories (or (∞,1)-categories) were exactly right, the Goldie Locks “just right” middle option. There are two kinds of “honest categories” of spectra: the homotopy category, and
one of several model categories. But I got the impression both were morally wrong in some sense, the first for carrying too little information, the latter for carrying too much. It’s clear modern homotopy theory needs more than the boring old homotopy category, but having a model
category is like carrying around resolutions of all of your objects, which is cumbersome, noncanonical, hard to prove basic things with etc.

So e.g. for brave new algebra (doing algebraic geometry over categories such as spectra) quasicategories have just the right level of information, with the slight disadvantage (at least to people who don’t like higher categories, so presumably aren’t reading this blog!) that they’re not usual categories. E.g. Lurie’s DAG III spells out the theory of “commutative rings” in this world in a clean conceptual way, compatible with the references (EKMM and Symmetric Spectra) John mentioned on the model level but easier to work with. And there is an increasing body of results which can’t (or shouldn’t) be done on the category level, but really lives in an infinity-categorical world. There are a lot of really simple algebraic ideas that just don’t work on the model level (or not without enormous pain) and are clear and conceptual on the quasi level – this I take to be one of the messages of derived algebraic geometry, and one that I would imagine would resonate well with the denizens of this cafe.

Posted by: David Ben-Zvi on December 19, 2007 10:49 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Without attempting to add anything substantive to this very interesting thread, let me just supply some links to references mentioned by David B-Z (at least, I think these are close to what he had in mind):

Posted by: Todd Trimble on December 20, 2007 2:45 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

John wrote:

If schemes don’t even admit all coequalizers, no wonder Grothendieck became dissatisfied with them and wanted to switch to general presheaves over CommRing.

It shouldn’t be too surprising that schemes don’t admit coequalizers, since they are analogous to manifolds. And this can certainly be awkward. One approach for dealing with this is to study (possibly simplicial) sheaves or pre-sheaves on a category of schemes, as a way to introduce limits and colimits. This is what is done by Jardine, Morel, Voevodsky and others in order to study the homotopy theory of schemes. It is nicely motivated by a paper of Dan Dugger called Sheaves and Homotopy Theory and in the published papers by Dugger cited in the abstract. One motivation is that if you do this construction starting with the category of manifolds, you end up with a homotopy theory equivalent to the homotopy theory of spaces.

About the other question of determining which presheaves on CommRing opCommRing^{op} are schemes, I highly recommend the already mentioned book by Gabriel and Demazure which takes this approach.

Posted by: Dan Christensen on December 22, 2007 3:32 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Several months ago Dan Christensen wrote the following:

One approach for dealing with this is to study (possibly simplicial) sheaves or pre-sheaves on a category of schemes, as a way to introduce limits and colimits. This is what is done by Jardine, Morel, Voevodsky and others in order to study the homotopy theory of schemes. It is nicely motivated by a paper of Dan Dugger called Sheaves and Homotopy Theory and in the published papers by Dugger cited in the abstract.

I must have missed that comment back then. Did have a look at Dugger’s notes here now and see that they provide a very nice discussion, so nice that I now included this as the first reference in the entry on [[model structure on simplicial presheaves]].

Dan next said:

One motivation is that if you do this construction starting with the category of manifolds, you end up with a homotopy theory equivalent to the homotopy theory of spaces.

Here I would just like to highlight two things:

a) the “this construction” referred to here is not just the localization of simplicial presheaves at their natural weak equivalences that models \infty-stacks, but this followed by further localization at maps that contract a cylinder to its base (def. 3.4.1, p. 29).

It is this further loclizatoin step that then makes \infty-stacks on manifolds homotopy invariant and then equivalent to just topological spaces. Otherwise \infty-stacks on manifolds are much richer than topological spaces: they are generalized smooth spaces.

b) often homotopy invariance is thought of as a defining property of cohomology theory, as in Dugger’s introduction to these nie notes. But then there are many things traditionally called cohomology theories which are not homotopy invariant. Like, as in a), cohomology with coefficients in (\infty-)sheaves such as notably things like Deligne cohomology and other “differential cohomoloy” theories.

So it’s actually good and useful that \infty-stacks on manifolds are not equivalent to just topological spaces, that’s not something one needs to fight.

Posted by: Urs Schreiber on June 18, 2009 2:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Here are some intro texts to schemes I guess to be useful:

Levine’s “Algebraic geometry background”.

Mumford’s famous Red Book.

The first chapter of Demazure, Gabriel “Groupes Algébriques” also contains a very nice and condensed introduction into schemes.

Best, Thomas

Posted by: Thomas Riepe on December 10, 2007 11:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Could Lambda-rings help?

Posted by: Kea on December 11, 2007 11:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Kea wrote:

Could Lambda-rings help?

I like λ\lambda-rings, and I keep meaning to talk about them in This Week’s Finds. But, I don’t know what a ‘module’ of a λ\lambda-ring should be.

However, it’s easy to believe that just as a ring RR can be thought of as an algebraic theory whose nn-ary operations are ‘RR-linear combinations’

(x 1,,x n)r 1x 1+r nx n(x_1, \dots, x_n) \mapsto r_1 x_1 + \cdots r_n x_n

so that an RR-module is just a model of this algebraic theory, so too the modules of a λ\lambda-ring are models of some algebraic theory. This would mean that λ\lambda-rings are generalized rings in Durov’s sense.

That would be nice.

Sadly, googling under ‘lambda-scheme’ turns up stuff that seems utterly irrelevant.

Posted by: John Baez on December 15, 2007 2:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

It seems hard even to get a definition of λ\lambda-rings. There is one on p. 26 of Fauser and Jarvis’s The Dirichlet Hopf algebra of arithmetics.

Can anyone give us a hint as to why they’re interesting?

From the paper’s abstract:

The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

Posted by: David Corfield on December 15, 2007 3:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

There’s too much to say about λ\lambda-rings to know where to start (or more precisely, where to end). But how about this:

In many situations, we can take direct sums of representations of some algebraic gadget. So, decategorifying, the set of isomorphism classes of representations becomes a commutative monoid.

But nobody likes commutative monoids. So, we throw in formal negatives and get an abelian group — the so-called Grothendieck group.

In many situations, we can also take tensor products of representations. Then our Grothendieck group becomes something better than an abelian group. It becomes a ring: the representation ring.

But, we’re not done! In many situations we can also take exterior and symmetric powers of representations. Indeed, we can often apply any Young diagram to a representation and get a new representation!

Then our representation ring becomes something better than a ring. It becomes a λ\lambda-ring!

More generally, the Grothendieck group of a monoidal abelian category is always a ring… but if we start with a symmetric monoidal abelian category, we get a λ\lambda-ring.

So, λ\lambda-rings are all about getting the most for your money when you decategorify a symmetric monoidal abelian category — for example the category of representations of a group, or the category of vector bundles on a space.

Allen Knutson’s dad wrote the book on λ\lambda-rings:

Donald Knutson, λ\lambda-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, Vol. 308, Springer, Berlin, 1973.

Posted by: John Baez on December 15, 2007 11:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Λ\Lambda-rings! My favorite subject.

I think Kea is referring to some talks I’ve given on Λ\Lambda-algebraic geometry. By this I mean the solution to the following analogy: commutative rings are to algebraic geometry as Λ\Lambda-rings are to what? In the one I gave at the Morgan—Phoa workshop, I also hinted at the relation to the subject of TWF 259. I have a few papers on this that are almost done. There are some ways in which this is more appealing than the rather formal approaches using abstract monoids and algebraic theories, for instance there are close relations with several existing, meaty subfields of arithmetic algebraic geometry. On the other hand, there are ways in which it is less appealing, for instance the relation with qq-deformation is not clear.

David Corfield wrote:

It seems hard even to get a definition of λ-rings.

I can’t tell whether you mean the definition is hard to find, hard to state, or hard to comprehend. I wrote a paper with Ben Wieland whose goal was to get ahold of an abstract framework for understanding Λ\Lambda-rings and similar constructions. (Part of this is a rediscovery of work of Tall—Wraith and Bergmann—Hausknecht. See the references in the paper.) You might try looking there, but I’ve been told that it’s hard to read. In the first of the papers I mentioned up top, I give a (hopefully) more direct approach to defining Λ\Lambda-rings and some analogous objects. It should be done very soon. In fact it was done, until a colleague had the gall to point out a few errors.

Can anyone give us a hint as to why they’re interesting?

John gave the orthodox point of view on why they’re interesting. To get a taste of their arithmetic nature (and hence a non-orthodox reason as to why they’re interesting), you could see the paper I wrote with Bart de Smit.

Posted by: James on December 18, 2007 7:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

James has out today Λ\Lambda-rings and the field with one element:

The theory of Λ\Lambda-rings, in the sense of Grothendieck’s Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring \mathbb{Z} of integers to produce Λ\Lambda-algebraic geometry. We show that Λ\Lambda-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than \mathbb{Z} and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

Links to papers mentioned in the comment above seem to be dead, but you can visit his homepage to read them.

Posted by: David Corfield on June 18, 2009 10:18 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Thanks for the mention, David! I guess I missed that last month.

Posted by: James on July 23, 2009 2:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Cool! More stuff to read…

Posted by: John Baez on June 18, 2009 10:58 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

John wrote, “I like λ-rings, and I keep meaning to talk about them in This Week’s Finds. But, I don’t know what a ‘module’ of a λ-ring should be.”

There’s a good reason for that: Λ\Lambda-rings are not monoids in some monoidal category of Λ\Lambda-modules. They are fundamentally non-linear objects! If they weren’t they’d be pretty boring. They are linear (and boring) if you’re looking at Λ\Lambda-rings that contain the rationals. Then a Λ\Lambda-ring is just a ring together with an action of the multiplicative monoid Z +Z_{+} of positive integers (this is freely generated by the prime numbers aka the Adams operations), in which case a Λ\Lambda-module could reasonably be defined to be a QQ-vector space together with an action of Z +Z_{+}.

But if your rings aren’t assumed to contain the rationals, then this won’t work. This is ‘because’ the power sum symmetric functions generate the ring of all symmetric functions, but only if you allow yourself denominators. In the language of “Plethystic Algebra”, you could say that the additive operations in the plethory Λ\Lambda of symmetric functions (i.e. linear combinations of power sums) do not generate all of Λ\Lambda.

Posted by: James on July 23, 2009 3:03 PM | Permalink | Reply to this

generic points

Kevin Buzzard’s comment (an addendum to TWF, not on this page) about generic points singles out Weil (and Borel) as predecessors to Grothendieck who didn’t have precise definitions of generic points. But Weil’s Foundations did define generic points. These were quite different from the generic point of Grothendieck, but they gave precise notions of generic phenomena.

The idea, famously promoted by Grothendieck for other reasons, was that if you worked over, say, Q-bar, you should not just look at Q-bar points, but points with values in transcendental extensions. The more transcendentals you need, the more generic. If you work over C, with its infinite transcendence degree, there’s no need to add more transcendentals.

I think I saw the same idea in Zariski, but I don’t know if he was precise nor if there is yet an earlier source.

Posted by: Douglas Knight on December 13, 2007 5:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

T.R. mentioned it in the TWF257 discussion, but let’s include it in this one too. Shai Haran has an article Non-additive geometry:

We develop a language that makes the analogy between geometry and arithmetic more transparent. In this language there exists a base field 𝔽\mathbb{F}, ‘the field with one element’; there is a fully faithful functor from commutative rings to 𝔽\mathbb{F}-rings; there is the notion of the 𝔽\mathbb{F}-ring of integers of a real or complex prime of a number field KK analogous to the pp-adic integers, and there is a compactification of SpecO K\operatorname{Spec}O_K; there is a notion of tensor product of 𝔽\mathbb{F}-rings giving the product of 𝔽\mathbb{F}-schemes; in particular there is the arithmetical surface SpecO K×SpecO K\operatorname{Spec} O_K\times\operatorname{Spec} O_K, the product taken over 𝔽\mathbb{F}.

Posted by: David Corfield on December 17, 2007 9:34 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Thanks for reminding me, David! I’ll add Haran’s paper as an official Addendum to week259, since I want that to be my main location for references to F 1F_1, at least until I get my math wiki going.

Posted by: John Baez on December 17, 2007 8:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

The abstract of a course by Haran suggests relations between the field with 1 element, the diagonal in Spec()× Spec(F 1)Spec()Spec(\mathbb{Z}) \times_{Spec(F_1)} Spec(\mathbb{Z}), and the Riemann hypothesis — exactly along the lines Minhyong was asking for!

I don’t understand this stuff at all, but it sounds cool, so I should give it a try. Any heuristic help would be greatly appreciated.

Posted by: John Baez on December 17, 2007 8:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

The 𝔽 un\mathbb{F}_{un} continues with Mapping 𝔽 1\mathbb{F}_1-land: An overview of geometries over the field with one element (including the map on p.19)

ABSTRACT. This paper gives an overview of the various approaches towards 𝔽 1\mathbb{F}_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar’s 𝔽 1\mathbb{F}_1-schemes, Toën and Vaquié’s 𝔽 1\mathbb{F}_1-schemes, Haran’s 𝔽\mathbb{F}-schemes, Durov’s generalized schemes, Soulé’s varieties over 𝔽 1\mathbb{F}_1 as well as his and Connes-Consani’s variations of this theory, Connes and Consani’s 𝔽 1\mathbb{F}_1-schemes, the author’s torified varieties and Borger’s Λ\Lambda- schemes. In a second part, we will tie up these different theories by describing functors between the different 𝔽 1\mathbb{F}_1-geometries, which partly rely on the work of others, partly describe work in progress and partly gain new insights in the field. This leads to a commutative diagram of 𝔽 1\mathbb{F}_1-geometries and functors between them that connects all the reviewed theories. We conclude the paper by reviewing the second author’s constructions that lead to realization of Tits’ idea about Chevalley groups over 𝔽 1\mathbb{F}_1.

Also a survey of Durov’s thesis.

Posted by: David Corfield on September 2, 2009 11:33 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

THERE IS A SCHOOL/CONFERENCE IN LEIDEN ON FROBENIUS LIFTS AND LAMBDA RINGS 5-10. October 2009 featuring

* Pierre Cartier: Lambda-rings and Witt vectors

* Lars Hesselholt: The de Rham-Witt complex

* Alexandru Buium: Arithmetic differential equations

* James Borger: Lambda-algebraic geometry

conference site

participants

Posted by: Zoran Skoda on September 22, 2009 9:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Another paper for the list – (non)commutative f-un geometryby Lieven Le Bruyn:

Stressing the role of dual coalgebras, we modify the definition of affine schemes over the ‘field with one element’. This clarifies the appearance of Habiro-type rings in the commutative case, and, allows a natural noncommutative generalization, the study of representations of discrete groups and their profinite completions being our main motivation.

Posted by: David Corfield on September 23, 2009 8:43 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

Yet more:

We ought to form a list of references, including those from here.

Posted by: David Corfield on November 19, 2009 5:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 259)

We ought to form a list

Yes, here.

Posted by: Urs Schreiber on November 19, 2009 11:38 PM | Permalink | Reply to this

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