This Week's Finds in Mathematical Physics (Week 284)

November 25, 2009

This Week’s Finds in Mathematical Physics (Week 284)

Posted by John Baez

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In week284 of This Week’s Finds, see Greg Egan’s new proofs of a theorem that goes back at least to 300 BC. One proof uses the fact that these triangles are congruent:

Another uses golden triangles. Then, hear about one day of the special session on homotopy theory and higher algebraic structures at UC Riverside. Categorified quantum groups, the Halperin-Carlsson conjecture, real Johnson-Wilson theories, Picard 2-stacks, quasicategories, motivic cohomology theory, and toric varieties — we got bombarded by all these concepts, and now you will too!

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I’ve got a bunch of questions. Can you help me out with them?

Aaron Lauda wrote: “It turns out, at least in the simply-laced case, that our algebras are also isomorphic to the Ext algebras between simple perverse sheaves on the Lusztig quiver variety. Lusztig’s bilinear form can be seen as taking the graded dimension of this Ext algebra, so it is natural that there is a relationship between the two constructions.” Can someone say more about what’s going on here? Please don’t assume I understand what Aaron told me!
How does the representation Licata describes, involving the cohomology of the co tangent bundle of the Grassmannians $Gr(n,k)$ for $k$ between $0$ and $n$ , fit into a more general story? I think the disjoint union of these Grassmannians should be thought of as the space of 1-stage ‘Springer flags’ in $n$ dimensions - where an $m$ -stage Springer flag is a chain of $m$ subspaces of $\mathbb{C}^n$ . I vaguely recall that it’s interesting to generalize by letting m be arbitrary. And I think that an even more general story - where we pass from $sl(2)$ to $sl(N)$ - involves Springer flags in the category of quiver representations. Is this right? What’s the big picture?
Is my account of Johnson–Wilson theories accurate? What are the most important things that I left out?
What’s ‘motivic’ about Voevodsky’s motivic cohomology? Does he propose a definition of motives? How is it related to Grothendieck’s conception of motives? How, from this viewpoint, can we see that motivic cohomology should be bigraded?
What other things should I have said, but didn’t?

Posted at November 25, 2009 1:27 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 284)

I seem to have broken the UCR website trying to download videos of the talks.

Posted by: Toby Bartels on November 25, 2009 2:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

Are you joking, or just overestimating your powers? It still works fine for me.

(I’ll admit I’ve never gotten the video of Licata’s talk to work on the UCR website. The supposedly identical file on my laptop works fine, and all the other talks seem to work fine on the UCR website.)

Posted by: John Baez on November 25, 2009 3:46 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

Everything works fine for me except the video of Licata’s talk. Strange…maybe it got corrupted during the upload?

Posted by: Chris Rogers on November 25, 2009 4:11 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

Tony’s talk seems to work from over here on my side.

Posted by: Alex Hoffnung on November 25, 2009 12:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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I considered the possibility that Licata’s talk had gotten corrupted during the uploading. So I uploaded it a second time, and the effect didn’t go away. Yet the file works on my laptop before uploading. I don’t know what’s going on. I’m glad Alex can see it. It’s a nice talk.

Posted by: John Baez on November 25, 2009 5:08 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

I can see Licata's talk now.

Posted by: Toby Bartels on November 25, 2009 7:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

Are you joking, or just overestimating your powers?

All I know is that it stopped serving me anything for a while, just hanging indefinitely. It's fine now.

By the way, your link to arXiv:0809.1760 is broken; you're missing a slash.

Posted by: Toby Bartels on November 25, 2009 5:38 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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Thanks for catching that. Fixed!

That could have been your thesis. But I’m glad that you did yours, and Dupont did that.

Posted by: John Baez on November 25, 2009 6:17 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

Lusztig says the linear group is a quantum version of a symmetric group is a quantum version of the symmetric group…

Eight word repeated with ‘the’ instead of ‘a’.

Posted by: David Corfield on November 25, 2009 9:09 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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Thanks! Curiously, I was thinking of you when I added this quote, taken from the MIT math department alumni magazine:

There are two fundamental and completely different examples in group theory: the “symmetric group” of permutations of n objects, and the “linear group” of n by n matrices over a field. Lusztig says the linear group is a quantum version of the symmetric group, with the value of Planck’s constant telling you which field you’re looking at. He has made that idea precise in a thousand beautiful ways for the past 30 years. - David Vogan

It’s a nice way to summarize what’s cool about Lusztig’s work — he’s just won another award. And it’s a great summary of this $q$ -deformation business.

Posted by: John Baez on November 25, 2009 5:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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The first sentence of that quotation puzzled me. The symmetric groups are the automorphism groups of finite sets. The general linear groups are the automorphism groups of finite-dimensional vector spaces. ‘Completely different’?? I’m sure David Vogan knows infinitely more about groups than me, but in my mental landscape they’re certainly not ‘completely different’ — rather, they’re closely analogous. And that’s before taking into account any of the quantum/field-with-one-element story.

Posted by: Tom Leinster on November 25, 2009 6:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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Tom wrote:

I’m sure David Vogan knows infinitely more about groups than me, but in my mental landscape they’re certainly not ‘completely different’ — rather, they’re closely analogous.

They’re both automorphism groups of something we know and love: an $n$ -element set in one case, an $n$ -dimensional vector space in the other. But as far as their representation theory goes — and I’m sure that’s on Vogan’s mind — they do seem ‘completely different’. Or maybe ‘dual’ would be a better word!

To understand the representation theory of $GL(n)$ , we definitely need to understand the representation theory of $S_m$ — but not particularly for $m = n$ . Why? Because representations of the permutation groups act on the category of representations of any group, via ‘Schur functors’. In the case of the group $GL(n,\mathbb{C})$ or especially $SL(n,\mathbb{C})$ , this allows for a complete analysis of the representations. This is called ‘Schur–Weyl duality’. Classic stuff.

But the stuff about $q$ -deformation and ‘the field with one element’ paints a shocking new picture, in which $GL(n,F_q)$ seems like a $q$ -deformed version of $S_n$ .

Posted by: John Baez on November 25, 2009 7:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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But the stuff about $q$ -deformation and ‘the field with one element’ paints a shocking new picture, in which $GL(n,F_q)$ seems like a $q$ -deformed version of $S_n$ .

In parts this is a bit by the very “definition” of $F_{un}$ , though, isn’t it? This is really what the business about the “field with one element” seems to boil down to: that one de-linearizes linear theories.

For instance a variety over the field with one element is – in some of the approaches, like that by Connes, by its very definition – something that does not have a function ring but just a function monoid.

So to some extent the $F_{un}$ -theorists take $Ab$ -enriched category theory and try out which of their constructions and results still make sense in a plain $Set$ -enriched context.

From this perspective, we indeed pass from $GL(n)$ as the automorphism group of the object in isomorphism class $[n]$ in the $Ab$ -category $Vect$ to $S_n$ as the automorphism group of the object in isomorphism class $[n]$ in the $Set$ -category $Set$ quite unspectacularly and quite as Tom indicates.

I know that this is making it sound less mysterious than it is. But on the other hand I feel much talk about $F_{un}$ is more mysterious than it needs to be.

Posted by: Urs Schreiber on November 25, 2009 7:55 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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Urs wrote:

I know that this is making it sound less mysterious than it is. But on the other hand I feel much talk about $F_{un}$ is more mysterious than it needs to be.

I had to run when writing my reply to Tom so I didn’t include the punchline, namely:

Soaring very high like the category theorist he is, Tom sees the group of automorphisms of the $n$ -element set as ‘closely analogous’ to the group of automorphisms of an $n$ -dimensional vector space. Down in the trenches, people battling with the representation theory of these groups have traditionally seen them as ‘completely different’. But Lusztig’s work is proving that Tom’s attitude is more correct than people had thought.

I agree that some talk about $F_{un}$ makes it more mysterious than it needs to be. But let’s face it: if all our dreams about the field with one element come true, we’ll have a proof of the Riemann Hypothesis! And this is not likely to arise simply from straightforward — a skeptic might say ‘trivial’ — reflections on the analogy between $Ab$ -enriched categories to $Set$ -enriched categories.

On the other hand, these ‘trivial’ reflections could well be the seed of some highly nontrivial work! And they already are. For example, who’d have thought that quantum groups were deeply related to ordinary groups over finite fields when $q$ is a prime power? In terms of mathematical physics, this is saying that special stuff happens when the exponential of Planck’s constant is a power of a prime number!

So, I think there’s a lot of very mysterious stuff going on here, but also a lot of very unmysterious stuff. And that’s what makes this subject so tantalizing: the hope that someday the mysterious will become unmysterious.

Posted by: John Baez on November 25, 2009 8:53 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

I’ve listed a few questions about the talks — you can see them here. I hope people will try answering them! Any help would be appreciated!

Posted by: John Baez on November 25, 2009 9:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

One could view the existence of realization functors as justifying the motivicy of Voevodsky’s construction.

Posted by: Thomas on December 2, 2009 1:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

On the motivic cohomology questions:

1. Voevodsky constructs a triangulated category of motives DM such that motivic cohomology can be recovered by morphisms in that category. The basic morphisms come from correspondences (ie, multi-valued maps), just like for Grothendieck motives. If “all conjectures” (Grothendieck standard conjectures, Beilinson-Soule vanishing, etc.) hold, then this category is presumably equivalent to the derived category of the category of mixed motives.

2. Motivic cohomology was previously defined as a universal arithmetic cohomology for varieties, and V.’s definition is isomorphic to the previous ones (higher Chow groups, or the one using K-theory with gamma filtration).

This is the cohomology coming up in the Beilinson et al. conjectures.

3. The relationship between DM and the “motivic stable homotopy category” is pretty much analogous to that between the derived category of abelian groups and the ordinary stable homotopy category.

4. The bi-grading comes from the “Tate twists”. To give a feeling for this, if the base field in the reals, then there’s a realization functor to Z/2 - equivariant homotopy, and the bigrading corresponds (with a shift) to the RO(2)-grading you get there.

Posted by: Christian Haesemeyer on November 25, 2009 9:45 PM | Permalink | Reply to this

motivic cohomology

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By the way, we started collecting some – currently quite rough – information along these lines at motivic cohomology and category of motives.

If nothing else, these entries link various keywords to page and verse in Voevodsky’s lectures where the details are given.

All expert help in improving these two entries is highly appreciated.

Posted by: Urs Schreiber on November 26, 2009 12:50 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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With regard to your first question: Aaron was actually explaining how Christopher Walker’s work from your TWF is actually a decategorification of his (well, more accurtely of Lusztig).

The point is that one can think of all the groupoids Chris describes are really the $\mathbb{F}_q$ points of a groupoid scheme (well, I think you might need to do a groupoid equivalence first). When you have a groupoid scheme, it has an instant categorification, given by looking at the category of perverse sheaves on the simplicial scheme given by successive fiber products of the morphisms over the objects.

What Aaron was explaining is that the KLR-algebras (or if you prefer to leave names out of it, the “quiver Hecke algebras”) are a combinatorial way of understanding that categorification, which is how Lusztig discovered the canonical basis.

In fact, you should think of q not as a formal parameter, but as the motivic integral of the affine line. Then the fact that you can think of it as the size of the field when you specialize to a finite field is no surprise at all.

Posted by: Ben Webster on December 1, 2009 4:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

By the way, I’ve expanded this to an n-Lab entry.

Posted by: Ben Webster on December 1, 2009 11:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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Thanks for your comments, and even more for your $n$ Lab entry. It’ll take me a while to understand this stuff, but I’m getting there.

Posted by: John Baez on December 2, 2009 12:08 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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With regard to the second, it’s a very similar story. You use Nakajima quiver varieties, and this gets you the picture for all simply-laced groups. I believe Cautis, Kamnitzer and Licata have a upcoming paper about this.

Your comment about Springer fibers is somewhat connected to this story, since the quiver varieties for type A are all homotopy equivalent to Spaltenstein varieties (the generalization of Springer fibers where you take the moment map for $T^*G/P$ and look at its fibers), and the Spaltensteins control a lot of their structure.

Posted by: Ben Webster on December 1, 2009 5:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 284)

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The Nakajima quiver varieties are also probably the answer to Christopher’s question at the end of his talk about how to get the whole quantum group. I know for a fact that you were in the audience when I conjectured how to do this by quantizing affine quiver varieties my talk at the Joint Meetings, but I think for groupoidification purposes you just have look at these affine varieties as groupoids. (or alternately, just the $\mathbb{F}_q$ -points of a Nakajima quiver variety).

In fact, this probably hidden somewhere in the paper of Zheng on categorifying quantum groups using quiver varieties.

Posted by: Ben Webster on December 1, 2009 6:06 PM | Permalink | Reply to this

cohomology

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A while ago, at the end of the above entry John asked:

How […] can we see that motivic cohomology should be bigraded?

Here is the answer:

Motivic cohomology – which is simply the cohomology in the $(\infty,1)$ -topos of $\infty$ -stacks on the Nisnevich site is bigraded (or rather: may naturally be taken to be bigraded) because it is the cohomology theory of a lined Grothendieck $(\infty,1)$ -topos, one that is equipped with an object $I$ that plays the role of the standard line:

Apart from the categorical 1-sphere $S^1 = \Delta^1/\partial \Delta^1$ that is present in any $(\infty,1)$ -topos $\mathbf{H}$ and defines integer grading on cohomology by

$H^{-p}(X,A) := \pi_0 \mathbf{H}(X, \Omega^p A)$

in the presence of a line object $I$ – which canonically carries the structure of an interval object – there is then a scond kind of 1-sphere, the geometric 1-sphere

$S^1_I = \Delta^1_I / \partial \Delta^1_I \,.$

Accordingly we may shift degrees with respect to both kinds of spheres and arrive at the bigrading

$H^{-p,-q}(X,A) := \pi_0 \mathbf{H}(X, \Omega^p \Omega_I^q A) \,.$

In motivic cohomology, the line object is the $\mathbb{A}^1$ from $\mathbb{A}^1$ -homotopy-theory fame. The corresponding geometric sphere is called the Tate sphere here and looping with both categorical and geometric loops is known as the Tate twist.

But all these words hide a bit that what is going on is not specific to motivic cohomology at all. Every cohomology theory in an $(\infty,1)$ -topos with a line object is naturally bigraded this way.

For instance the cohomology of $\infty$ -Lie groupoids is. And people consider it whenever they compute the classes of gerbes and higher principal bundles on an ordinary smooth $n$ -sphere. But it seems so far nobody bothered to arrange this information explicitly into a 2-dimensionall table. But one could.

A bit more about the different notions of gradings in cohomology is now at

$n$ Lab: Cohomology – Gradings

Posted by: Urs Schreiber on January 28, 2010 11:57 AM | Permalink | Reply to this

Re: cohomology

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Thanks, that’s all very useful. I like the idea of a ‘lined’ $(\infty,1)$ -topos. I especially like the idea of treating Lie $\infty$ -groupoids in this bigraded fashion, because I’ve been thinking about this idea — less formally — for quite a while.

When I first started thinking about topological 2-groups, James Dolan was intrigued and seemingly a bit annoyed by the presence of a ‘topological dimension’ as well as ‘categorical dimension’, and urged me to think of them as examples of ‘double $\infty$ -categories’, or something like that. In short, some sort of bigraded structure.

In the paper you and I wrote, From loop groups to 2-groups, we figured out a way to suppress this bigraded structure by taking a topological 2-group $G$ and turning it into a topological group, namely by nerve of the underlying topological groupoid of $G$ , geometrically realizing that, and noticing the resulting topological space had a group structure.

In the paper I later wrote with Danny Stevenson, The classifying space of a topological 2-group, we studied this in more detail. All the hard work of the paper involved three different ways of turning a topological 2-group $G$ into a topological space $B G$ , the classifying space for $G$ -2-bundles. One involves taking a topological 2-group, converting it into a topological group as I described above, then taking the classifying space of that. But all three are very important. They give homotopy-equivalent spaces. However, it seems that nobody has ever written up a complete proof!

Around this time I asked on the topology mailing list if anyone knew a model category structure on simplicial spaces such that geometric realization gives a Quillen equivalence to the usual model category of topological spaces. This could be the slick approach to the problem of crushing a bigraded structure down to a graded structure. I got some replies from model category experts who said ‘sure, we could do that’ — but it wasn’t sitting there in the literature already.

Danny later did a lot more work on this sort of thing, and his paper on this issue will appear sometime this year, I hope.

Anyway, I think it would be very good to explicitly study Lie $\infty$ -groups in an explicitly bigraded manner.

As for motivic cohomology, I’m familiar with how $\mathbb{A}^1$ -homotopy theory makes use of the line object $\mathbb{A}^1$ to define bigraded cohomology groups. But my question was about how the bigrading arises naturally from Grothendieck’s original hopes and dreams about ‘motives’ — namely, motives as category formed by ‘splitting idempotents’ in the category with varieties as objects and correspondences as morphisms. Voevodsky’s formulation of motivic cohomology may fulfill Grothendieck’s hopes and dreams — but I don’t understand why it does, because the original hopes and dreams didn’t involve using $\mathbb{A}^1$ to define homotopies, as far as I know.

My question:

What’s ‘motivic’ about Voevodsky’s motivic cohomology? Does he propose a definition of motives? How is it related to Grothendieck’s conception of motives? How, from this viewpoint, can we see that motivic cohomology should be bigraded?

So in other words, I’m looking for some lessons in algebraic geometry, which will bridge the gap between Grothendieck’s questions and Voevodsky’s answers. I’m actually interested in algebraic geometry, not just algebraic geometry as a launching pad for investigations of $(\infty,1)$ -categories.

What you are hinting is that somehow there must be a line object hiding in Grothendieck’s hopes and dreams. And that might be a good way to make my question more precise. Why must there be this line object?

If I understood ‘Tate twists’, maybe the answer would be obvious. But I understanding nothing about ‘Tate twists’.

Posted by: John Baez on January 28, 2010 5:53 PM | Permalink | Reply to this

Re: cohomology

the presence of a ‘topological dimension’ as well as ‘categorical dimension’,

This is something I had to combat in my thesis, when defining homotopy groups of topological groupoids in such a way that they are isomorphic to the homotopy groups of the geometric realisation of the nerve. (The full result is only partially proved, but the special case of a delooping of a topological group is complete, but only in my thesis for homotopy groups up to dimension 2)

One idea which did occur to me, but I did not follow up is that there could be a sub-?-category of line objects in the (oo,1)-category at hand (for me this was the 2-category of topological groupoids and ordinary functors), not all equivalent, but each somehow a (sequential) limit of objects (weakly) equivalent to a line object. I’m not sure I can make this precise, or if it even can be made precise, but all this talk about line objects and homotopy brought it back to mind.

Posted by: David Roberts on January 29, 2010 1:58 AM | Permalink | Reply to this

Re: cohomology

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I won’t have time to say much for a few days, but here are some words of warning:

Most importantly, motives are a cohomology theory, but this should not be confused with motivic cohomology! At one time, it might have been reasonable to call the functor which associates a variety to its motive the ‘motivic cohomology’ functor (being a cohomology functor which outputs a motive), but at some point someone (Beilinson?) started calling something else motivic cohomology, and that’s what it means now.

The relation is that the category of motives contains a certain object called the Tate motive $Q(1)$ , which can be thought of as the dual of $H^2$ of the Riemann sphere. It is one-dimensional but it is not isomorphic to $H^0$ of the point. For instance, they carry different Hodge structures. The $n$ -th tensor power of $Q(1)$ is denoted $Q(n)$ . From what I remember, the various motivic cohomology groups of $X$ are supposed to be the various Ext groups in the category of motives of the various powers of that Tate motive viewed as some sort of sheaves of motives over $X$ . I can’t remember now the formula relating the numbering of the movitic cohomology groups to the number of the ext groups and the powers of the Tate motives.

In particular, motivic cohomology is a bigraded theory taking values in plain old abelian groups, whereas the universal Weil cohomology should be a theory takes values in the category of motives, which is much richer. For instance it has projections to the category of Hodge structure, Galois representations, and crystals, while there is no interesting way to get a Galois representation from a graded abelian group.

Posted by: James on January 29, 2010 10:10 AM | Permalink | Reply to this

Re: cohomology

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In particular, motivic cohomology is a bigraded theory taking values in plain old abelian groups, whereas the universal Weil cohomology should be a theory takes values in the category of motives, which is much richer. For instance it has projections to the category of Hodge structure, Galois representations, and crystals, while there is no interesting way to get a Galois representation from a graded abelian group.

I need to better learn this, but it makes me wonder:

is it clear that the non-abelian information you are alluding to is not contained in $Sh_{(\infty,1)}(Nis)$ ? Because a priori this $\infty$ -topos knows more than just abelian cohomology (bigraded or not): for non-stable coefficient objects it knows nonabelian cohomology.

I suppose there are good discussions in the literature on the relation between “universal Weil cohomology” and “motivic cohomology”. If you, or somebody, could point me to something concretely, I’d be very grateful.

Posted by: Urs Schreiber on January 29, 2010 2:27 PM | Permalink | Reply to this

Re: cohomology

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I don’t know what $\mathrm{Sh}_{(\infty,1)}(\mathrm{Nis})$ means exactly, but I imagine it’s some infinity-category version of the category of sheaves of sets on the Nisnevich topology. If so, then I should be able to answer your question by giving two varieties with equivalent Nisnevich topologies but different motives. I would imagine that two elliptic curves over the complex numbers would have isomorphic Nisnevich sites. (This is true for the Zariski site, which is coarser than the Nisnevich site, and for the analytic site, which finer, but that doesn’t look like a proof yet.) But two elliptic curves are isomorphic if their $H^1$ vector spaces with their Hodge structures are isomorphic. So it appears that the Hodge structure (and hence the motive) contains more information than the Nisnevich site.

I’m not sure what you mean by the “non-abelian information”. The motive is purely abelian information in the sense that any definition of the category of motives would be an abelian category. So the abelian/non-abelian distinction between $\pi_1$ and $H^n$ is orthogonal to the distinction between abelian-group-valued cohomology theories and cohomology theories with values in other abelian categories, such as any proposed category of motives. For instance, there are motivic versions of $\pi_1$ .

I don’t know any good references for the relation between motivic cohomology and motives. It’s essentially given by Chern character maps. For instance the Picard group of a complex variety (which is a certain motivic cohomology group) maps via the Chern class map to $H^2$ .

Posted by: James on January 30, 2010 9:21 AM | Permalink | Reply to this

Re: cohomology

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Sorry for the slow reply, James. It seems I missed your reply for a while. You wrote:

I don’t know what $Sh_{(\infty,1)}(Nis)$ means exactly, but I imagine it’s some infinity-category version of the category of sheaves of sets on the Nisnevich topology.

Yes. Supposed to be the notation for the intrinsic object that is modeled by the local model structure on simplicial presheaves on the Nisnevich site.

If so, then I should be able to answer your question by giving two varieties with equivalent Nisnevich topologies but different motives. […]

Hm, okay. The beginning of the sentence made me expect a somewhat different end of it:

Given any variety or scheme or other $\infty$ -stack $X$ on the abstract Nisnevich site (objects are all smooth schemes of finite type), and given yet another such object $A$ , the motivic cohomology of $X$ with coefficients in $A$ of bidegree $(p,q)$ is

$H^{p,q}(X,A) := \pi_0 Maps_{Sh_\infty(Nis)}(X, \Omega^\infty_T \Sigma^{p-q} \Sigma_t^q) \,,$

where $\Omega_T$ denotes looping with respect to the Tate sphere and $\Sigma_t$ denotes suspension with respect to the geometric 1-sphere $\mathbb{A}^1-\{0\}$ .

Often, I suppose, the motivic cohomology of $X$ will be that where $A = \mathbb{Z}$ . But in fact $A$ could be any object whatsoever in $Sh_\infty(Nis)$ . If it is not a stable object, then I would call $H(X,A)$ a nonabelian cohomology of $X$ .

I gather that typically motivicists don’t consider unstable coefficient objects. But I mentioned them since you said something like that motivic cohomology sees too little information. My remark or question was just that using cohomology, as above, with coefficients just in $\mathbb{Z}$ clearly contains less information than if we allow $A$ to run over more general coefficient objects (more general abelian ones or even nonabelian ones). Alltogether that’s quite a bit of information.

I don’t know any good references for the relation between motivic cohomology and motives.

Yeah. I suppose that’s part of the reason why the subject appears a bit intransparent from the outside.

My first encounters with motivic stuff strongly reminded me of my first encounters with string/M-theory, in the following sense: all the expositions would say at the beginning would be:

it’s highly important and a huge unification – it is bigshot xyz’s dream;
it’s very mysterious and we don’t know what it is. And that’s all we gonna say for a while…

Posted by: Urs Schreiber on February 3, 2010 5:34 PM | Permalink | Reply to this

Re: cohomology

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First, regarding the point about the cohomology of $\mathbf{Z}$ , I think it’s important to remember that it typically contains almost no information in algebraically defined sites, rather than analytically defined ones. I haven’t studied the Nisnevich site, but this is definitely true for the Zariski site, which is coarser, and the etale site, which is finer. For instance $H^1(X,\mathbf{Z})=0$ in either topology when $X$ is an elliptic curve.

Question: Do the functors $\Omega_T$ and $\Sigma_t$ depend only on the underlying site, or do they also depend on the variety that gave rise to the site? In other words, suppose $X$ and $Y$ are varieties; then does any equivalence between their Nisnevich sites commute with these two functors? I ask because different elliptic curves (over $\mathbf{Q}$ , say) can have nonisomorphic Picard groups.

Regarding references, you might try Spencer Bloch’s paper “Lectures on Mixed Motives”. See section 0.1.8. There are also two volumes “Motives” edited by Uwe Jannsen, but Bloch’s article is more focused on the point you’re interested in. There’s also the book “Beilinson’s conjectures on special values of L-functions”, edited by Rapoport, which probably discusses this all extensively. (I never actually looked to closely at it.)

Posted by: James on February 4, 2010 11:11 PM | Permalink | Reply to this

Re: cohomology

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First, regarding the point about the cohomology of $\mathbb{Z}$ , I think it’s important to remember that it typically contains almost no information in algebraically defined sites, rather than analytically defined ones.

I thought integral motivic cohomology gives the Chow groups. Could you comment on the brief account of that reproduced here?

Question: Do the functors $\Omega_T$ and $\Sigma_t$ depend only on the underlying site, or do they also depend on the variety that gave rise to the site?

Just so that we are sure we are talking about the same thing: by the “variety that gave rise to the site” you are referring to the scheme denoted “ $S$ ” here?

We might be talking past each other here regarding “gros toposes” versus “petit toposes”. I am probably assuming implicitly mostly that $S = Spec k$ is the ground field or the like, so that we are talking about a “gros site” of all schemes (smooth, of finite type,…). Every scheme in the world becomes an object on the $\infty$ -stack $\infty$ -topos on that site, and its cohomology is given by homotopy classes of maps of $\infty$ -stacks out of it.

It seems to me now that you are thinking of computing the cohomology of a given scheme by first forming its petit Nisnevich site and then doing something there. Of course that’s not unrelated, but let’s straighten out what exactly we are talking about here Please have a brief look at the above two webpages and let me know to what extent the perspective given there matches what you have in mind.

Regarding references, you might try Spencer Bloch’s paper “Lectures on Mixed Motives”. See section 0.1.8. There are also two volumes “Motives” edited by Uwe Jannsen, but Bloch’s article is more focused on the point you’re interested in. There’s also the book “Beilinson’s conjectures on special values of L-functions”, edited by Rapoport, which probably discusses this all extensively. (I never actually looked to closely at it.)

Thanks! Will have a look.

Posted by: Urs Schreiber on February 5, 2010 12:20 AM | Permalink | Reply to this

Re: cohomology

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OK, I do think there’s a good chance we’re talking past each other. I had assumed you were computing some cohomology group on the petit Nisnevich site associated to some variety. But again, I know a very limited amount about motivic cohomology.

My understanding is that Chow groups $\mathrm{CH}^i(X)$ are the motivic cohomology groups $H^{2i}(X,\mathbf{Z}(i))$ , where $\mathbf{Z}(i)$ is a certain complex of sheaves on the small site corresponding to $X$ . (All this with respect to the Zariski topology, say.) The point is that to get the Chow groups you need to take coefficients in a complex, not the sheaf $\mathbf{Z}=\mathbf{Z}(0)$ . That’s all I meant. Maybe when you were talking about the cohomology of $\mathbf{Z}$ , you implicitly meant all its twists $\mathbf{Z}(i)$ .

As for the links you gave, I think it’s unlikely I’ll be able to comment on the first one. I’m able to understand almost nothing there. The second one says much less, but is easier for me to read and doesn’t appear to contradict anything I remember about the big Nisnevich site.

Posted by: James on February 5, 2010 12:58 AM | Permalink | Reply to this

Re: cohomology

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the presence of a ‘topological dimension’ as well as ‘categorical dimension’,

Yes, exactly, this is what this is all about: when we pass from plain $\infty$ -groupoids to $\infty$ -groupoids modeled on test objects in a site $C$ , then the geometry of $C$ merges with that of $\infty Grpd$ .

if anyone knew a model category structure on simplicial spaces such that geometric realization gives a Quillen equivalence to the usual model category of topological spaces.

I’ll provide you with a reference for that. A much more general statement is well known, I think: for every model category $C$ the category of simplicial objects in that category carries a model structure whose fibrant objects are those fibrant simplicial objects whose face and degeneracy maps are weak equivalences, and this is Quillen equivalent to the original model category. That’s one way to realize the simplicial enrichment of a model category!

This could be the slick approach to the problem of crushing a bigraded structure down to a graded structure.

So in general we want to be looking at the geometric morphism

$\mathbf{H} \stackrel{\leftarrow}{\to} \infty Grpd$

from our $(\infty,1)$ -topos $\mathbf{H}$ of “ $\infty$ -groupoids modeled on $C$ ” to just plain $\infty$ -groupoids, which forgets of all the “ $C$ -structure” and remembers just the underlying topological structure. In terms of $\infty$ -stacks this means taking global sections, i.e. evaluating on the terminal object. For instance a 2-groupoid internal to diffeological spaces is sent to the underlying groupoid of plots from the point.

For $\mathbf{H} = Sh_{(\infty,1)}(Top)$ , i.e. for topological $\infty$ -groupoids this is quite nicely discussed in the article by Daniel Dugger that I keep pointing to. See the above link for the reference.

My question:

[…] How, from this viewpoint , can we see that motivic cohomology should be bigraded?

Yeah, I know. I took that “from this viewpoint” out of your question, intentionally. ;-)

I found all the motivic story quite frustratingly intransparent and confusing for a long while. I was very pleased when I finally learned about the Morel-model in which everything became so clear and in fact just a simple special case of a simple general story. In fact, in this formulation, it turns out that I have been thinking about “motivic cohomology” for a long time – just having a different kind of site in mind. So I am very much inclined to think that this is the right point of view, and it suppresses my interest in understanding by which convoluted ways this perspective was arrived at.

As far as splitting idempotents goes: i see that you are probably hoping to see a connection to your groupoidification program. But it would seem to me even that should now be clearer using the Morel model and the insight that quasicoherent oo-stacks are just fiberwise stabilized overcategories. I am suspecting that in your groupoidification program, you get the desired linearity (which you can achieve by splitting idempotents) by not using just overcategories as you do, but fiberwise stabilized overcategories: not just groupoids over groupoids, but abelian $\infty$ -groupoids over groupoids.

What you are hinting is that somehow there must be a line object hiding in Grothendieck’s hopes and dreams. And that might be a good way to make my question more precise. Why must there be this line object?

I am not too familiar with Grothendieck’s dreams. But that there is a line object in the site $CRing^{op}$ is a tautology. And given any site, that studying cohomology means mapping $\infty$ -groupoids modeled on the site into each other has somehow become quite clear. From this point of view Morel-Voevodsky’s $\infty$ -stack formulation of motivic cohomology is the most obvious thing in the world to do when faced with $CRing^{op}$ , and the only remaining question to me seems to be why why this obvious answer isn’t better advertized. Luckily the $n$ Lab entry on motivic cohomology now tells everyone! :-)

Posted by: Urs Schreiber on January 28, 2010 7:17 PM | Permalink | Reply to this

Re: cohomology

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I wrote:

So in general we want to be looking at the geometric morphism $\mathbf{H} \stackrel{\leftarrow}{\to} \infty Grpd$

I’ll say something better. What I had mentioned isn’t actually the functor you were talking about.

In prop. 3.2.8, p. 22 Dugger discusses how – what for our purposes is the forgetful funtor $\gamma : Diff \to Top$ – induces a Qullen adjunction

$sPSh(Diff)_{proj} \stackrel{\leftarrow}{\to} Top$

that takes a smooth $\infty$ -groupoid $A$ (you may want to think of simplicial presheaves appearing here as being in particular nerves of $n$ -groupoids internal to diffeological spaces = concrete sheaves on $Diff$ ) to a topological space by (going step by step through the construction in the proof on p.22)

first degreewise forming the topological space

$A^{Top}_n := \int^{U \in Diff} A_n(U)\cdot \gamma(U)$

underlying the presheaf/diffeological space;
then collecting that to the simplicial topological space $A^{Top}_\bullet$
and then forminng the geometric relalization of that

$L(A) := \int^{[n] \in \Delta} \Delta^n_{Top} \times A^{Top}_n \,.$

Later on the top of p. 33 this is discussed further. Dugger has a somewhat different goal here than what we are talking about, as he wants to find those smooth $\infty$ -groupoids for which this operation induces an equivalence, sort of the opposite of the interesting point we are talking about: in p. 29, def 3.4.1 he introduces the $I$ -homotopy localization of the local Cech model structure on $sPSh(Diff)$ (that which truly models smooth $\infty$ -groupoids) at the line object $I = \mathbb{R}$ . Notice the remark below the definition on notation: the $I$ -subscipt in the following denotes both Cech-localization (which we want here) and homotopy localization (which is a bit of a distraction for the present point of the discussion).

Posted by: Urs Schreiber on January 28, 2010 9:14 PM | Permalink | Reply to this

Re: cohomology

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I mentioned something and wrote:

(which is a bit of a distraction for the present point of the discussion).

Or is it?

Re-reading your question

if anyone knew a model category structure on simplicial spaces such that geometric realization gives a Quillen equivalence to the usual model category of topological spaces.

my impression now is that this is pretty much the answer to the question:

there is a model category structure on simplicial (presheaves on) manifolds, such that geometric realization

$sPSh(Diff) \stackrel{realization}{\to} Top$

is a Quillen equivalence. This model structure is the left Bousfield localization of the global projective structure at Cech covers and at the projections $\{X \times I \to X\}$ .

And the statement is in exercise 3.4.12 in Dugger’s notes.

Posted by: Urs Schreiber on February 3, 2010 5:11 PM | Permalink | Reply to this

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