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June 14, 2012

Cohomology in Everyday Life

Posted by David Corfield

I have been looking for examples, accessible to a lay audience, to illustrate the prevalence of cohomology. Here are some possibilities:

  • Penrose’s impossible figures, such as the tribar
  • Carrying in arithmetic
  • Electrical circuits and Kirchhoff’s Law
  • Pythagorean triples (Hilbert’s Theorem 90)
  • Condorcet’s paradox (concerning the impossibility of combining comparative rankings)
  • Entropy, but I think we never quited nailed this.

Anyway, I’d be grateful for any other cases of cohomology in everyday life.

There’s a related MathOverflow question on this. One of the answers notes a cohomological interpretation of mass. Following this up, I see Santiago García in Hidden invariance of the free classical particle writes that mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” Is this an interesting point of view?

Posted at June 14, 2012 12:27 PM UTC

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Re: Cohomology in Everyday Life

I’m happy with
Electrical circuits and Kirchhoff’s Law
but are the others really accessible to a LAY audience?
Perhaps Gauss’ linking number?

Posted by: jim stasheff on June 14, 2012 1:07 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Well, I’m not looking to give any very deep description. In the case of Penrose’s impossible figures, such as the tribar, I would only talk of an impossibility of patching the locally consistent data into a consistent whole. The Pythagorean triples case wouldn’t involve an introduction to Galois cohomology.

Posted by: David Corfield on June 14, 2012 1:27 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

“ In the case of Penrose’s impossible figures, such as the tribar, I would only talk of an impossibility of patching the locally consistent data into a consistent whole.”

Oh, that is neat!

Posted by: jim stasheff on June 15, 2012 2:05 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Being a great believer in pilot projects, I’d like to see a 1-2 page description of the carrying in arithmetic case, targeted to the bright primary school student. You guys can easily target the bright high school student that Urs mentions; but it will take some hard thinking to make the case clearly to a more naive audience. And it would force one to focus on only the most fundamental concepts. That would be a valuable contribution.

Posted by: Charlie Clingen on June 15, 2012 2:18 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Isaksen’s paper would make for a fairly early undergraduate account.

Posted by: David Corfield on June 15, 2012 2:35 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

The referenced paper starts with the assumption that the reader is familiar with elementary group theory, including subgroups, homomorphisms, isomorphisms, cosets, and quotient groups and then, by the third page, moves into cocycles and then onward and upward from there. Actually, it is a fascinating article and a great way for a stubborn beginner like me to get familiarized with cocycles and coboundries and the beginnings of cohomology. So I will definitely spend too much time studying it (probably a couple of weeks), as a part of my never-ending quest to understand gauge theory. Thanks, David, for pointing out this article; I really appreciate it. However, I fear that this paper is not a useful presentation of cohomology in everyday life for the truly naïve reader. Learning from this paper, as clearly written as it is, is a bit like trying to learn a foreign language using a foreign-language-only dictionary. In fact, it is really written for people with non-trivial mathematical training. The problem is not necessarily that the concepts involved are too difficult to understand. The problem is that the concepts are described using a language that the naïve person does not speak, and a style of thinking that a naïve person is not fluent in. So the non-expert trying to fully understand this article is like the speaker of kitchen Chinese trying to engage with a Chinese-only speaker in a discussion about health insurance or wedding customs or cancer treatment. I know that it can be done, but only with great effort by all involved. So, explaining to smart but truly naïve people how cohomology occurs in daily living, as, for example in carrying in addition, is in my opinion a very difficult task that requires the dedication of a highly knowledgeable, highly skilled mathematician and communicator/teacher. It isn’t cutting edge research, but it would be an important demonstration to a wide audience that far-reaching, fundamental mathematical concepts are part of our daily lives and are not only beautiful, but easy to grasp. I can’t do the writing, but I surely have the experience and the lack of knowledge required to critique it and contribute to it. David has a great idea. This would be an extremely valuable contribution.

Posted by: Charlie Clingen on June 16, 2012 6:59 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I agree: Isaksen’s article points out that there is a useful example of group cohomology in elementary school arithmetic, but the article is certainly not a school exposition of this fact.

What I can’t quite tell from what you write is whether you have yourself understood the point that Isaksen makes and are suggesting that somebody else should write such an exposition, or if you are still yourself wondering how to understand the example of group cohomology in the operation of carrying.

Sorry if I am misunderstanding you here. But if the latter is the case, let us know at which point you get stuck and we should be able to explain the rest in simple words. Then when everything is clear, you can write the kind of exposition that you rightly say should exist!

Posted by: Urs Schreiber on June 18, 2012 2:41 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks for the offer to help, Urs. Let me explain. My understanding is that David is looking for examples of cohomology that a) occur in everyday life, and b) that the layman can understand. So I have been trying to think of examples that satisfy both criteria. Carrying in addition is a great example of a), but the paper certainly does not satisfy b); in fact, even if the detailed exposition can be followed, it’s not obvious to me that it will lead to a basic understanding of what cohomology actually is.

So there seem to be two choices: choice 1) can the explanation given in the paper be streamlined somehow, and at the same time can the essential background mathematics (homomorphisms, quotient groups, etc.) be kept simple to the extent that the example becomes easy to grasp while still being useful; or if this is not possible, choice 2): is there some other example that will serve the desired purpose? I fear that simplifying the paper, which is already very clearly presented, while simultaneously teaching enough of the required underlying math may be quite difficult.

As for choice 1), I am willing to take a deeper look at that challenge and see, with a little help from you folks, if there is an approach to do the job. Perhaps writing an introduction to the paper, giving a road map outlining what is to come and explaining some of the key mathematical tools in simple terms would be useful. However, this choice gives an algebraic view of some cohomological concepts, and as such may not be easy for the layman to visualize and comprehend.

For choice 2, I tried to imagine a simple physical situation, a static, scalar temperature field that just might barely support some sort of simple cohomology demonstration. The intent is to use a geometrical presentation of some basic cohomology concepts that might be easier to understand conceptually. I need help here, because I don’t have the necessary background yet. My example was simply to suggest a starting point. I assume it isn’t complex enough. But the example, or some extension thereof, does have the benefit that it can easily be visualized, and therefore “overlaying” it with a cohomology argument might be easier for the layman to understand. Some expert thoughts on choice 2, or a replacement for it, would be helpful. What do you think?

Posted by: Charlie Clingen on June 19, 2012 2:58 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

giving a road map outlining what is to come and explaining some of the key mathematical tools in simple terms would be useful.

Hm, do we need a road map? I thought we are just looking for the best way to say the following to somebody very young and/or very lay:

You have ten fingers on your hands, call them

Fingers={1,2,3,4,5,6,7,8,9,10} Fingers = \{1,2,3,4,5,6,7,8,9,10\}

You also have a ten toes, call them similarly

Toes={1,2,3,4,5,6,7,8,9,10}. Toes = \{1,2,3,4,5,6,7,8,9,10\} \,.

You count with your fingers, going right to left. So if I say “6” you raise the thumb of your left hand. When I say “add 3”, you hide the thumb and instead raise the second but last finger of the left hand. When I now say “add 5” you count one more finger on the left hand, then switch back and count four on the right hand. And so on.

To remember when you ran out of fingers on the left hand and switched back to the right hand, raise a toe!

So when I say: “add 6 and 3” you just count nine fingers, done. When I say “add 9 and 5” you raise the fourth finger of the right hand… and one toe. That’s 14 together.

So what’s the rule when to raise a toe? The rule is:

Carrying:Fingers×FingersToes Carrying : Fingers \times Fingers \to Toes

Carrying(countathenb){raisetoe ifcountingbfromarunsoutoffingers dontraisetoe otherwise Carrying(count a then b) \coloneqq \left\{ \array{ raise\;toe & if\; counting\;b\;from\;a\;runs\;out\;of\;fingers \\ don't\;raise\;toe & otherwise } \right.

Challenge: convince yourself that this rule happens to satisfy the following magic rule: for any three numbers a,b,ca,b,c that I tell you, we have graphically that

b a c = b a c \array{ \bullet &\stackrel{b}{\to}& \bullet \\ {}^{\mathllap{a}}\uparrow &\nearrow& \downarrow^{\mathrlap{c}} \\ \bullet &\to& \bullet } \;\; = \;\; \array{ \bullet &\stackrel{b}{\to}& \bullet \\ {}^{\mathllap{a}}\uparrow &\searrow& \downarrow^{\mathrlap{c}} \\ \bullet &\to& \bullet }

meaning that

Carrying(countathenb)+Carrying(counta+bthenc)=Carrying(countathenb+c)+Carrying(countbthenc). Carrying(count a then b) + Carrying(count a+b then c) = Carrying(count a then b+c) + Carrying(count b then c) \,.

A map Fingers×FingersToesFingers \times Fingers \to Toes satisfying this magic rule is called a 2-cocyle on the group of finger-counting with values in the group of toe counting.

And that’s it!

Posted by: Urs Schreiber on June 19, 2012 6:27 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Beautiful example, Urs! Thank you! Your example certainly makes the first pages of the paper very concrete. My hope is that I could build upon your example, explain away the magic of the cocycle condition, possibly move on to coboundries, and then sketch a bare-bones view of cohomology that illustrates its importance “in everyday life”. This area is new to me, so it’s going to take a while for me become more familiar with it, even in its simplest forms. It seems that explaining the real value of cohomology demands getting into some areas of group theory or differential geometry; teaching those concepts to the very young and very lay is a real challenge. So back to your example and the paper. I’m astounded by the depth to which this simple example leads. I may have some dumb questions to ask.

Happy Summer Solstice!

Posted by: Charlie Clingen on June 20, 2012 10:00 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

explain away the magic of the cocycle condition

That’s what the “graphical” version of the cocycle condition that I indicated above is supposed to achive:

the condition on a 2-cocycle on a group says precisely: given a tetrahedron whose edges are consistently labeled by elements in the group, in that going around any triangle gives the neutral element, the assignment of the 2-cocycle to each face similarly has the property that “going around the tetrahedron” yields the neutral element in the coefficient group. That’s the 2-cocycle condition.

This is explained in more detail at nLab:group cohomology.

I’ll be happy to add further explanation to that page (I guess you want to be looking at the Examples-section of that page), if you let me know which bits are unclear.

Posted by: Urs Schreiber on June 21, 2012 12:52 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Urs, I see how the equation under your diagrams above relate to Formula 2.1, the cocycle condition, in Isaksen’s paper. But apparently I don’t understand completely; I’m embarrassed to admit that I don’t see how the diagrams correspond to your equation. Can you say a few words about the diagrams? Are they commutative diagrams? What do the vertices in the diagrams represent? If this level of questions is too elementary pursue here, just let me know. I’ve already learned a lot and have some ideas about how to move ahead.

Posted by: Charlie Clingen on June 24, 2012 9:19 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

A map satisfying this magic rule is called a 2-cocyle on the group of finger-counting with values in the group of toe counting.

Sentences like that are what make the n-category cafe cool.

Posted by: Bruce Bartlett on June 22, 2012 1:47 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

How true!

Posted by: Charlie Clingen on June 22, 2012 2:43 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Charlie wrote:

Being a great believer in pilot projects, I’d like to see a 1-2 page description of the carrying in arithmetic case, targeted to the bright primary school student.

David wrote:

Isaksen’s paper would make for a fairly early undergraduate account.

Charlie wrote:

However, I fear that this paper is not a useful presentation of cohomology in everyday life for the truly naïve reader.

Isaksen thanks Jim Dolan for telling him about this idea in the first place. Jim is the kind of guy who would enjoy explaining 2-cocycles and their role in carrying with a minimum of extra concepts (subgroups, homomorphisms, isomorphisms, cosets, quotient groups…). This takes real work. If you allow yourself to use all these concepts the expository task becomes much simpler. It’s not unfair to allow yourself these concepts, because you can’t really define a 2-cocycle in general without a bunch of these concepts.

You can however understand the example that comes from carrying without all that stuff. Unfortunately for Charlie, Jim prefer to explain things in person, not by writing. So, let me give a quick summary:

There’s a rule for carrying numbers when you add.
This rule needs to obey some equation to guarantee that addition is associative. You can write down that equation yourself. This equation is called the 2-cocycle equation.

Posted by: John Baez on June 25, 2012 5:46 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks, John! Following Isaksen’s paper, I have worked through that part. I’m looking at extensions now and came across an interesting tangential discussion in T. Tao’s blog, which uses principle bundles to structure the problem. This means I will be reviewing fibers, projections, etc. for a while. Amazing how these “unexpected” connections keep coming up! I realize that I am dealing with these topics at the most superficial possible level; even so every day brings new and exciting discoveries. And I haven’t even gotten into cohomology in differential geometry yet, which is where I am headed!

Posted by: Charlie Clingen on June 25, 2012 11:03 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I’m looking at extensions now and came across an interesting tangential discussion in T. Tao’s blog, which uses principle bundles to structure the problem.

Which discussion was that? Maybe it is not so tangential. There is an intimate relation between the theory of (higher) bundles and cohomology. To a large extent, these theories are equivalent. (That is, incidentally, the topic of the latest post) above.

For instance for the example of the carrying 2-cocycle if we regard it as a map B 10carryingB 2 10B \mathbb{Z}_{10} \stackrel{carrying}{\to} B^2 \mathbb{Z}_{10}, as we should, then it classifies a B 10B \mathbb{Z}_{10}-principal 2-bundle over B 10B \mathbb{Z}_10, and this is B 100B \mathbb{Z}_{100}:

B 100 B 10 carrying B 2 10 \array{ B \mathbb{Z}_{100} \\ \downarrow \\ B \mathbb{Z}_{10} &\stackrel{carrying}{\to}& B^2 \mathbb{Z}_{10} }

Looping this once gives the extension of group 100 10\mathbb{Z}_{100} \to \mathbb{Z}_{10}.

This relation between cocycles and higher bundles is universal. The only problem is that it is often not made explicit.

Posted by: Urs Schreiber on June 26, 2012 8:00 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks, Urs. Here is the posting I mentioned:

http://terrytao.wordpress.com/2011/06/07/central-extensions-of-lie-groups-and-cocycle-averaging/

I am now taking Isaksen’s and your descriptions of carrying and restructuring them using Tao’s description. Very instructive!

Posted by: Charlie Clingen on June 26, 2012 1:31 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Here is the posting I mentioned: http://terrytao.wordpress.com/2011/06/07/central-extensions-of-lie-groups-and-cocycle-averaging/

Okay, thanks, so I can amplify my above remark: the bundles appearing there are indeed not tangential to the theory, but at its core.

I’ll expand a bit on this. The very general setup is the following:

for AA any abelian group, an nn-cocycle with values in AA on some XX is a morphism

XB nA X \to \mathbf{B}^n A

to the nn-fold delooping of AA.

The corresponding bundle PXP \to X classified by this morphism is the homotopy fiber of this morphism

P X B nA. \array{ P \\ \downarrow \\ X &\to& \mathbf{B}^n A } \,.

This construction establishes an equivalence

{AcocyclesandhighercoboundariesonX}{higherAbundlesandtheirhigherequivalencesonX}. \left\{ A-cocycles\;and\;higher\;coboundaries \; on\;X \right\} \simeq \left\{ higher\;A-bundles\;and\;their\;higher\;equivalences\;on\;X \right\} \,.

In particular on equivalence classes this says that degree-nn AA-cohomology of XX classifies AA-nn-bundles on XX:

H n(X,A)ABund n(X) /. H^n(X,A) \simeq A Bund_n(X)_{/\sim} \,.

For the special case that X=BGX = \mathbf{B}G is itself a delooping, a cocycle

X=BGcB nA X = \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A

is a group nn-cocycle on GG and the corresponding bundle

BG^ BG c B nA \array{ \mathbf{B}\hat G \\ \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A }

is the delooping BG^\mathbf{B}\hat G of the corresponding group extension G^\hat G. Conversely, under looping, the group extension G^G\hat G \to G is the AA-(n1)(n-1)-bundle classified by Ωc\Omega \mathbf{c}

G^ G Ωc B n1A. \array{ \hat G \\ \downarrow \\ G &\stackrel{\Omega\mathbf{c}}{\to}& \mathbf{B}^{n-1} A } \,.

The situation is conceptually even better than the way it is presented in that blog entry:

the above story holds true for groups equipped with any notion of geometry, if only we consider all the constructions in the corresponding context of higher geometry. Notably the situation of Lie group cocycles and Lie group extensions works verbatim as above, there is no need for an extra concept of “local cocycle” and “local coboundary” (of course it can still be very useful in computations). This is discussed a bit at nLab:Lie group cohomology.

Posted by: Urs Schreiber on June 26, 2012 3:19 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

How about this as a candidate for an everyday situation that should be pretty easy to understand? Start with a simple physical situation: a room enclosing a static temperature field, a scalar field. There is a tiny heat source hanging from the ceiling in the middle of the room. The room can have any shape and the walls can be at whatever convenient temperatures facilitate the discussion, and there is some magic way to display the temperature at every point in the room simultaneously. So we can imagine (actually see) isotherms centered at the heat source by scanning the temperature displays. This allows us to talk about manifolds, functions, gradients, directional derivatives, and one-forms. Can this picture be used, or can it be further embellished to be used to discuss, say a trivial example of De Rham cohomology, without demanding a significant mathematical background? Better yet, is there a way to do this in only two dimensions? The idea would be to demonstrate, not prove, the simplest possible aspects of cohomology so that the newbie can understand, physically, a fundamental example illustrating what cohomology is about and how it can help us to understand the world around us.

Posted by: Charlie Clingen on June 17, 2012 9:49 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Carrying as a cocycle is pretty awesome; why didn’t anyone tell me that when I was learning about group cohomology?

What I want to know now is, what about multiplication? Suppose I have an ideal II (like (10)(10)) in a ring RR (like /100\mathbb{Z}/100); is the “extension”

IRR/I I \to R \to R/I

classified by some sort of cohomology of R/IR/I with coefficients in II?

Posted by: Mike Shulman on June 17, 2012 10:25 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Hi Mike. Yes, there is such a theory for extensions by square-zero ideals. The homology functor is usually called Exalcomm. It’s explained EGA and Weibel’s book on homological algebra, for example. The general subject is called Andre-Quillen homology or the theory of the cotangent complex.

Posted by: James Borger on June 19, 2012 4:31 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

We have at nLab a page on deformation theory, which says

Formal deformation theory studies the obstruction theory of extensions to infinitesimal thickenings.

A typical example of an infinitesimal thickening is a square-0-extension of a ring.

I see that page links to unwritten pages on André-Quillen homology and cohomology.

Posted by: David Corfield on June 19, 2012 7:50 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks! Extensions only by square-zero ideals seems like a very arbitrary restriction, though; is it essential to the theory?

Posted by: Mike Shulman on June 19, 2012 4:22 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Well, if II isn’t square-zero, then you can’t make sense of how RR acts on it until you have RR itself, rather than R/IR/I. But classifying all possible RR’s given II and R/IR/I was the whole point. So there is a chicken and egg problem. When you’re classifying square-zero extensions, the RR-module structure, for whatever RR, will always factor through R/IR/I. So you can pin it down before you have RR in hand.

Similarly, in group cohomology, you look at extensions of groups by abelian normal subgroups. So the conjugation action of the yet to be produced GG on NN factors through G/NG/N and so can be specified in advance.

In principle, you can iterate this to get some control over extensions by nilpotent ideals. But if you don’t have any nilpotence condition, you’re just not doing deformation theory anymore.

Posted by: James Borger on June 20, 2012 12:41 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

OK, OK. I guess I don’t deny that it’s possible. These things are always pretty tautological, so if it’s possible, you should be able to just sit down and write out some cocycle/coboundary formalism. But I do find it hard to imagine it being a useful thing, from the geometric point of view.

Posted by: James Borger on June 20, 2012 11:05 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I don’t care about whether I’m doing deformation theory; I care about classifying extensions! And there’s a perfectly good version of group cohomology that classifies extensions by nonabelian groups: it’s called (oddly enough) nonabelian group cohomology. Surely there is something analogous for rings.

Posted by: Mike Shulman on June 20, 2012 8:31 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Oops. This was meant to be a reply to Mike’s nonabelian cohomology comment below.

Posted by: James Borger on June 20, 2012 1:12 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Mike asked:

Suppose I have an ideal II (like (10)) in a ring RR (like /100\mathbb{Z}/100); is the “extension” IRR/II \to R \to R/I classified by some sort of cohomology of R/IR/I with coefficients in II?

Yes, this is called Everett’s theorem, a ring analog of Schreier theory. The account Ideal extensions of rings by Petrich seems to be useful.

Posted by: Urs Schreiber on June 20, 2012 12:03 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I wanted to say thanks for this (Everett’s theorem); I haven’t had time to read it yet (and I don’t know when I will) but it makes me happy to know that it exists. (-:

Posted by: Mike Shulman on June 21, 2012 5:36 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I had made a note at nLab:carrying (before seeing the above request, so it is not designed to be a reply to that).

Posted by: Urs Schreiber on June 15, 2012 2:57 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

A reasonable point to make then is that, if all I’m doing is pointing to phenomena whose underlying explanation can be thought about in cohomological terms, I may as well just point to the universe, and refer the audience to your Cohomological Physics.

I see you mention there Bargmann on extensions of the Galilean group.

Posted by: David Corfield on June 14, 2012 1:42 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks for publicizing that note.

Posted by: jim stasheff on June 15, 2012 2:01 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

if all I’m doing is pointing to phenomena whose underlying explanation can be thought about in cohomological terms, I may as well just point to the universe

Yes! Fundamental physics is all controled by cohomology.

What kind of lay audience exactly do you mean to speak to? Entirely lay, or with some mathematical training, just not in cohomology?

Concerning very simple examples of cohomology: Cohomology in degree 0 is often very simple to describe, nevertheless sometimes of some interest. For instance the degree-0 cohomology of a groupoid is just the set of invariant functions on it, i.e. the functions on connected components. If the groupoid is a Lie groupoid then this is equivalently the degree-0 cohomology of the corresponding Lie algebroid, computed by the BRST complex, and again in degree 0 it is simply the invariant functions: the “gauge invariant” functions. While simple, that’s certainly important.

Posted by: Urs Schreiber on June 14, 2012 2:40 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I will have such little time to go into mathematical detail. From experience, any attempt to go into such detail swallows up the time. I need to achieve too much else. One useful handle, though, might be to talk about cohomology in terms of obstruction and extension.

The talk is to be a sketch of the thinking which starts here. You’ll see a conversation of yours with Mike on the equivalence principle is involved. It’s amazing how your summaries of the conversation end up so close to Friedman’s way of describing the change of status of statements in physics.

Posted by: David Corfield on June 14, 2012 3:37 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

The talk is to be a sketch of the thinking which starts here.

Ah, so the audience will be philosophers? People not necessarily trained in math but also not shy about abstract reasoning?

There might then be the following route that gets one to the heart of the matter with really only very few and very fundamental elements of reasoning: just combine the basic idea of homotopy type theory with that discussed at cohomology. The result is: cohomology is nothing but the theory of functions f:XAf : X \to A.

That explains both how it works, and why it is of fundamental importance. For instance…

One useful handle, though, might be to talk about cohomology in terms of obstruction and extension.

…for instance obstruction theory becomes this simple statement:

suppose you have a function (“universal characteristic class”)

c:AB \mathbf{c} : A \to B

and a point pt BBpt_B \in B (“universal trivial cocycle”). Write then i:FAi : F \to A for the function that includes the fiber of ff over pt Bpt_B (“Whitehead stage of AA”).

Then given any other function

f:XA f : X \to A

we can consider the “lifting problem” of finding f^\hat f such that if^fi \circ \hat f \simeq f.

F * f^ i pt B X f A c B. \array{ && F &\to& * \\ & {}^{\mathllap{\hat f}}\nearrow & \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{pt_B}} \\ X &\stackrel{f}{\to}& A &\stackrel{\mathbf{c}}{\to}& B } \,.

By construction of ff, this clearly can exist only if

cfpt B \mathbf{c}\circ f \simeq pt_B

hence if the composite

f *ccf f^* \mathbf{c} \coloneqq \mathbf{c}\circ f

(“obstruction class”) is equivalent to the trivial cocycle, hence when its cohomology class vanishes: “c\mathbf{c} is the universal obstruction class to lifts through ii”.

This is a trivial high-school level statement if one thinks in terms of functions. But it remains a theorem if the functions are actually morphisms of \infty-stacks. Homotopy type theory makes precise the sense in which nevertheless the way it works is still the trivial high-school level argument:

in homootopy type theory:

  • functions are cocycles;

  • equivalence classes of functions are cohomology classes.

That’s really all there is to cohomology, fundamentally.

Posted by: Urs Schreiber on June 14, 2012 4:48 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

But doesn’t that purely “internal” point of view on cohomology obscure its “external” meaning (parametrizing the failure of a local-to-global construction)?

Posted by: Mike Shulman on June 14, 2012 5:02 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

But doesn’t that purely “internal” point of view on cohomology obscure its “external” meaning (parametrizing the failure of a local-to-global construction)?

I wouldn’t think it does. It’s all still nicely there:

By the magic of homotopy type theory/\infty-topos theory, every function f:XBGf : X \to \mathbf{B}G is locally trivial, in that there is a cover UXU \to X such that f| Uptf|_U \simeq pt. (This is using that pt:*BGpt : * \to \mathbf{B}G is a cover).

It follows that the desired lifts exist over UU, and that hence the obstruction is equivalently that of descending (f |U)^\widehat {(f_{|U})} along UXU \to X.

One can play this game further (e.g. here) and give a nice (I think) account of twisted cohomology (and twisted higher bundles), in terms of sections of locally trivial but globally non-trivial bundles, which is all essentially expressible in just the high-school kind of argument about functions.

Posted by: Urs Schreiber on June 15, 2012 7:04 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

But now you’ve moved outside the internal language. What I was saying is that the glib statement “functions are cocycles” is what obscures the meaning of cohomology; you have to move outside the internal language to talk about functions existing on a cover.

Posted by: Mike Shulman on June 15, 2012 8:02 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

But now you’ve moved outside the internal language.

No: cover = effective epimorphism!

Posted by: Urs Schreiber on June 15, 2012 8:32 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

But the meaning of “cover” which has to do with local-to-global properties resides externally.

Posted by: Mike Shulman on June 15, 2012 10:06 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

But the meaning of “cover” which has to do with local-to-global properties resides externally.

Hm, not sure what you mean. If you fix a site of definition, then all the covers in the site are reflected internally as effective epimorphisms.

Posted by: Urs Schreiber on June 15, 2012 10:28 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Mike,

of course everything I said here is trivial and I know that you know it well. I am just saying it to reflect how I don’t understand what issue you have in mind.

Maybe you need to give me more details on which statement exactly you think exists only externally and not internally.

Posted by: Urs Schreiber on June 15, 2012 2:16 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I seems to me that the external part is exactly the choice of a site of definition. Can you explain what a principal bundle or connection is without one?

In general, the geometrical interpretation of cocycles/cohomology classes somehow involves “the external part”, whatever it may be, and all the applications rely on such interpretations.

Posted by: Stanisław Szawiel on June 17, 2012 7:57 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I seems to me that the external part is exactly the choice of a site of definition.

Modulo subtleties that Mike may be thinking of, the situation is this:

the external choice of a site is internally not reflected by a notion of descent (that is present also without choosing a site) but is internally reflected by a notion of scheme or manifold. In order to internally say that a space locally looks like drawn from a repository of test spaces, hence to be a test-space scheme/manifold, we need to know what these test spaces are, and that’s what the choice of site does. This is nicely formalized in Lurie’s Structured Spaces.

Can you explain what a principal bundle or connection is without one?

Yes! I have been working on a writeup on this question for a while, see here and the references listed there.

Right this moment I am busy finalizing a series of articles focused on princial bundles in an \infty-topos that I have been writing with Thomas Nikolaus and Danny Stevenson. You can find a preview handout here.

One of the main statements is (which may be evident to some experts, but deserves to be highlighted): a good theory of principal bundles exists in a presentable \infty-category already as soon as the Giraud axioms hold, hence as soon as it is an \infty-topos.

No specification of site is needed, not for this step.

Posted by: Urs Schreiber on June 18, 2012 2:14 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Hmm, evidently I am kind of confused. I feel like I often get confused about what is internal and what is external. But I still feel like there are some aspects of cohomology that only make sense in terms of comparing the internal to the external, e.g. this discussion.

Posted by: Mike Shulman on June 16, 2012 5:17 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Write then i:FAi:F \to A for the function that includes the fiber of ff over pt Bpt_B

So, ff should be c, yeah?

So I might have thought of the obstruction to lifting the identity function on the circle through the projection from the boundary of a Moebius strip, by thinking of the nontrivial element of H 1(S 1,/2)H^1(S^1, \mathbb{Z}/2\mathbb{Z}), or the nontrivial cocycle in Hom(S 1,K(/2,1))Hom(S^1, K(\mathbb{Z}/2\mathbb{Z}, 1)), picturing a pullback of the fibration S RP S^{\infty} \to RP^{\infty}, but it was all already there in the high -school argument.

Posted by: David Corfield on June 14, 2012 5:49 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

So, ff should be c\mathbf{c}, yeah?

Right, sorry, at this point the ff was meant to be a c\mathbf{c}, yes.

So I might have thought of the obstruction to lifting the identity function on the circle through the projection from the boundary of a Moebius strip, by thinking of the nontrivial element of H 1(S 1,/ 2)H^1(S^1, \mathbb{Z}/\mathbb{Z}_2), or the nontrivial cocycle in Hom(S 1,K(/2,1))Hom(S^1, K(\mathbb{Z}/2\mathbb{Z},1)),

So you assume your audience to be familiar a bit with ordinary groups?

There is this simple construction, which goes a long way and, I believe, brings the Moebius-strip example a bit closer to “everyday life”:

The orthogonal group O(n)O(n) has an evident group homomorphism O(n) 2O(n) \to \mathbb{Z}_2, which simply sends an orientation-reversing transformation to the nontrivial element and everything else to the trivial element. The ordinary fiber of that map over the trivial element is therefore the special orthogonal group

SO(n)O(n) 2. SO(n) \to O(n) \to \mathbb{Z}_2 \,.

It so happens that after passing to the coresponding one-object Lie groupoids (so I guess at this point one needs the audience to know what the 1-object groupoid of a group is), this remains a fiber sequence

BSO(n)BO(n)w 1B 2. \mathbf{B}SO(n) \to \mathbf{B}O(n) \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2 \,.

Now the original projection has a grand name: it is the universal first Stiefel-Whitney class w 1\mathbf{w}_1.

A function g:XBO(n)g : X \to \mathbf{B}O(n) is an orthogonal structure on XX. A lift to f^:XBSO(n)\hat f : X \to \mathbf{B} SO(n) is a choice of orientation of that orthogonal structure. The obstruction to its existence is f *w 1f^* \mathbf{w}_1.

Specifically for n=1n = 1 (the dimension of the circle) we have

  • O(1)= 2O(1) = \mathbb{Z}_2,

  • w 1=id\mathbf{w}_1 = id;

  • SO(1)=*SO(1) = *

and so the Moebius-strip example is now the special case of this situation where one starts with an orthogonal structure on the circle which is not orientable.

Posted by: Urs Schreiber on June 15, 2012 6:52 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

If, as Urs writes,

in homotopy type theory:

  • functions are cocycles;
  • equivalence classes of functions are cohomology classes.

That’s really all there is to cohomology, fundamentally.

What will happen after categorification?

  • functors are ???;
  • equivalence classes of functors are &&&.

So he might then tell us that just as via homotopy type theory, cohomology should be straightforward to the high-school student familiar with functions, so via categorified homotopy type theory, &&& should be straightforward to the university student familiar with functors.

But then, if cohomology is so very important, (“Fundamental physics is all controled by cohomology”, etc.), wouldn’t &&& be very important in physics too? And if so, won’t we have glimpsed it already?

Posted by: David Corfield on June 15, 2012 2:47 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

so via categorified homotopy type theory, &&& should be

In this discussion, let’s stick with the terminology Mike used in the previous thread, which seems more suitable:

“If (equivalence classes of) functions in homotopy type theory are cocycles (cohomology classes), what are functions in directed homotopy type theory?”

A famous example of a directed cohomological nature are:

directed (and symmetric monoidal) cocycles on n-Bord are… nn-dimensional topological quantum field theories!

wouldn’t &&& be very important in physics too? And if so, won’t we have glimpsed it already?

Yup. :-)

A central question that is still mainly open is how undirected cocycles induce directed ones via quantization:

an (undirected) cocycle of the form

c^:BG connB nU(1) conn \hat {\mathbf{c}} : \mathbf{B} G_{conn} \to \mathbf{B}^n U(1)_{conn}

encodes an “extended” action functional of higher Chern-Simons theory. For n=3n = 3 it is known now that by some variant of quantization this induces a 3-directed cocycle Bord 33VectBord_3 \to 3Vect.

It is to be expected that for general nn, the undirected cocycle c^\hat \mathbf{c} similarly induces an nn-directed cocycle Bord nnVectBord_n \to n Vect. Lots of hints for how this should work are available. But a clear picture is still missing, as far as I am aware.

Posted by: Urs Schreiber on June 15, 2012 3:24 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

How odd. Of course, I knew that TQFTs would be an answer, since HDA0 got me into this in the first place.

Maybe it’s just when you say things like

Fundamental physics is all controled by cohomology

I forgot you meant to include by this nn-directed cohomology.

Anyway, I didn’t quite get this before:

A central question that is still mainly open is how undirected cocycles induce directed ones via quantization.

Interesting.

Posted by: David Corfield on June 15, 2012 6:16 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Anyway, I didn’t quite get this before:

A central question that is still mainly open is how undirected cocycles induce directed ones via quantization.

Interesting.

A fairly comprehensive formulation of this quantization step for discrete cocycles (meaning: no geometry, cocycles on just bare \infty-groupoids), hence, in physics lingo, for “∞-Dijkgraaf-Witten Theory” is indicated in Freed-Hopkins-Lurie-Teleman.

An attempt to deal with the general geometric case is the discussion of Prequantization in cohesive homotopy type theory that we had here before. I have some further remarks on this on p.7 here

Of course there are plenty of further bits and pieces out there in the literature. Maybe it’s just a matter of putting them together in the right way.

Posted by: Urs Schreiber on June 15, 2012 7:05 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

So is there an obstruction story to tell about nn-directed cocycles, such as Bord nnVectBord_n \to n Vect?

Posted by: David Corfield on June 16, 2012 11:59 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

So is there an obstruction story to tell about nn-directed cocycles, such as Bord nnVectBord_n \to n Vect?

Yes! There is a notion of “twisted TQFT”, sometimes called an “anomalous TQFT”, and it is the nn-directed analog of twisted cocycles in twisted cohomology, hence of situations where “there is an obstruction”. Also “holography” is related to this.

This deserves a more comprehensive reply, but I must keep it brief right now, being absorbed with something else.

By the way, David, I find it quite amazing and enjoyable how you just keep asking the next relevant question. If more people around me would recognize these questions as central…

Posted by: Urs Schreiber on June 18, 2012 2:24 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

A funny thing which has occured to me recently is that the property of being an empty space is cohomological, in that a space is empty iff it has a map to the empty space. It might be hard for a non-specialist to tell when you’ve drawn a figure of the empty space, but then what are talks for?

A fairly different one — every tangent quadrilateral satisfies the kernel-cokernel equation, and by way of a partial converse, every convex quadrilateral satisfying the kernel-cokernel equation is a tangent quadrilateral. In a similar way, every triangle fits in essentially one period-3 long exact sequence, exactly the one given by its inscribed circle — and conversely, which also gives a precise expression of the euclidean triangle inequality. I’ve been wanting something written up about exact sequences of positive real numbers; they (or pushouts of real numbers) feel like right way to think about the inclusion/exclusion principle, for instance.

Posted by: Jesse McKeown on June 14, 2012 5:29 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Sorry to go off on a tangent but you hit on something this noobie was thinking about recently - it seems that you’re asserting there’s exactly one function from the empty set to itself, something I agree with but haven’t found official word on in any literature. If one adopts this as a truth, it strongly supports the definition 0^0 := 1, since in discrete terms, n^m counts the number of functions from a set of m elements to a set of n elements. Of course, 0^0 is usually regarded as an indeterminate form. Is there some system of math where 0^0 = 1 really makes sense, in which appropriate axioms keep you from running into those inconsistencies usually associated with 0^0? Does anybody think about this?

David, great concept for a talk! Makes a fearsome subject seem approachable.

Posted by: Franciscus Rebro on June 15, 2012 12:36 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Well, some people do write “0 ^ 0 = 1”; and most mathematicians are very careful to say that this may be true or false (or nonsense) — it depends on what one means by “0”, “1”, “^” and “=”. Particularly, if one means by “^” the set of functions from the right thing to the left thing, by “0” an empty set and by “1” a singleton, and by “=” that there is an isomorphism, then … most people are happy to believe that the equation as written is correct. General point-set topology with extensional equality of functions is one context where we would agree with the equation, though we probably wouldn’t have written it that way. The empty set is almost never named “0”, nor is a singleton usually named “1”, unless you’re secretly writing von Neuman ordinals.

However, if by “0” is meant the additive-neutral real number in your favourite real field, and by “1” is meant the multiplicative-neutral element, and “^” the ordinary exponential partial function, every calculus teacher ought to tell you that “0 ^ 0” is an undefined expression, so that “0 ^ 0 = 1” not only isn’t true, it isn’t even a sentence! But it gets worse: calculus being the algebraic conclusions of analysis, one would be content with a provisional definition for “0^0” that at least made something continuous — the trouble comes down to deciding what gets to be continuous, and what doesn’t? The expression “x ^ y” is quite reasonably definable in a continuous way as long as we provide that “x” signifies an indeterminate positive real number. We might be happy to note that in this case x ^ 0 is always 1, but this is actually a BAD thing (for continuing “^” to 0 and 0) because, obviously, if y is anything positive, then taking x small enough makes x^y as small as you like, and hence well separated from 1 — but for any small x, taking y small enough makes x ^ y as close to 1 as you like; even worser, for any small negative y, taking x small enough makes x ^ y as large as you like. The upshot of which is simply that “x ^ y” can’t be made continuous in two variables all the way to (0,0).

Yes, people have thought about these things a great deal. Just be careful with yourself when you take your turn to be clear from the outset what precise context you’re thinking in — what does “=” mean, what are you assuming, etc. Happy mathing!

Posted by: Jesse McKeown on June 15, 2012 2:25 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

The empty set is almost never named “0”, nor is a singleton usually named “1”

I write “0” and “1” all the time to mean the initial and terminal objects of a category.

Posted by: Mike Shulman on June 15, 2012 6:45 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Fair enough!

Posted by: Jesse McKeown on June 15, 2012 4:31 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks for the informative reply!

Posted by: Franciscus Rebro on June 15, 2012 8:06 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

A funny thing which has occured to me recently is that the property of being an empty space is cohomological, in that a space is empty iff it has a map to the empty space. It might be hard for a non-specialist to tell when you’ve drawn a figure of the empty space, but then what are talks for?

I like that example. \emptyset ist the moduli stack for empty structure.

It also fits nicely with the cocycles-are-just-HoTT-functions-picture: in logic this “funny thing” is a standard observation about the interpretation of negation (scroll down). The negation of a proposition ϕ\phi, regarded as the collection of its proofs is ϕ\phi \to \emptyset. A cocycle p:ϕp : \phi \to \emptyset is a proof that ϕ\phi is false.

Posted by: Urs Schreiber on June 15, 2012 7:20 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I didn’t mention (because it didn’t seem important) that the context where I noticed this again was aiming to define the is_inhab type for HoTT — that is, to minimize the axioms about it needed; and this leads one to write coq definitions like

Definition is_empty (X : Type) := X -> False.
Definition is_inhab (X : Type) := is_empty (is_empty X).

Definition inhab { X : type } : X -> is_inhab X := fun x f => f x.

Lemma is_inhab_paths { X : Type } : forall x y : is_inhab X, x == y.
...

For that last one you need funext; and it seems that to re-deduce something like is_inhab_rect one has to assume

Axiom inhab_prop_choice : forall X : Type, is_prop X -> is_inhab X -> X.

that there are no nontrivial proper subsingletons. Anyways; yes, the double-negation monad was very much in mind when that all got written.

Posted by: Jesse McKeown on June 15, 2012 4:53 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

With your definition of is_inhab, your inhab_prop_choice is just the law of double negation for h-props, and thus equivalent to the law of excluded middle for h-props. But it’s cute that in the Boolean case, at least, you can define is_inhab in that way. (In the non-Boolean case, your definition of is_inhab can’t be right, because not every h-prop would be a fixed point of it.)

Posted by: Mike Shulman on June 15, 2012 7:14 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

There I go revealing my Boolean bias again… the next naïve question I would ask is whether there is (or ought to be) an hprop FfFf (for “fake false”) such that X(XFf)Ff X\mapsto (X \to Ff) \to Ff is your is_inhab monad there. I’m half afraid that this will lead to talk of generic filters, while all the while I feel no compunction about presuming to pick one of not-none essentially-uniquely-identifiable things.

Posted by: Jesse McKeown on June 16, 2012 12:08 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

No, you can’t have a “fake false” that behaves that way either. For any proposition FF, we have F(XF)F F \vdash (X \to F) \to F so if FF is not the true false \bot, then the operation (()F)F((-)\to F) \to F doesn’t fix \bot.

Posted by: Mike Shulman on June 16, 2012 1:09 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I remember a frivolous discussion over coffee a very long time ago which imagined a sort of “categorical purity test”, that would be a list of statements asking you which of them are part of your mathematical consciousness. We never did the work to come up with a good continuum of such statements, but the conversation stuck in my mind because of the one example (which might have been the one that started the whole conversation): “The center of a set is ‘true’.”

Now I finally have another statement to go along with that one: “Falsity is a cocycle.” (-:

Posted by: Mike Shulman on June 15, 2012 7:09 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

A fairly different one — every tangent quadrilateral…

Jesse, could you point me to something about this being cohomological?

Posted by: David Corfield on June 15, 2012 9:01 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Well, perhaps it’s more about homological algebra than cohomology as-such (which I see now is broadly taken as meaning local-to-global questions); it’s an exaggerated way of saying that the two pairs of opposites sides have the same total length.

It can also be taken as an example of parity distinctions relating to choices available : one can choose the point of tangency (on the shortest side) by adjusting the angles of the quadrilateral. There is no choice in a triangle; I’ve just checked that similarly one can adjust a pentagon to be circle-tangent, but again one has no choice on the points of tangency — because the cyclic order on the sides interacts with the parity of sides.

Take or ignore the family of puzzles as you will.

Posted by: Jesse McKeown on June 15, 2012 4:28 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Well I’ve never noticed that before. So a quadrilateral may be inscribed in a circle if opposite angles are supplementary, so each pair sums to the same value. While a quadrilateral has an inscribed circle if opposite sides sum to the same value. Is this a shadow of some projective duality?

Posted by: David Corfield on June 19, 2012 8:46 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I don’t think so, or at least I’m not sure; perhaps a difference worth noting is that a quadrilateral is inscribed as soon as one pair of opposite angles are supplementary; there isn’t a condition you can impose on one pair of opposite sides’ lengths in a quadrilateral that entails the same condition for the other pair of sides; one seemingly must talk of all four sides at once. Anyways, the best way I’ve found to see it is

  • through any point on a given circle there is a unique tangent line
  • through any point PP outside a given circle there are two tangents; let the tangents meet the circle in QQ and RR resp. Then PQPRP Q \equiv P R. (Pythagoras, one way)
  • the length |PQ|=|PR|| P Q | = | P R | determines the distance from P to the circle. (Pythagoras, another way)

Then, considering an nn-gon tangent to some circle, one can specify n1n-1 of the sides and one point of tangency as a distance along one of the specified sides — there is a system of inequalities to consider too, but leave that be for now. Then all the other points of tangency are determined, as well as the decomposition of the nnth side. In the exact sequences language, one has a sequence of spans in the positive reals

s 1<min(A 1,A 2);;s n2<min(A n2,A n1) s_1 \lt \min (A_1, A_2) ; \dots ; s_{n-2} \lt \min (A_{n-2}, A_{n-1})

and equations (“exactness”)

A 2=s 1+s 2;;A n2=s n3+s n2 A_2 = s_1 + s_2 ; \dots ; A_{n-2} = s_{n-3} + s_{n-2}

which determine the initial “kernel” s 0=A 1s 1, s_0 = A_1 - s_1, and s n1=A n1s n2 s_{n-1} = A_{n-1} - s_{n-2} the final “cokernel”; so the remaining side is s 0+s n1 s_0 + s_{n-1} . To complete the construction of the tangent polygon, it remains only to specify the radius of the tangent circle; that there is such a circle, as soon as nn is at least 3, is essentially an application of the intermediate value theorem.

Anyways, the most cohomology-like-thing I can see (beyond the unnecessary exact sequence language) is the odd/even sides dichotomy — in both cases there are equations imposed; but in the even case, they say what the various sides can be, while in the odd case they say where the tangency can be — so there are obstructions and ambiguities in an integration problem; in this case they happen to be complementary.

Posted by: Jesse McKeown on June 19, 2012 4:26 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

The reason I mentioned projective geometry is that if you take the circle circumscribing a cyclic quadrilateral as the conic which allows you to relate pole and polar, then the dual of the quadrilateral will be a tangent quadrilateral.

Posted by: David Corfield on June 20, 2012 5:46 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

The original reason for my scepticism was the very metric feel of the vanishing alternating-side-length-sum equation; that and the fact that the dual of a circle tends not to be a circle anymore. To be sure, conic-tangent nn-gons are worth studying, but there isn’t much I can say about their sides; maybe in re-projecting the conic to a circle the cross-ratio will rescue us? So, perhaps! I should hate to push through all that, just the same.

It does turn out easy enough to check that the alternating sum of angles in a cyclic 2n2n-gon is zero (or equivalently, of the external angles), that this is one of only two equational constraints, and similarly that the shape of a cyclic (2n+1)(2n+1)-gon is determined by its angles; the key is that peripheral angles are proportional to the arcs they subtend.

Posted by: Jesse McKeown on June 20, 2012 4:20 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

David asks

Jesse, could you point me to something about this being cohomological?

I have to second that: I don’t understand yet how that geometric example is about cohomology.

Is this maybe supposed to be about cohomology with coefficients not in a group but in something like a tropical ring? I am trying to look at it from this angle, but I still don’t quite see it.

Is this just meant as something that “vaguely feels alike” involving exact sequences, or is the claim that there is some actual exact sequence of sorts playing a role?

Posted by: Urs Schreiber on June 20, 2012 5:02 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

As I’ve already apologized, I do rather see now that what was primarily intended by “cohomology in everyday life” at first doesn’t quite figure in the family of theorems I had in mind; at the same time I do suspect that the notion of “exact sequence of positive reals” I’ve alluded to can be put more precisely analogous to what we’re all used to from homological algebra than my grasp of the relevant language permits. Since that point in the conversation (though I didn’t say so) I’ve restricted myself to trying to answer David’s naturally tangential (… er… ahem…) questions.

I’m afraid I also don’t know how to answer your question about tropical rings! Perhaps that’s what I’m missing, though it would be a surprise… In any case I certainly don’t mean to make my confusions contagious, and you may feel free to ignore me in any case.

Posted by: Jesse McKeown on June 21, 2012 12:15 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

If a Kan lift is some best approximation to lifting a morphism through another, is there a cohomological account of it? I.e., does cohomology measure how ‘good’ the approximation is?

Then there’s a dual account of all this, linking homotopy to obstructions for extensions?

Posted by: David Corfield on June 19, 2012 8:57 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Back in 1974, the Physicist R. H. Atkin wrote a whole book, Mathematical structure in human affairs that applied simplicial homology theory to the social sciences (including geography). He is remembered for introducing the concept of q-connectivity as a structural metric.

Posted by: James Juniper on June 21, 2012 11:33 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks. I see Atkins developed Q-analysis in papers including

Atkin, R. (1972). From cohomology in physics to q-connectivity in social science. International Journal of Man-Machines Studies vol. 4, 341–362.

Posted by: David Corfield on June 22, 2012 6:22 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Some of the ideas of Q-analysis have been usefully applied in sensible situations, but there is also some feeling that there was excessive enthusiasm for this theory in some social science circles (see J. Legrand – “How far can Q-analysis go into social systems understanding” ,
in Fifth European Systems Science Congress, 2002).

Recently there has been a renewal of interest in these ideas from the point of view of combinatorics as H. Barcelo and R. Laubenbacher – “Perspectives on A-homotopy theory and
its applications” , Discrete Math. 298 (2005), no. 1-3, p. 39 – 61, is related to it.

Atkin’s theory was based on one of my favorite results, due to Dowker, that the nerve of a relation and of the opposite relation have the same homotopy type. This was proved by Dowker in order to explain the fact that the Cech and Vietoris definitions of (co)homology relative to an open cover are isomorphic.

Posted by: Tim Porter on June 22, 2012 7:20 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Another nice example of cohomology in everyday life is of course differential forms in the incarnation of electromagnetism.

To show that a cohomology group is nonzero, we need a witness. And that is what nature does. For example, to witness the fact that the 1st De Rham cohomology group of the punctured plane is nonzero, shove a wire through the paper and run an electric current through the wire. The fact that a magnetic field wraps around the wire (the “right hand rule”, which schoolkids are pretty much okay with) essentially witnesses the fact that the first De Rham cohomology is nonzero.

Posted by: Bruce Bartlett on June 22, 2012 8:36 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Bruce, I’m interested in following this further. Can you point me to a simple reference as a starter? Thanks.

Posted by: Charlie Clingen on June 23, 2012 12:58 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

If we count applications to fundamental physics (electromagnetism and other fundamental fields) as being “in everyday life” (which of course they are in one sense, but not in another), then there are loads and loads of examples of cohomology. Most all of fundamental physics can be neatly conceived of entirely in terms of cocycles and cohomology.

The fact that Bruce mentions is closely related to the fact that Maxwell’s equations identify, in the absense of magnetic current, the electromagnetic field strength as, in particular, a cocycle in degree-2 de Rham cohomology.

So already in as far as electromagnetism is everywhere (we are having this communication here via electromagnetism) cohomology is everywhere.

I’m interested in following this further. Can you point me to a simple reference as a starter?

You might enjoy looking at Frankel’s The Geometry of Phyiscs (contents). The effect that Bruce just mentioned is section 16.4.f there.

Posted by: Urs Schreiber on June 23, 2012 4:20 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks,Urs!

Posted by: Charlie Clingen on June 23, 2012 5:11 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks from me too, Urs. I also learnt from your answer.

Posted by: Bruce Bartlett on June 24, 2012 8:58 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

David wrote:

Following this up, I see Santiago García in Hidden invariance of the free classical particle writes that mass “has a cohomological significance, it parametrizes the extensions of the Galileo group.” Is this an interesting point of view?

Yes! The locus classicus for this viewpoint may be Guillemin and Sternberg’s book Symplectic Techniques in Physics.

Briefly: in classical mechanics, the Galilei group acts on the symplectic manifold of states of a free particle. But in quantum mechanics, we only have a projective representation of this group on the Hilbert space of states of the free particle. The cocycle is the particle’s mass.

Switching to a much more lowbrow way of talking: you can’t see the mass of a free classical particle by just watching its trajectory, since it goes along a straight line at constant velocity no matter what it’s mass is. But you can see the mass of a free quantum particle, because its wavefunction smears out faster if it’s lighter! So there’s some difference between classical and quantum mechanics. Ultimately this arises from the fact that the latter involves an extra constant, Planck’s constant.

Working out how the last two paragraphs are related is a fun exercise in taking some ideas from cohomology and seeing what they amount to in ‘real life’.

Posted by: John Baez on June 25, 2012 5:34 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

In slight disguise, one can see this cocycle also control already the classical free non-relativistic particle, in the sense that its action functional is of the form of a 1d WZW model with that cocycle being the “WZW term” that however comes down to be the ordinary free action.

Posted by: Urs Schreiber on June 25, 2012 5:59 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

That’s sounds like a good exercise for the Summer. Understand Guillemin and Sternberg’s point of view, and then see how to to understand the cocyle in terms of the WZW model.

First step, retrieve book from the library.

Posted by: David Corfield on June 26, 2012 11:44 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Hi All,

I have an example from basic thermodynamics. Recall there is an abstract notion of systems A and B being in thermal equilibrium. The first law states that this relation is transitive. Another perspective on the first law is that temperature is an exact 1-form on the complete graph of thermal systems.

A priori, we’re allowed to define temperature for some system A with respect to system B. Call this T_A/B. Then we can also define T_B/A. Temperature is so defined so that T_A/B = T_B/A iff A and B are in thermal equilibrium. Roughly speaking, T_A/B defines a 1-form on the complete graph of thermal systems. What the first law says is that this 1-form is the differential of a temperature function defined for each thermal system (not just relative to a second thermal system).

This perspective on the first law I learned from Kardar’s statistical physics of particles chapter 1.

Posted by: Ryan Thorngren on June 27, 2012 10:54 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks. With all those differential forms around, you’d think cohomology would have a natural place in thermodynamics.

What’s “the complete graph of thermal systems”?

By the way, to display symbols you need to choose a Text Filter in the reply form with ‘itex’. This behaves almost the same way as LaTex.

Posted by: David Corfield on June 28, 2012 8:47 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Thanks for the tip.

By the complete graph of thermal systems, I mean the graph whose vertices are thermal systems and with edges between every two vertices. The notion of cohomology I am using is just simplicial cohomology of this graph.

By the way, I should have said T A/B=0T_{A/B}=0.

Posted by: Ryan Thorngren on June 28, 2012 6:51 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Entropy, but I think we never quited nailed this.

there is a new note of Misha Gromov describing a category theoretic approach to entropy. I do not think he mentions cohomology, though.


In a Search for a Structure, Part 1: On Entropy. June 19, 2012

Posted by: t on June 29, 2012 8:40 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

there is a new note of Misha Gromov describing a category theoretic approach to entropy.

After a second glance at it I discovered the definition:

the article proposes to study entropy in the presheaf topos PSh(Meas fin)PSh(Meas_{fin}) over the category of finite measure spaces.

I may need a third glance to see what exactly it is that this topos-theoretic perspective achieves here. Maybe somebody who read the article in detail can help me?

I do not think he mentions cohomology, though.

Should we not really be considering the sheaf topos Sh(Meas fin,J)Sh(Meas_{fin}, J) with respect to some Grothendieck topology JJ on this site? Any idea?

If so, then we are to expect the corrsponding sheaf cohomology to play a big role.

By the way, I like the typo on p. 3:

We manipulate with spaces PP as with their underlying sets […] in-so-far as it does lead to confusion.

Posted by: Urs Schreiber on June 30, 2012 3:54 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

In the context of Bohrification we meet reductions.

Let C subset D be commutative subalgebras and f a state.

Then we have a surjection S(D) \twoheadrightarrow S(C) which preserves the measure defined by f.

Posted by: Bas Spitters on July 1, 2012 7:51 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Full-text search does not seem to find the word ‘shea’ in the paper…

Maybe v-category aka measure space on page 4? That is, the
x-fans define coverings on $P^{op}$? Or perhaps minimal fans?
I do not see what sheaf condition would mean wrt x-fans, does
it make any sense ?

Let me quote the relevant part of the paper:

Spaces over P. A space X over P is, by definition, a covariant functor from
P to the category of sets, where the value of X on P ∈ P is denoted X (P ).

Minimal Fans and Injectivity. An x-fan over $b_i$ in a category is called mini-
mal if every a between x and $\{b_i \}$ is isomorphic to x. (More precisely, the arrow
x → a that implements ”between” is an isomorphism.)


An essential feature of minimal fans, say fi Q → Pi, a feature that does not
depend on X (unlike the ∨-product itself) is the injectivity of the corresponding
(set) map from Q to the Cartesian product ∏i Pi (that, in general, is not a
reduction).

Posted by: t on July 2, 2012 2:15 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Hey t,

you write:

That is, the x-fans define coverings on P opP^{op}?

Maybe, I don’t know. I don’t really have the time to look into this right now, unfortunately (and not too much of a motivation either, to be frank: somebody should state clearly what the gain of these developments is meant to be…). But it would be good if somebody pursued this idea of Gromov’s and formulated it in the proper topos-theory language that apparently Gromov has in mind.

By the way, to make math typesetting appear as expected on this blog, you need, unfortunately, to choose, before sending a comment, above the edit pane in a pulldown menu called “Text filter” an item that contains the word “MathML”.

Posted by: Urs Schreiber on July 2, 2012 7:31 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

We ought to compare what he’s saying with all our work on entropy, such as that in Tom’s post. Come to think of it, we ought to add the latter material to the nLab entropy entry.

Are Gromov’s ‘reductions’ the same as coarse-graining or Markov morphisms, mentioned here.

No time to check now.

Posted by: David Corfield on July 2, 2012 11:31 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I feel I’d like to understand the paper better:

…the cone + I\mathbb{R}_+^I is ugly, it breaks the Euclidean/orthogonal symmetry of I.\mathbb{R}^I.

We had such a cone in view over here.

Now we have the orthogonal symmetry, even better the unitary symmetry of I\mathbb{C}^I, and may feel proud upon discovering the new world where entropy “truly” lives. Well…, it is not new, physicists came here ahead of us and named this world “quantum”. (p. 14)

Posted by: David Corfield on July 3, 2012 12:48 PM | Permalink | Reply to this

Re: Cohomology in Everyday Life

hi Urs,

I am a laymam. How does kirchoffs law come into this topic? It was mentioned at the beginning.

Posted by: anonymous on July 7, 2012 5:24 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Try John Baez’s page Circuit Theory I.

Posted by: David Corfield on July 7, 2012 8:30 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

How does kirchoffs law come into this topic?

Kirchhoff’s laws are a special case of Maxwell’s equations, obtained by a kind of coarse-graining. The nice thing about them is that they preserve the cohomological nature of Maxwell’s equations: where the latter are formulated in de Rham cohomology (i.e. involing “infinitesimal cochains”), Kirchhoff’s laws are naturally understood in terms of cochains on finite cell complexes.

This is nicely discussed in Appendix B Harmonic Chains and Kirchhoff’s Circuit Laws of Theodore Frankel’s book The Geometry of Phyiscs.

Posted by: Urs Schreiber on July 7, 2012 11:56 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

In fact, isn’t that how the Maxwell Equations were first worked out? From Faraday’s work and Gauss’ etc. it was essentially understood how magnetic and mutual induction worked in wires, and how build-up of charge separation (e.g., at capacitors) induced electric fields; and in a sense what Maxwell did was to smooth-out and abstract away the physical wires; not to mention how his introduction of the displacement current term was quite explicitly argued from what nowadays should be called a homotopical invariance principle.

Posted by: Jesse C. McKeown on July 8, 2012 1:32 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

In fact, isn’t that how the Maxwell Equations were first worked out?

Yes, I guess so.

On the other hand, the cohomological nature of Kirchhoff’s laws was probably only noticed – or at least made explicit – after that of Maxwell’s equations was understood.

Posted by: Urs Schreiber on July 8, 2012 8:42 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

I learnt of the cohomological nature of Kirkhoff’s laws in a geometric topology course in 1967. I think it was well known from a graph theoretic viewpoint well before that, so it would be interesting historically to verify the relative dates. When was `Maxwell as cohomology’ first noticed?

Posted by: Tim Porter on July 9, 2012 7:29 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

When was `Maxwell as cohomology’ first noticed?

I am not sure. One data point that I happen to have easily available right now is this:

Frankel on p. 5 here attributes it to Hargreaves, pointing for reference to that in turn to

  • E. T. Whittaker, A history of the theories of aether and electricity, Vol. 2, Harper and Brothers, New York, (1960).
Posted by: Urs Schreiber on July 9, 2012 9:47 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Ah, this is just what I need to find out.

Frankel on p. 5 here attributes it to Hargreaves…

‘It’ refers to the observation that the electromagnetic field tensor forms a skew-symmetric matrix, and so defines an exterior form. It’s probably too early to call this cohomological. He appears to be working in the early years of the 20th century. E.g., from here

A formulation of Maxwell’s equations involving integrals over two-dimensional surfaces in time-space was given by R. Hargreaves as early as 1908 (contemporaneously with the famous publications of Minkowski) in the Cambridge Philosophical Society Transactions, vol. 21, p. 116.

Of course, there will be a great deal of arbitrariness in how much to read of current understanding into early formulations.

If Weyl is telling a cohomological story of Kirchhoff in 1923, you’d think he’d realise the full electromagnetic theory could be treated this way around that time.

De Rham would be a good person to look at too.

Posted by: David Corfield on July 9, 2012 10:07 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

E.g., from here

Thanks. True, that is probably before differential forms were understood as cocycles.

According to

Weibel, History of homological algebra (pdf)

that was (p. 5 there) first written down in 1929 by Elie Cartan (and thus before de Rham’s thesis in 1931).

So at least in principle the cohomological nature of Maxwell’s equations was known to mankind by max(1908,1929)=1929max(1908,1929) = 1929. The remaining question is maybe: who, after 1929, first put the existing pieces together explicitly.

Posted by: Urs Schreiber on July 9, 2012 10:37 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Oh, but in any case, it seems to imply that the cohomological nature of Kirchhoff’s equations was known at least 6 years before that of Maxwell’s equations. So my guess to the contrary above was wrong.

Posted by: Urs Schreiber on July 9, 2012 10:44 AM | Permalink | Reply to this

Re: Cohomology in Everyday Life

Have you seen that explanation of why Weyl’s paper wasn’t noticed? It seems that Hilbert disapproved of Poincaré’s Analysis Situs, so Weyl submitted in Spanish to a Mexican journal.

Posted by: David Corfield on July 9, 2012 11:02 AM | Permalink | Reply to this

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