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January 18, 2010

The Sacred and the Profane

Posted by David Corfield

The categories mailing list recently engaged in some soul-searching as to why category theory remains such a well-kept secret and why its wonders are not better known. Many people proposed theories to explain this, while others were quick to deny the premise, pointing out that it is no longer a secret, as the mainstream work of Voevodsky, Lurie, and many others bears witness. I wonder, though, if this was quite the point at stake.

When I met up with Minhyong Kim before Christmas he made the very interesting remark that there’s a difference between treating some system of mathematics in a ‘sacred’ way and in a ‘profane’ way. This distinction was introduced by Dirk van Dalen in the Preface to his Logic and Structure:

Logic appears in a ‘sacred’ and in a ‘profane’ form; the sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. Some early catastrophes such as the discovery of the set theoretical paradoxes or the definability paradoxes make us treat a subject for some time with the utmost awe and diffidence. Sooner or later, however, people start to treat the matter in a more free and easy way. Being raised in the ‘sacred’ tradition my first encounter with the profane tradition was something like a culture shock. Hartley Rogers introduced me to a more relaxed world of logic by his example of teaching recursion theory to mathematicians as if it were just an ordinary course in, say, linear algebra or algebraic topology. In the course of time I have come to accept this viewpoint as the didactically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason this introductory text sets out in the profane vein and tends towards the sacred only at the end. (p. V)

When you see that XX can be used to achieve YY, you may react in two ways. You may say that it reflects to the greater glory of XX that YY is made possible, or you may just be grateful to XX for helping to achieve the desired YY. If I show you how my new knife cuts through some wood, I may do so to allow you to admire the sharpness of the blade, rather than because I want two pieces of wood. You may respond by asking whether the knife can do what you take to be useful. We will then feel we haven’t quite understood one another. Perhaps then something similar is going on in the difference between a sacred attitude towards (higher) category theory and a profane one aiming at other ends.

Of course, you would like to ask your profane user about the choice of those other ends. Why are you so keen to see whether my machinery can help you? At some point in our questioning we should expect to be able to proceed no further, and to hear the reply, “That is simply what I want to know.” Gian-Carlo Rota called this the ‘bottom line’.

How do mathematicians get to know each other? Professional psychologists do not seem to have studied this question; I will try out an amateur theory. When two mathematicians meet and feel out each other’s knowledge of mathematics, what they are really doing is finding out what each other’s bottom line is. It may be interesting to give a precise definition of a bottom line; in the absence of a definition, we will give some examples.

To the algebraic geometers of the sixties, the bottom line was the proof of the Weil conjectures. To generations of German algebraists, from Dirichlet to Hecke and Emil Artin, the bottom line was the theory of algebraic numbers. To the Princeton topologists of the fifties, sixties, an seventies, the bottom line was homotopy. To the functional analysts of Yale and Chicago, the bottom line was the spectrum. To combinatorialists, the bottom lines are the Yang-Baxter equation, the representation theory of algebraic groups, and the Schensted algorithm. To some algebraists and combinatorialists of the next ten or so years, the bottom line may be elimination theory.

I will shamelessly tell you what my bottom line is. It is placing balls into boxes, or, as Florence Nightingale David put it with exquisite tact in her book Combinatorial Chance, it is the theory of distribution and occupancy. (Indiscrete Thoughts, 51-52)

We could say then the mathematicians hold their bottom line to be sacred, and as such display a sense of duty in their actions towards it.

On the subject of duty, the philosopher R. G. Collingwood had this to say about choice of action,

…of the three reasons for choice [because it is useful; because it is right; and, because it is my duty], the third alone is a complete reason. Choice is always choice to do an individual action. Why do I do it? The answer ‘because it is useful’ explains only why I do an action leading to a certain end: not why, among the various possible actions which might have led to that end, I choose this and not another. The answer ‘because it is right’ explains only why I do an action of a certain kind, specified by the rule which I obey; not why, among the various possible actions conforming to that specification, I choose this and not another. But the answer ‘because it is my duty’ is a complete answer. What I do is an individual action; what it is my duty to do is an individual action; if what I do is my duty these two individual actions are one and the same. I do this and no other action because this and no other is the action it is my duty to do. (‘Duty’ in Essays in Political Philosophy, p. 151)

His idea is that there’s nothing further to be said, as when you reply to your beloved’s question “Why do you love me?”, with “Because I do.” In the case of duty, we might say “Here I stand. I can do no other.”

More radical than a shift of mathematical bottom line, would be a move out of mathematics altogether, as has cropped up regarding John’s career changing decision. One effect of watching someone wrestle with their duty is to question whether you are following your own.

Posted at January 18, 2010 12:17 PM UTC

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Re: The Sacred and the Profane

The van Dalen quote is fascinating, because I “grew up” seeing proof theory as the “profane”, practical end of logic, and model theory and categorical logic as the “sacred” form. Categorical logic is what explains wild and mysterious things like Lawvere-Tierney topologies on categories of sheaves, whereas proof theory is what we use to design languages to keep web browsers from crashing. :)

What’s interesting is that how naturally my profane viewpoint arises naturally out of the philosophical ideas of Martin-Lof, which are explicitly sacred in motivation (albeit a different one than the categorical logicians). He explains his judgmental methodology as an attempt to reject the Fregean view of logic as being about truth only, and to return to the older Aristotelian vew of logic as being about all of reasoning, broadly construed. (He wants to do this while retaining all the technical advances of the Fregean revolution, of course.)

But when you start designing logical judgements to talk about how computer programs move around networks, this can’t help but demystify the machinery of logic, even though it might be “sacredly” justified as an example of bringing life and reason into closer harmony.

(As an aside, Martin-Lof’s judgmental methodology is something that has proven resistant to explanation via the standard tricks of categorical proof theory. I know Peter Lumsdaine has talked to Bob Harper about this, but I don’t know if they ever found a good story.)

Posted by: Neel Krishnaswami on January 18, 2010 1:57 PM | Permalink | Reply to this

Re: The Sacred and the Profane

One effect of watching someone wrestle with their duty is to question whether you are following your own.

Do you remember why you chose to go to college and study mathematics or physics or philosophy? I remember Stephen Hawking saying something like “physics was the topic that the smart guys chose” - if I remember correctly the question was “why not biology or medicine?”. I think he meant to say that physics was considered to be a topic for the people who could understand things, and biology and medicine were for people who could not, but who were good at memorizing facts (that’s a prejudice that I have heard from time to time, too).

The first years of my time at university I was very enthusiastic about the correlations of mathematics and physics (the concurrent development of the concepts of KMS states in statistical physics and modular theory for von Neumann algebras is a good example of this “prestabilized harmony”), which made me visit a lot of mathematics classes. But later I grew more and more disappointed at the lack of impact and interest that anything that was produced at the math and physics departments had/got - please don’t be offended, this was the highly subjective, irrational feeling I had at that time. That’s the main reason I left academia and took a job as a software developer.

But for some reason I don’t quite understand I kept and keep thinking about various topics in mathematical physics - eventually I tried to understand a bit about quantum gravity and string theory, mainly because there was a promise of the justifiable use of sophisticated mathematics. Well, if you want to learn some string theory you have to sip at algebraic topology and geometry, and this means you cannot escape category theory and voilà , there I am.

But I think it is not quite unusual to do one job for a living, and another one you are really interested in, that means more to you than just a hobby.

Posted by: Tim vB on January 18, 2010 2:57 PM | Permalink | Reply to this

Re: The Sacred and the Profane

This sacred/profane dichotomy could be used to frame a phenomenon which I’d imagine has affected most fields of mathematics. Tim Gowers’s essay on the two cultures, much loved on this blog, discusses Banach space theory in a similar light. The example that comes to my mind most readily is algebraic topology. I only know a little algebraic topology and learned it from the profane point of view that it exists to solve certain kinds of problems. I know a number of algebraic topologists for whom the subject presumably has a more sacred character, but I personally am baffled as to what motivates the esoteric things they study.

Posted by: Mark Meckes on January 18, 2010 3:32 PM | Permalink | Reply to this

Re: The Sacred and the Profane

A lot of algebraic topology is all about the beautiful ways that spaces (or more precisely, homotopy types) can be seen as algebraic structures. The dream would be to understand spaces in a completely thorough way, algebraically. Pretty much all of algebra winds up getting used in pursuit of this goal. This quest is a bit of a ‘holy grail’, in the sense that nobody really seems to think we’ll ever reach the end — but nonetheless, our understanding is transformed by the process.

Posted by: John Baez on January 18, 2010 10:07 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I just can’t forget a conversation between two topologists at lunch one day, arguing about whether some algebraic-topological statement was true or not. Topologist A insisted that it was nearly trivial, but topologist B inisisted that it couldn’t possibly be true. After several minutes of this it transpired that A was only making this claim about “spaces”, to which B replied that well, yes, of course it’s true for spaces. The nontopologists in the room responded to this with various mixtures of laughter and incredulity.

I readily grant that although I’m not familiar with them, there are good reasons with solid mathemtical motivations behind them that such a conversation among experts may make perfect sense. But hearing that conversation didn’t do much to dispel laymen’s impressions of an esoteric field which has moved far beyond its historical roots.

NB: I don’t mean this as criticism of algebraic topology! I think my topologist friends are probably equally baffled as to what motivates the things I’m interested in.

Posted by: Mark Meckes on January 19, 2010 2:54 PM | Permalink | Reply to this

Re: The Sacred and the Profane

You could imagine an argument between A and B on whether or not every bounded set of numbers is finite, concluding with ‘Well of course it’s true if you’re talking about *natural* numbers!’ This would have been in the days when other kinds of numbers were still esoteric. Now that I think of it, the analogy with [spaces vs. spectra] seems quite good.

Posted by: Minhyong Kim on January 21, 2010 1:15 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Well, one interesting question is this: what entities, more general than ‘spaces’, was topologist B concerned with?

Spectra, perhaps? You’ll find plenty of topologists interested in these. And indeed, many statements that are trivially true for spaces are false for spectra. On the other hand, many problems become much easier for spectra! And that’s why they were invented: they are like spaces, but with a lot of the tricky wrinkles smoothed out.

I think that spectra are one of those things that’s bound to seem esoteric to people who aren’t algebraic topologists.

Mike Shulman’s advisor, Peter May, is one of the great gurus when it comes to spectra!

Posted by: John Baez on January 21, 2010 8:01 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Well, one interesting question is this: what entities, more general than ‘spaces’, was topologist B concerned with?

That is a good question. The fact that the rest of us never knew is what made the experience funny. Otherwise it would have been merely educational.

Posted by: Mark Meckes on January 21, 2010 2:41 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Mircea Eliade’s The Sacred and The Profane (1957) is not about Math, but is (I suspect) the origins of this repurposing. Mircea Eliade, in analyzing the Sociology and Anthropology and Metaphysics of Reklgion, describes two fundamentally different modes of experience: the traditional and the modern. He has a Triality beyond the Duality.

A Synopsis of Eliade’s The Sacred and the Profane:

“Crucial to an understanding of Eliade’s The Sacred and the Profane are three categories: the Sacred (which is a transcendent referent such as the gods, God, or Nirvana), hierophany (which is the breakthrough of the sacred into human experience, i.e. a revelation), and homo religiosus (the being par excellence prepared to appreciate such a breakthrough). One of Eliade’s aims is to acquaint readers with the idea of the numinous, a concept provided in Rudolf Otto’s The Idea of the Holy. The numinous experience is that experience of the Sacred which is particular to religious human beings (homo religiosus) in that it is experientially overwhelming, encompassing the mysterium tremendum et fascinans, both the awesomely fearful and the enthrallingly captivating aspects of the Holy, or, the Wholly Other . In expanding and expounding the phenomenological dimensions of the Sacred, Eliade points out that the Sacred appears in human experience as a crucial point of orientation at the same time it provides access to the ontological reality which is its source and for which homo religiosus thirsts. According to Eliade, homo religiosus thirsts for being. In terms of space, the Sacred delineates the demarcation between sacred and profane and thus locates the axis mundi as center.”

Posted by: Jonathan Vos Post on January 18, 2010 6:30 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Minhyong also mentioned the sacred and the profane in October, and said something about how it pertained to the changing perceptions of category theory (and similarly, I’d add, all kinds of ‘abstract mathematics’).

Posted by: Tom Leinster on January 18, 2010 10:49 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Ah yes. I’d missed that. To repeat part of it:

My feeling is that many mathematicians’ view of category theory went through a similar transition in the course of our times, perhaps the last several decades. Regardless of what input there was from philosophy, it’s hard to avoid the impression that Maclane’s categories were somewhat on the sacred side, conjuring up images of whole universes of some profound sort, and a functor was really an awful transformation. (Consequently, one couldn’t allow too many of them.) These days, a typical category looks much more like a graph.

This perspective will probably have matured when people think nothing of defining single-use categories and functors for solving concrete problems, pretty much in the way that an analyst cooks up an auxiliary function to prove a bound.

There isn’t the idea here though, unlike in the van Dalen quotation – “this introductory text sets out in the profane vein and tends towards the sacred only at the end”, of a return to the sacred after a trip through the profane.

Even if you start using category theory for very practical concrete problems, I would still say there’s a difference of degree as to how much you see it as a useful tool and how much you take its success to reflect the glory of category theory. There are still people whose ‘bottom line’ is category theory, who will be more inclined to take categories seriously.

Posted by: David Corfield on January 19, 2010 9:00 AM | Permalink | Reply to this

Re: The Sacred and the Profane

“Sacred” in mathematics obviously refers to “platonic” in contrast to other parts of mathematics. The reference to Eliade fits very well, because “platonism” finally goes back to “shamanism” about which Eliade wrote. Thomas McEvilley wrote a fascinating book on ancient greek platonism, orphism etc. relating it with ancient indian and other traditions (“The Shape of Ancient Thought”), in which he brilliantly gives an account of the connection platonism/shamanism and how that developed. E.g. when Yuri Manin writes about “mathematics chooses us” and “emotional platonism”, that are the main characteristica of shamanism: shamans did not chose their profession, neither did their skills restrict to learned handycraft independent from “emotion”. When K. Kato writes about “Mysterious properties of zeta values seem to tell us (in a not so loud voice) that our universe has the same properties: The universe is not explained just by real numbers. It has p-adic properties … We o u r s e l v e s may have the same properties.”, this fits precisely into the shamanistic way of thinking (“knowing something is becoming it”). But does that really fit to bureaucratized science since WW2? I think, since WW2 mathematics slowly changes in a similar fashion as remarked in McEvilley’s book on the end of shamanism when religious bureaucraties emerged. Even the homeopathic dose of platonism in “intrinsic motivation/curiosity in the things for themself” may become unusual. Perhaps the majority of mathematicians feels towards remnants of platonistic attitutes like career-priests in temple bureaucracies towards the wandering orphics, fakirs, sorcerers etc.?

Posted by: Thomas on January 18, 2010 11:58 PM | Permalink | Reply to this

Re: The Sacred and the Profane

No wonder I feel like a wandering sorcerer.

Posted by: John Baez on January 19, 2010 7:08 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Where does Kato say that?

Posted by: David Corfield on January 19, 2010 9:51 AM | Permalink | Reply to this

Re: The Sacred and the Profane

“Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. Part I”, page 159. Lots of other fun Katoisms in that paper.

Posted by: James on January 19, 2010 10:17 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Unfortunately we don’t subscribe. But I can see the first page:

As the night sky, mathematics has two hemispheres: the archimedean hemisphere and the non-archimedean hemisphere. For some reasons, the latter hemisphere is usually under the horizon of our world, and the study of it is historically always behind the study of the former.

Great opening!

Posted by: David Corfield on January 19, 2010 10:56 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Kato ends with

Mysterious properties of zeta values seem to tell us (in a not so loud voice) that our universe has the same properties: The universe is not explained just by real numbers. It has pp-adic properties (as is claimed by some people in physics) … We ourselves may have the same properties.

Are there physical meanings of zeta elements?

This is reminiscent of Manin in ‘Reflections on Arithmetical Physics’

On the fundamental level our world is neither real, nor pp-adic; it is adèlic. For some reasons reflecting the physical nature of our kind of living matter (e.g., the fact that we are built of massive particles), we tend to project the adèlic picture onto its real side. We can equally well spiritually project it upon its non-Archimedean side and calculate most important things arithmetically.

The relation between “real” and “arithmetical” pictures of the world is that of complementarity, like the relation beween conjugate obsrevables in quantum mechanics.

Is there any sense in which they are possibly right? Manin adds, “Of course, one is not obliged to take this metaphysics seriously.”

Can you see purely from the pp-adic side that the other completion of the rationals is going to be Archimedean and ordered?

Posted by: David Corfield on January 22, 2010 3:31 PM | Permalink | Reply to this

Re: The Sacred and the Profane

The society which scorns excellence in plumbing because plumbing is a humble activity, and tolerates shoddiness in philosophy because philosophy is an exalted activity, will have neither good plumbing nor good philosophy. Neither its pipes nor its theories will hold water.

- John W. Gardner, Secretary of Health, Education, and Welfare under President Lyndon Johnson (1912-2002)

Posted by: jim stasheff on January 23, 2010 1:12 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Excellent! I’ll suggest to the department that we adopt it as our motto.

We might say that Kato and Manin are projecting outwards (none too carefully) the profound experience of number theorists, expressed very nicely by Minhyong here:

Given the discussion above, one might expect all mathematicians to be number theorists, and maybe in some sense they are. However, there is an even more essential reason why most practicing mathematicians can get by with a rather naive understanding of numbers and might be better off doing so. This is because of the extremely useful geometric picture of the real and complex numbers. Much of the time, it is perfectly reasonable to visualize the complex numbers as a geometric plane, and base all other constructions upon that basic picture, oblivious to the fine structure of our objects, pretty much as one can do plenty of classical physics without worrying about the fact that the macroscopic objects we are considering arise from the complicated interaction of elementary particles. Now, my claim is that the role of a number theorist in mathematics is exactly analogous to the role of a particle theorist in physics. That role being to probe the nature of the ultimate constituents of the objects that others study from a far coarser perspective. Thus, in contrast to the continuum picture of the complex plane, a number theorist is more likely to perceive of each individual number or groups of numbers in a discrete fashion, and in nested hierarchies reflecting various complexities, and even attach a symmetry group to each individual number. It is not that number theorists avoid the plane model, since it is also an important tool in much of number theory. It is just that the plane has a much more grainy and elaborate shape, with many levels of microscopic detail and structure.

As for the physics side of things - Why do we appear to live in a real manifold? - I couldn’t imagine how to proceed.

Posted by: David Corfield on January 23, 2010 9:30 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Why do we appear to live in a real manifold?

I thought it was because we invented the notion of ‘real manifold’ with the space we live in in mind.

Posted by: Mike Shulman on January 23, 2010 11:32 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I meant rather why do we appear to live in the kind of space which we describe as a real manifold - the physics question to which Manin hints at an answer in terms of our constitution by massive particles.

To the question “Why do I live in England?” I don’t want the answer “Because that’s what they call the place where you live”, but rather “Your recent ancestors lived there, and you have chosen not to emigrate”.

Posted by: David Corfield on January 24, 2010 12:09 AM | Permalink | Reply to this

effective physical space

why do we appear to live in the kind of space which we describe as a real manifold - the physics question

Depending on what you allow the answer to presuppose, there is a good answer to this.

Let me assume that we do allow ourselves to presuppse that whatever particles we consider about which we wonder what they may perceive as the ambient space that they inhabit, are particles whose dynamics is determined by quantum mechanics.

Such dynamics then is encoded in a suitably well behaved symmetric monoidal functor 1Cob Riem λVect1Cob_{Riem}^\lambda \to Vect from the symmetric monoidal category generated from 1-dimensional Riemannian cobordism and one trivalent vertex (where three intervals touch in one singular point).

Any such functor determines an effective background geometric space which is such that the energy spectrum of the particle encoded by the functor is that one would expect of particles propagating in that geometric space. In this sense this geometric space is the space as perceived by the internal state of the particle.

Namely one shows (as discussed here) that any such functor is determined by three pieces of data:

  • its value on the single generating object: this is a vector space AA;

  • its value on the trivalent vertex, this is a linear map AAAA \otimes A \to A equipping AA with the structure of a unital and associative algebra;

  • its value on the Riemannian interval of infinitesimal length: this is a self-adjoint operator 1+Δ1 + \Delta on the Hilbert space completion AHA \hookrightarrow H of AA (or the like, depending a bit on technical details that I am glossing over).

Together this data is called a spectral triple. A well developed theory of spectral triples shows that they are formal duals to generalized Riemannian manifolds traditionally called (somewhat imprecisely) “noncommutative spectral geometries”. This is the effective geometry of ambient space as seen by the particle encoded by our functor.

Not every such effective geometry is a real manifold. But there is a large scale limit describing the geometry as seen not by a single one of our particles but by a whole condensate of them, in which these geometries do look like real manifolds.

This way physical space emerges from pure abstract nonsense.

You might complain that I didn’t use full abstract nonsense, as 1Cob Riem λ1Cob_{Riem}^\lambda involves quite a bit of by-hand input – notably the notion of 1-dimensional real space!. But that’s to some degree just for simplicity of presentation here and can be improved upon:

First we can get rid of the by-hand insertion of the interaction vertex by passing to 2Cob Riem2Cob_{Riem} or higher: 2-dimensional cobordisms automatically have the required trivalent interaction: called the trinion or pair-of-pants.

A nice discussion of how such 2-functors give rise to effective spectral geometries as indicated above should appear in the upcoming book

Mathematical foundations of quantum field and perturbative string theory

as the contibution by Yan Soibelman, that you are probably all already looking forward to.

But secondly we can also get rid of the Riemannian structure on the cobordisms and consider just topological cobordisms. We may think of functors on these as obtained from averaging over all Riemannian metrics in a suitable sense, where the suitable sense boils down to an integral over the compactified moduli space of Riemann surface structures.

This leaves us just with the (,2)(\infty,2)-category Bord 2Bord_2. And that is, as you know, despite its superficial appearance, something determined by pure abstract nonsense.

Posted by: Urs Schreiber on January 24, 2010 3:23 PM | Permalink | Reply to this

Re: effective physical space

Wow, thanks! So no sign of the rest of Manin’s adelic world?

Your comment about Bord 2Bord_2 brings me back to something I’ve been wondering for a while. Can we see from its abstract nonsense description how the reals (as topological space or smooth space) emerge? Are they implicit in the very notion of a morphism?

Posted by: David Corfield on January 25, 2010 10:01 AM | Permalink | Reply to this

Re: effective physical space

So no sign of the rest of Manin’s adelic world?

I have to admit that I do not feel I understand the meaning of that quote by Manin. I understand the individual terms that he uses, but I don’t see what connection between them he is meaning to indicate. I haven’t tried to look at the context that it comes from, though. Maybe you can say in your words what it is that you think Manin is saysing? And why you find it interesting?

something I’ve been wondering for a while. Can we see from its abstract nonsense description how the reals (as topological space or smooth space) emerge? Are they implicit in the very notion of a morphism?

I might be wrong, but I feel inclined to think this is another incarnation of the familiar fact how manifolds may also model topological spaces, hence \infty-groupoids.

A good fact to keep in mind for unravelling the way the notion of space and higher category interact is this:

the collection of those \infty-Lie groupoids (= \infty-stacks on the category of smooth manifolds) with the property that

  • their generalized smooth/diffeological structure is such that the \infty-groupoid of plots by a manifold XX is equivalent to the \infty-groupoid of plots by the smooth manifold X×IX \times I

are equivalent to topological spaces!

This is discussed and referenced at topological \infty-groupoid.

So this means that every \infty-groupoid and every topological space may be thought of as an \infty-Lie groupoid modeled on smooth manifolds.

Using this identification, you can form for any topological space XX – after regarding it as an \infty-Lie groupoid! – its smooth path \infty-groupoid Π(X)\Pi(X).

This is closely related to forming Sullivan differential forms on a topological space: we first regard it as an \infty-Lie groupoid and then take ordinary differentials forms of that (“of the \infty-stack” as many like to say).

But now notice that Π(X)\Pi(X) is much like the \infty-groupoid Bord (X)Bord_\infty(X) of bordisms equipped with a map to XX: heuristically, Π(X)\Pi(X) consists of all those cobordisms that are topologically trivial (nn-disk-shaped). It contains just paths no loops, no spheres.

I expect that Bord (X)Bord_\infty(X) can be refined to a \infty-Lie groupoid, too, and that we then have an inclusion

Π(X)Bord (X), \Pi(X) \hookrightarrow Bord_\infty(X) \,,

but I haven’t really tried to construct this concretely.

Then we have the canonical projection

Bord (X)Bord (*)=Bord Bord_\infty(X) \to Bord_\infty({*}) = Bord_{\infty}

and we can ask question like wether a local system :Π(X)Mod\nabla : \Pi(X) \to \infty Mod extends through this somehow

Π(X) Mod exp(S ()) Bord (X) [,X]exp(S ()) Bord , \array{ \Pi(X) &\stackrel{\nabla}{\to}& \infty Mod \\ \downarrow & \nearrow_{\exp(S_\nabla(-))} \\ Bord_\infty(X) \\ \downarrow & \nearrow_{\mathrlap{\int_{[-,X]}\exp(S_\nabla(-))}} \\ Bord_\infty } \,,

(where I added labels only to tease my readers).

In such a kind of picture, with the above equivalence between homotopy invariant \infty-Lie groupoids and topological spaces in mind, manifolds and \infty-categories interact by determining each other, back and forth.

I don’t know if that explains anything, but when I regard it from this perspective, the fact that Bord Bord_\infty is something like a free abstract nonsense guy on one generator feels less mysterious than otherwise.

Posted by: Urs Schreiber on January 25, 2010 4:57 PM | Permalink | Reply to this

Re: effective physical space

Manin starts from the observations that it makes number theoretic sense to take the whole family of completions of \mathbb{Q} simultaneously in the shape of the topological ring of adèles, A A_\mathbb{Q}.

For example, if you normalize the left-invariant measure on SL 2(A )SL_2(A_\mathbb{Q}), requiring that the integral over SL 2(A )/SL 2()SL_2(A_\mathbb{Q})/SL_2(\mathbb{Q}) be 1, then decomposing over the valuation places (the primes and \infty) gives you Euler’s

1=π 2/6× p(1p 2). 1 = \pi^2/6 \times \prod_p (1 - p^{-2}).

He then speculates as to whether adèles are showing themselves in string theory. For instance,

I conjecture that one can define on the space of adèlic points of a universal moduli space an adèlic Polyakov measure whose Archimedean component is the ordinary Polyakov measure. hopefully, the corresponding total adèlic volume would … giv[e] an arithmetic expression of the string partition function.

I haven’t a clue how promising this is. I see Marcolli is continuing to think along these lines in her Number Theory and Physics in the section ‘Speculations on arithmetical physics’. It ends by pointing to motives:

A similar adelic philosophy was taken up by other authors, who proposed ways of introducing non-archimedean and adelic geometries in quantum physics. A recent survey is given in [25]. For instance, Volovich [26] proposed space-time models based on cohomological realizations of motives, with étale topology “interpolating” between a proposed non-Archimedean geometry at the Planck scale and Euclidean geometry at the macroscopic scale. In this viewpoint, motivic L-functions appear as partition functions and actions of motivic Galois groups govern the dynamics.

Seems to be plenty of work going on, such as this and that.

As for my own interest, I guess there’s something intriguing about the thought that a number system we take for granted is merely a projection onto one component of something larger and more intrinsic.

Posted by: David Corfield on January 26, 2010 9:29 AM | Permalink | Reply to this

Re: effective physical space

This is an interesting argument, but I’m not convinced. In particular, consider this paragraph:

“Not every such effective geometry is a real manifold. But there is a large scale limit describing the geometry as seen not by a single one of our particles but by a whole condensate of them, in which these geometries do look like real manifolds. “

It seems to me like you are taking the large scale only in the archimedean sense, that is, the condensate is of a large number of particles in the archimedean metric, but is of an insignificant amount in the p-adic metrics. Unfortunately, to make sense of a p-adic large scale here requires a fractional number of particles, something which cannot be defined as far as I know. However, I suspect that if fractional particles can somehow be defined, this p-adic limit will be some kind of p-adic space.

Posted by: Itai Bar-Natan on April 17, 2011 4:19 AM | Permalink | Reply to this

Re: The Sacred and the Profane

The idea of mathematician as shaman* is quite intriguing. Kato definitely fits the bill, especially if you’ve been to as many Shinto shrines with him as I have. I do recommend this article very highly to everyone. Even if you skip entirely the technical bits, there should be a lot to hold your interest. Non-experts might find it somewhat surprising that Kato’s results discussed in that article are actually among the most spectacular in number theory of the last two decades.


*I should insert the proviso, however, that my sense of a shaman might be quite different from many others here, since these characters were quite commonplace when I was growing up.

Posted by: Minhyong Kim on January 21, 2010 1:24 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I copied this entry to a non-math friend who is interested into shamanism. he then asked:

Do you have some books you’d recommend - readable by a non-mathematician, and when talking about mathematics and mathematical research, relating it to “platonism/(shamanism)” etc? Considered respectable by scholars and philosophy of science types?

re Eliade

re Thomas McEvilley - who is he, or what’s his background? Is “The Shape of Ancient Thought” the book he wrote you’re referring to? What year was it?

Who is Yuri Manin? What book are you citing?

Who is K. Kato and what are zeta values? What book are you citing?

I believe you once cited Godel as a “platonist” - is there a book, readable by a non-mathematician, which is useful about this?

I’ve read a little about orphism and neopythagorean thought in the centuries on both sides of 1CE, but don’t know how reliable it is.

Very interesting,

thanks,

Byron

Can someone help me provide the references?
Don’t worry about zeta values; that’s not appropriate at this point.

jim

Posted by: jim stasheff on January 24, 2010 1:35 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Thomas McEvilley’s book is online (youtube interview, thanks to J.H.). It is about connections between ancient greek (esp. platonic) and ancient indian philosophy, but it’s summaries of single theories are very excellent for themself. Lots of other interesting infos are in that book. The relation of antique greek philosophy to (a creatively adapted) earlier shamanism is of course well known since ages and has been frequently discussed in the past centuries. E.g. Nietzsches Friend Rohde’s great work “Psyche. Seelencult und Unsterblichkeitsglaube der Griechen” and the small books on the greeks and Nietzsche by G. Colli. But one finds lost of very good discussions in many books on early greek philosophy. The only thing I dislike in most text is their use a confuse and pathetic language. Conc. Gödel, I have no idea if he considered himself a platonist and if that would be of interest. And I doubt that there ever was a set of doctrines etc. for “platonism”. I guess that ancient platonistic or pythagorean groups were less focused on specific contents of belief, but more on a certain mentality and lifestyle (maybe better called “mindstyle”).

Posted by: Thomas on January 24, 2010 11:43 AM | Permalink | Reply to this

Re: The Sacred and the Profane

To which another member of the same e-list responds

Greek philosophy especially Plato is my professional focus.
The above assertions are pifle, nonsense. There is no
demonstrable influence of Indian “philosophy” on Greek
philosophy. Godel in calling himself a platonist was referring
to mathematical ontology, Plato’s belief that anything
that was one, i.e. one thing, was in the set of things which
exist (this point is made in Theaetetus) As opposed to the
mathematicians who believe that numbers have no ontological
dimension. The beliefs of Pythagoras whatever they were
have left no written record. There are late testimonia (by late
I mean in Roman times, 2nd and 3d century A.D.) which list sets
of opposite categories said to be Pythagorean. Since nothing
sure is known about Pythagoras’ beliefs, it is safe to assert
his shamanistic origins. Hey, why not? Like Humpty Dumpty
there will always be those who can believe in up to 6 impossible
things BEFORE BREAKFAST!

jim

Posted by: jim stasheff on January 25, 2010 1:21 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Your friend would enjoy some surprises by browsing in some of the many good books on history of antique philosophy. What surprises me is that the well-known greek philosophy/shamanism still agitates people.

Posted by: Thomas on January 25, 2010 5:06 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I’m not sure I understand the sacred/profane distinction. (I’m not even sure everyone above is using these terms in a consistent way.) Do all subjects have it or is it only foundational subjects, where people tend to make philosophical interpretations? For instance, do group theory, number theory, and differential geometry have both aspects?

Posted by: James on January 19, 2010 7:16 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Maybe reference to the sacred is a little distracting when thinking about the final end of your work as a mathematician. There certainly are those who infuse their writings with a sense of awe and mystery, but Rota brings us nicely down to earth with his bottom line of ‘placing balls into boxes’.

If you chase along your interests (“This seems promising since it might help me do that. I want that because it will tell me something about the other thing.”), do you reach a bottom line?

Posted by: David Corfield on January 19, 2010 9:17 AM | Permalink | Reply to this

Re: The Sacred and the Profane

Are you asking about me personally or are you using the ‘indeterminate you’? If the first, then Of course! It’s to prove the Riemann hypothesis for all motives. (Yes, me and everyone else.) Actually, the reason that’s interesting to me is that it’s clearly accounted for by some basic cohomological theory for systems of polynomial equations with rational coefficients, but no one knows what it is. (See for instance the beginning of Deninger’s recent paper.) So for me it’s mostly about discovering new worlds which everyone can feel but no one can see.

I’m not sure what here is sacred and what is profane. Maybe I have a more circular view of things. The Riemann hypothesis is important because it points the way towards the mysterious cohomology theory, which is important because it would let us solve deep problems like the Riemann hypothesis.

But I do feel the distinction that Minhyong pointed out (so well) in category theory. But I think that’s because a domain of discourse feels so much grander to me than a graph– and closer to my own use of categories– and not so much because of any bottom lines there.

Posted by: James on January 19, 2010 10:07 AM | Permalink | Reply to this

Re: The Sacred and the Profane

James, I agree with you and perhaps a good example of a lack of a fundamental distinction would be the case of “graph groupoids”:

Adrian Wilson shows that an axiomatic approach to groupoids is needed to define a general graph groupoid, but then he proves [1] that the resulting general graph groupoid is equivalent to the usual defintion of a groupoid as a small category with inverses.

[1] The monograph by Adrian Wilson entitled “Graph Groupoids and their Topology” available from Amazon and others.

Also, see the monograph by Ilwoo Cho entitled “Graph Groupoids and Partial Isometries”.

Posted by: Charlie Stromeyer on January 19, 2010 1:52 PM | Permalink | Reply to this

Re: The Sacred and the Profane

The mysterious cohomology theory seems then to be the more important thing. You would expect an explanatory proof of the Riemann hypothesis for all motives to express features of this cohomology, but it could be that an unsatisfying calculation works. On the other hand, if you have a proper grip on the cohomology, then you have the Riemann hypothesis.

Perhaps in turn you only want that mysterious cohomology to better understand \infty-categorical hom-spaces.

Posted by: David Corfield on January 19, 2010 2:54 PM | Permalink | Reply to this

cohomology

The mysterious cohomology theory seems then to be the more important thing. You would expect an explanatory proof of the Riemann hypothesis for all motives to express features of this cohomology […] Perhaps in turn you only want that mysterious cohomology to better understand \infty-categorical hom-spaces.

Turns out, by the way, that following Morel’s defintion, motivic cohomology is – just like pretty much every other notion of cohomology – precisely \infty-categorical hom-spaces in some (,1)(\infty,1)-topos: in this case the (,1)(\infty,1)-topos of \infty-stacks on the Nisnevich site.

This is described explicitly now in the new version of nnLab: motivic cohomology.

See the introduction and then particularly the section Homtopy stabilization of the (,1)(\infty,1)-topos on NisNis.

As the nnPOV slogan goes:

Thousands of definitions of notions of cohomology and its variants. From the nnPOV, just a single concept: an ∞-categorical hom-space in an (∞,1)-topos.

Of course I can’t stop anyone from defining somthing that doesn’t fit this bill and call it “cohomology”. There are certainly “wrong” definitions of cohomology out there. For instance smooth group cohomology is often defined in terms of morphisms out of the usual bar resolution of the Lie group, mimicking the case of discrete groups. But this is too restrictive: in the (,1)(\infty,1)-topos of smooth \infty-groupoids only some of the maps out of BG\mathbf{B}G have this form. But there are more . And these do capture certain smooth group cocycles that are otherwise mysteriously missing (such as the one classifying the smooth String 2-group).

So the corrected definition of smooth group cohomology does fit the above slogan again.

One other case that might not exactly fit the above slogan is possibly the standard definition of Bredon cohomology and equivariant stable cohomology. This case, and what to do about it, I am currently discussing with Mike Shulman at the nnForum, here.

Posted by: Urs Schreiber on January 21, 2010 3:30 PM | Permalink | Reply to this

Re: cohomology

Equivariant cohomology is given by hom-spaces in a stable (,1)(\infty,1)-category, namely the equivariant stable category. The question under discussion is just whether that stable (,1)(\infty,1)-category is the stabilization of any (,1)(\infty,1)-topos. I’m not sure why you want to restrict cohomology to only happen in topoi.

Posted by: Mike Shulman on January 21, 2010 6:46 PM | Permalink | Reply to this

Re: cohomology

I’m not sure why you want to restrict cohomology to only happen in topoi.

That’s an important point, let’s discuss this. We had a little bit of related discussion here and there before: for instance once there was a little exchange here beetween Denis-Charles Cisinksi, you and me, on to which extent stable (,1)(\infty,1)-categories serve the “same purpose” as (,1)(\infty,1)-toposes.

Back then I was (and still am) interested in understanding to which extent the grand program of Kontsevich et al. on characterizing “noncommutative geometries” as formal duals of stable (,1)(\infty,1)-categories is on par with the grand program of characterizing concrete spaces as formal duals of (structured) (,1)(\infty,1)-toposes.

Accordingly, whatever I call cohomology in the latter case, to which extent should I still call that cohomology in the former case? We had some exchange on this over this here. There was a claim there that one “should” think of these stable (,1)(\infty,1)-categories as stabilizations of corresponding (,1)(\infty,1)-toposes.

I certainly don’t want to impose artifical restrictions on concepts, but I am also a bit wary of loosening the context too much.

So for instance one question one should ask is: if I call something a cohomology class, what do I expect to come with this notion?

In an (,1)(\infty,1)-topos, I am guaranteed that cohomology will have all of the abstract characteristics that we are used to from TopTop. It is not clear to me which of these we should allow ourselves to throw out of the window.

For instance: if the ambient (,1)(\infty,1)-category does happen to be an \infty-stack (,1)(\infty,1)-topos, then the (,1)(\infty,1)-Giraud’s axioms/Rezk’s axioms, do guarantee that morphisms interpreted as cocycles do have the familiar classification property: they classify something, namely they classify principal \infty-bundles – as describeed there (I have just expanded the discussion there a bit to amplify this more).

This is a pretty important aspect. And what makes it work is the fact that in an \infty-stack (,1)(\infty,1)-topos every groupoid object is effective. So this fails in a stable (,1)(\infty,1)-category.

Maybe this shouldn’t worry me in general. But it does certainly in the example of GG-equivariant stable homotopy theory. There is a non-standard but “natural” version of GG-equivariant stable homotopy theory, such that cohomology in there classifies GG-equivariant AA-principal bundles, for AA some stable object. But this is not the “standard” theory. I’d probably be happy to accept this, if I would see better what motivation one has for considering it.

Posted by: Urs Schreiber on January 21, 2010 8:47 PM | Permalink | Reply to this

Re: cohomology

There is a paper by Lashof-May-Segal

‘Equivariant bundles with abelian structure group’,Contemporary Math. 19 (1983), 167-176

which may provide some insights, see, e.g. here.

Posted by: David Roberts on January 21, 2010 11:26 PM | Permalink | Reply to this

Re: cohomology

to which extent stable (∞,1)-categories serve the “same purpose” as (∞,1)-toposes.

Just for completeness, this needs to go along with a nod towards the stable Giraud theorem.

Posted by: Urs Schreiber on January 22, 2010 1:52 AM | Permalink | Reply to this

Re: cohomology

which may provide some insights

I can’t access this right now, for various stupid reasons. One of them being that I go to bed now! :-)

But do you have any particular insight in mind, or is this just meant as a pointer to equivariant bundles in general?

Do you have a pdf of the thing?

Posted by: Urs Schreiber on January 22, 2010 1:58 AM | Permalink | Reply to this

Re: cohomology

I read the paper a couple of years ago, so I can’t recall much. I was hoping it would shed some light on the dilemma with the difference ‘natural’ and the ‘standard’ theory, but now that I skim over it again, the paper probably falls on the side of the ‘natural’ cohomology theory. Although, being a Segal paper, there are probably some subtle gems waiting to be teased out. :)

Posted by: David Roberts on January 22, 2010 6:06 AM | Permalink | Reply to this

Re: cohomology

Actually later on in the paper there are some interesting things, so have a look anyway, and sorry, I don’t have a pdf (at the moment - I’m at work)

Posted by: David Roberts on January 22, 2010 6:08 AM | Permalink | Reply to this

Re: cohomology

Okay, after a very helpful discussion with my wife Megan, who studies equivariant cohomology theory, I think I understand things better. According to the nPOV on cohomology, if XX and AA are objects in an (,1)(\infty,1)-topos, the 0th cohomology H 0(X;A)H^0(X;A) is π 0(Map(X,A))\pi_0(Map(X,A)), while if AA is a group object, then H 1(X;A)=π 0(Map(X,BA))H^1(X;A)= \pi_0(Map(X,B A)). More generally, if AA is nn times deloopable, then H n(X;A)=π 0(Map(X,B nA)H^n(X;A) = \pi_0(Map(X, B^n A). In TopTop, this gives you the usual notions if AA is a (discrete) group, and in general, H 1(X;A)H^1(X;A) classifies principal ∞-bundles in whatever (,1)(\infty,1)-topos.

Now consider the (,1)(\infty,1)-topos GTopG Top of GG-equivariant spaces, which can also be described as the (,1)(\infty,1)-presheaves on the orbit category of GG. For any other group Π\Pi there is a notion of a principal (G,Π)(G,\Pi)-bundle (where GG is the group of equivariance, and Π\Pi is the structure group of the bundle), and these are classified by maps into a classifying GG-space B GΠB_G \Pi. So the principal (G,Π)(G,\Pi)-bundles over XX can be called H 0(X;B GΠ)H^0(X;B_G \Pi). If we had something of which B GΠB_G \Pi was a delooping, we could call the principal (G,Π)(G,\Pi)-bundles “H 1(X;?)H^1(X;?)”, but I don’t think there is such a thing. As far as I can tell, B GΠB_G \Pi is not connected, in the sense that B GΠ∗\to B_G \Pi is not an effective epimorphism and thus B GΠB_G \Pi is not the quotient of a group object in GTopG Top.

Okay, so far so interesting, where do the spectra come in? If we have an object AA of our (,1)(\infty,1)-topos that can be delooped infinitely many times, then we can define H n(X;A)H^n(X;A) for any integer nn by looking at all the spaces Ω nA=B nA\Omega^{-n} A = B^n A. These integer-graded cohomology groups are closely connected to each other, e.g. they often have cup products or Steenrod squares or Poincare duality, so it makes sense to consider them all together as a cohomology theory. We then are motivated to put together all of the objects {B nA}\{B^n A\} into a spectrum, a single object which encodes all of the cohomology groups of the theory. A general spectrum is a sequence of objects {E n}\{E_n\} such that E nΩE n+1E_n \simeq \Omega E_{n+1}; the stronger requirement that E n+1BE nE_{n+1} \simeq B E_n restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the (,1)(\infty,1)-topos. In TopTop, the most “basic” spectra are the Eilenberg-Mac Lane spectra produced from the input of an ordinary abelian group.

Now we can do all of this in GTopG Top, and the resulting notion of spectrum is called a naive GG-spectrum: a sequence of GG-spaces {E n}\{E_n\} with E nΩE n+1E_n \simeq \Omega E_{n+1}. Any naive GG-spectrum represents a cohomology theory on GG-spaces. The most “basic” of these are “Eilenberg-Mac Lane GG-spectra” produced from coefficient systems, i.e. abelian-group-valued presheaves on the orbit category. The cohomology theory represented by such an Eilenberg-Mac Lane GG-spectrum is called an (integer-graded) Bredon cohomology theory.

Megan and I don’t know whether there is some sense in which Eilenberg-Mac Lane GG-spectra can be produced by successively “delooping” a coefficient system, but it seems possible.

It turns out, though, that the cohomology theories arising in this way are kind of weird. For instance, when you calculate with them, you see torsion popping up in odd places where you wouldn’t expect it. It would also be nice to have a Poincare duality theorem for GG-manifolds, but that fails with these theories. The solution people have come up with is to widen the notion of “looping” and “delooping” and thereby the grading: instead of just looking at Ω n=Map(S n,)\Omega^n = Map(S^n, -), we look at Ω V=Map(S V,)\Omega^V = Map(S^V,-), where VV is a finite-dimensional representation of GG and S VS^V is its one-point compactification. Now if AA is a GG-space that can be delooped “VV times,” we can define H V(X;A)=π 0(Map(X,Ω VA)H^V(X;A) = \pi_0(Map(X,\Omega^{-V} A). If AA can be delooped VV times for all representations VV, then our integer-graded cohomology theory can be expanded to an RO(G)RO(G)-graded cohomology theory, with cohomology groups H α(X;A)H^\alpha(X;A) for all formal differences of representations α=VW\alpha = V - W. The corresponding notion of spectrum is a genuine GG-spectrum, which consists of spaces E VE_V for all representations VV such that E VΩ WVE WE_V \simeq \Omega^{W-V} E_W. A naive Eilenberg-Mac Lane GG-spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a Mackey functor, and in this case we get an RO(G)RO(G)-graded Bredon cohomology theory.

RO(G)RO(G)-graded Bredon cohomology has lots of formal advantages over the integer-graded theory. For instance, the torsion that popped up in odd places before can now be seen as arising by “shifting” of something in the cohomology of a point in an “off-integer dimension,” which was invisible to the integer-graded theory. Also there is a Poincare duality for GG-manifolds: if MM is a GG-manifold, then we can embed it in a representation VV (generally not a trivial one!) and by Thom space arguments, obtain a Poincare duality theorem involving a dimension shift of α\alpha, where α\alpha is generally not an integer (and, apparently, not even uniquely determined by MM!). Unfortunately, however, RO(G)RO(G)-graded Bredon cohomology is kind of hard to compute.

So I guess what I’m saying now is that the cohomology itself can, and perhaps should, still be viewed using hom-spaces in an (,1)(\infty,1)-topos. In fact, for nonabelian cohomology, such as the classification of principal bundles with nonabelian structure group, this is the only option. For abelian cohomology that exists at all degrees, we can “package up” the spaces representing it at all degrees into an object called a “spectrum” and then study spectra in their own right. But depending on the (,1)(\infty,1)-topos we started with, there may be other kinds of “degree” that should be taken into account, in addition to the usual integers.

What’s still not clear to me is whether the notion of “RO(G)RO(G)-grading” is somehow canonically determined by the (,1)(\infty,1)-topos GTopG Top, maybe even in a way that could be applied to other (,1)(\infty,1)-topoi. Peter May has commented that in some sense, the RO(G)RO(G)-grading isn’t really “fully general” either. There’s a larger class of “homotopy spheres” which are already invertible in the equivariant stable homotopy category, so that all genuine GG-spectra can be “looped and delooped” by them, and the RO(G)RO(G)-graded cohomology theories they represent could be extended to a grading on that larger class of spheres. But identifying precisely what those spheres are, for any particular GG, is apparently pretty hard.

Posted by: Mike Shulman on January 23, 2010 4:22 AM | Permalink | Reply to this

Re: cohomology

So where we marvelled at the bold attempt to go beyond the bold theory of \infty-groupoids by considering spectra to be \mathbb{Z}-groupoids, you’re telling us that this is the miniscule part of the theory of RO(G)RO(G)-groupoids (or some even larger grading) where GG is trivial?

Roll on the study of categories graded by representations of the monster group.

Posted by: David Corfield on January 24, 2010 10:30 AM | Permalink | Reply to this

Re: cohomology

But this is too restrictive: in the (∞,1)-topos of smooth ∞-groupoids only some of the maps out of BG have this form.

which form?

Posted by: jim stasheff on January 22, 2010 1:33 PM | Permalink | Reply to this

Re: cohomology

I wrote

But this is too restrictive: in the (∞,1)-topos of smooth ∞-groupoids only some of the maps out of BG\mathbf{B}G have this form.

Jim Stasheff asks

which form?

Jim will know in a moment what I meant, but I say it in full detail anyway, in case anyone else is wondering:

For GG a Lie group, regarded as the sheaf on the category CartSpCartSp of cartesian smooth manifolds that it represents, the corresponding simplicial sheaf is the usual

BG:=(G×GG*). \mathbf{B}G := \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\stackrel{\to}{\to} {*} \right) \,.

In the standard literature the cohomology of the Lie group GG is (equivalently, but usually more implicitly than I say it now) defined simply as the collection of homotopy classes of morphisms of simplicial sheaves BGA\mathbf{B}G \to A from the simplicial sheaf above to some coefficient sheaf.

This is blindly copied from the way one can compute the cohomology of discrete groups: when GG is discrete, then BG\mathbf{B}G happens to be cofibrant in the relevant model category structure, and this is what makes morphisms out of this BG\mathbf{B}G indeed model cocycles in cohomology.

But for GG Lie, the simplicial sheaf BG\mathbf{B}G above in general fails to be cofibrant. This means that morphisms of simplicial sheaves out of it will fail to see all classes of morphisms in Ho(BG,A)Ho(\mathbf{B}G,A). And this is what cohomology really is.

Notably (for simple, simply connected compact GG), there is the 3-cocycle in Ho(BG,B 3U(1))Ho(\mathbf{B}G,\mathbf{B}^3 U(1)) whose homotopy fiber is the string 2-group BString(G)BG\mathbf{B}String(G) \to \mathbf{B}G. This is not a morphism of simplicial sheaves out of BG\mathbf{B}G directly, one needs a bigger resolution.

For instance let [Ω eGP eG][\Omega_e G \to P_e G] be the evident (up to some non-essential choices) crossed module of based loops and based paths in GG and write B[ΩGPG]\mathbf{B}[\Omega G \to P G] for the corresponding simplicial sheaf

B[ΩGPG]:=(PG×PG×ΩGPG*). \mathbf{B}[\Omega G \to P G] := \left( \cdots P G \times P G \times \Omega G \stackrel{\to}{\stackrel{\to}{\to}} P G\stackrel{\to}{\to} {*} \right) \,.

Then we have an acyclic fibration B[ΩGPG]BG\mathbf{B}[\Omega G \to P G] \stackrel{\simeq}{\to} \mathbf{B}G, hence this is a somewhat bigger model for BG\mathbf{B}G. Take μ 3CE(𝔤)\mu_3 \in CE(\mathfrak{g}) be the canonical Lie algebra 3-cocycle on 𝔤\mathfrak{g} normalized such that it represents the image in de Rham cohomology of one of the generators of H 3(G,)H^3(G,\mathbb{Z}) \simeq \mathbb{Z}. Then the operation of integrating μ\mu over 3-balls in GG cobounding chosen disks between the loops appearing in [ΩGPG][\Omega G \to P G] provides a morphism of simplicial sheaves

B[ΩGPG] μ 3 B 3U(1) BG \array{ \mathbf{B}[\Omega G\to P G] &\stackrel{\int \mu_3}{\to}& \mathbf{B}^3 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G }

that represents a class in Ho(BG,B 3U(1))Ho(\mathbf{B}G, \mathbf{B}^3 U(1)) which is not representable by any morphism of simplicial sheaves out of BG\mathbf{B}G directly.

And the extension of BG\mathbf{B}G classified by this is BString(G)\mathbf{B}String(G).

If GG had been discrete, the whole construction would have worked by mapping just out of BG\mathbf{B}G itself. This is what people are used to. This is why one finds sometimes the attitude that cohomology of groups is defined to be morphisms out of the bar-resolution BG=G ×\mathbf{B}G = G^{\times \bullet}. But cohomology is more abstractly and more intrinsically defined than that. This is just one algorithm for computing it, applicable in only some situations.

Posted by: Urs Schreiber on January 22, 2010 3:48 PM | Permalink | Reply to this

Re: cohomology

Okay, after a very helpful discussion with my wife Megan, who studies equivariant cohomology theory, I think I understand things better.

Thanks, Mike, that’s very useful! And thanks to Megan!

The most “basic” of these are “Eilenberg-Mac Lane G-spectra” produced from coefficient systems, i.e. abelian-group-valued presheaves on the orbit category […] Megan and I don’t know whether there is some sense in which Eilenberg-Mac Lane G-spectra can be produced by successively “delooping” a coefficient system, but it seems possible.

Maybe I am misunderstanding what you are saying here, but if we are talking about abelian-group-valued (,1)(\infty,1)-presheaves, then these deloop simply objectwise, no? Let me say what I am thinking of in detail, just so we are sure we are talking about the same thing:

we model PSh (,1)(O G)PSh_{(\infty,1)}(O_G) by sPSh(O G) projsPSh(O_G)_{proj}. In there all simplicial preseheaves that are objectwise in the image of the Dold-Kan nerve Ξ\Xi are Kan-complex valued and hence fibrant. In particular this is true for Eilenberg-MacLane presheaves Ξ(A[n])\Xi(A[n]) for AA an abelian-group valued presheaf on O GO_G, regarded as a complex of abelian-group valued sheaves concentrated in degree 0.

Given any such, its loop object is the homtopy pullback of the point inclusion *Ξ(A[n]){*} \to \Xi(A[n]) along itself. Since everything is fibrant, this is computed as the pullback of the co-mapping cone that I like to write EΞ(K[n])\mathbf{E}\Xi(K[n]) to the point (as it is the corresponding “universal Ξ(A[n])\Xi(A[n])-principal \infty-bundle”). Writing out what this means it gives under Dold-Kan just the down-shift of the corresponding complex of sheaves, Ξ(A[n1])\Xi(A[n-1]), as usual, since Ξ\Xi is right adjoint and preserves limits.

[…] a naive GG-spectrum: […] The cohomology theory represented by such an Eilenberg-Mac Lane GG-spectrum is called an (integer-graded) Bredon cohomology theory.

Oh, I see. From our discussion on the nnForum I had gotten away with the impression that Bredon cohomology needs the “non-naïve”/”genuine” GG-spectra as coefficients instead. But I hadn’t unwrapped this in detail for myself to actually check. Okay, very good!

The solution people have come up with is to widen the notion of “looping” and “delooping” and thereby the grading:

A naive Eilenberg-Mac Lane GG-spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a Mackey functor,

Aha, very useful to know. I need to think more about this, but it feels as if one should be able to understand this in terms of the “stable Giraud theorem” that asserts that every locally presentable stable (,1)(\infty,1)-category arises as a geometric localization of an (,1)(\infty,1)-category of spectrum-valued (,1)(\infty,1)-presheaves:

As Schwede and Shipley discuss on p 12/13 of their Classification of stable model categories (and as we discussed previously on the nForum), using their main theorem 3.3.3, the equivariant stable (,1)(\infty,1)-category is presented by topological Mackey functors on the homotopy orbit category. I suppose this must be a model theory version of the stable Giraud theorem that ensures that the equivariant stable (,1)(\infty,1)-category is a geometric localization of the collection of spectrum-valued (,1)(\infty,1)-presheaves. Maybe the geometric localization is modeled by the passage from sSetsSet-enriched SpSp-valued presheaves to SpSp-enriched SpSp-valued presheaves? I don’t know. But it looks to me like there ought to be a connection along such lines.

So I guess what I’m saying now is that the cohomology itself can, and perhaps should, still be viewed using hom-spaces in an (∞,1)-topos

Good.

For abelian cohomology that exists at all degrees, we can “package up” the spaces representing it at all degrees into an object called a “spectrum” and then study spectra in their own right.

Yes. I belive one way to think about it is that we do want think of cohomology as taking place exclusively in an (,1)(\infty,1)-topos, but that it is of interest to look at systems of coefficient objects that arise from acting with certain monads on the (,1)(\infty,1)-topos.

So for the case of Eilenberg-Steenrod style abelian cohomology, what we really do is look at coefficient objects of the form

Σ Σ nΩ A \Sigma^\infty \circ \Sigma^n \circ \Omega^\infty A

for all nn \in \mathbb{Z}, where

HΩ Σ Stab(H) \mathbf{H} \stackrel{\stackrel{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}} Stab(\mathbf{H})

is the stabilization adjunction.

Somehow it is really important to keep both ends of this adjunction in sight: we want to get out an object on the left, but we may also do want to have threaded that first through the right.

Maybe choosing a different adjunction here could account for non-standard grading such as RO(G)RO(G)-grading?

Posted by: Urs Schreiber on January 23, 2010 2:14 PM | Permalink | Reply to this

Re: cohomology

if we are talking about abelian-group-valued (,1)(\infty,1)-presheaves, then these deloop simply objectwise, no?

Yes, you should be able to do that. What I meant was, I don’t know whether the Eilenberg-Mac Lane G-spectrum associated to a coefficient system is equal to what you would get by regarding the coefficient system as an abelian-group-valued (,1)(\infty,1)-presheaf and delooping it. Or equivalently, whether the 0-space of the Eilenberg-Mac Lane spectrum associated to a coefficient system is equal to that coefficient system so regarded. It seems possible, but there are so many traps in the equivariant theory that I don’t trust the construction that seems “obvious” to a category theorist to always be the one that the equivariant homotopy theorists are interested in.

From our discussion on the nForum I had gotten away with the impression that Bredon cohomology needs the “non-naive”/”genuine” G-spectra as coefficients instead.

That’s possibly because I myself was under that impression as well. But Megan straightened me out.

it feels as if one should be able to understand this in terms of the “stable Giraud theorem”

What confuses me about the stable Giraud theorem is that it’s claiming that any locally presentable stable (,1)(\infty,1)-category is a left-exact localization of the stabilization of a category of presheaves on an (,1)(\infty,1)-category. But the Schwede-Shipley result for stable model categories is about model structures on a category of spectrum-valued presheaves on a spectrally enriched category. In other words, in the latter case the domain category of the presheaves is already “stabilized” in some sense. How are those two related?

Maybe choosing a different adjunction here could account for non-standard grading such as RO(G)-grading?

Something like that may be it. You have the functors Σ \Sigma^\infty and Ω \Omega^\infty switched, by the way; Σ \Sigma^\infty goes to the stabilization and Ω \Omega^\infty from it. I would say that in general the coefficient objects are of the form Ω Σ nA\Omega^\infty \Sigma^n A for some stable object AA. (Most AA we’re interested in will not be of the form Σ \Sigma^\infty of anything.) We can then get RO(G)-grading by replacing “nn” by some nontrivial representation sphere in the “genuine” equivariant stable category. Of course then the question is, how do you know that the “genuine” stable category is the adjunction you want to put there.

Posted by: Mike Shulman on January 23, 2010 11:50 PM | Permalink | Reply to this

Re: cohomology

What confuses me about the stable Giraud theorem is that it’s claiming that any locally presentable stable (∞,1)-category is a left-exact localization of the stabilization of a category of presheaves on an (∞,1)-category. But the Schwede-Shipley result for stable model categories is about model structures on a category of spectrum-valued presheaves on a spectrally enriched category. In other words, in the latter case the domain category of the presheaves is already “stabilized” in some sense. How are those two related?

Yeah, so that’s what I meant: the fact that Schwede-Schipley have

  • SpSp-enriched SpSp-valued presheaves

instead of

  • sSetsSet-enriched SpSp-valued presheaves

must be the fact that accounts for the left-exact localization.

Stable Giraud tells us, tanslated to model category models, that we need a left-Bousfield localization of the model category of sSetsSet-enriched SpSp-valued presheaves. Reading Schwede-Schipley’s result with this in mind must imply that their passage to SpSp-enrichment (i.e. stabilization of the domain) somehow models this Bousfield localization. In some unfamiliar way. Not that I understand it.

You have the functors Σ \Sigma^\infty and Ω \Omega^\infty switched, by the way;

Ah, right, thanks for catching that.

I would say that in general the coefficient objects are of the form ΩΣ nA\Omega \Sigma^n A for some stable object AA. (Most AA we’re interested in will not be of the form Σ \Sigma^\infty of anything.)

Ah, right, thanks for cathching that, too!

Of course then the question is, how do you know that the “genuine” stable category is the adjunction you want to put there.

Yeah, it’s still somewhat mysterious. But much less than it was yesterday. I am even feeling inclined to now get back to my orginal motivation of looking at dendroidal presheaves on the orbit category…

Posted by: Urs Schreiber on January 24, 2010 1:12 AM | Permalink | Reply to this

Re: cohomology

I wrote:

Yeah, so that’s what I meant: the fact that […]

That was a bit vague. Let me try again, something more precise:

it looks as if the Schwede-Schipley result in light of the stable Giraud theorem indicates that it ought to be true that

there is an adjunction

R:SpCat(O G st,Sp)sSetCat(O G,Sp):L R : Sp Cat(O_G^{st}, Sp) \stackrel{\leftarrow}{\to} sSet Cat(O_G, Sp) : L

which is a Quillen adjunction with respect to the pertinent model category structures such that the left adjoint LL is a left localization of model categories.

Here on the left the category SpSp of spectra is regarded as enricched over itself, while on the right it is regarded as a simplicially enriched category. And on the right we have the orbit category O GO_G and on the left its “stabilization” to an SpSp-enriched category.

If this is true at all, then maybe we should build the adjunction by pre- and postcomposing hom-object-wise with the adjoint functors sSetΣ Ω SpsSet \stackrel{\stackrel{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}} Sp.

Posted by: Urs Schreiber on January 24, 2010 4:22 PM | Permalink | Reply to this

Re: cohomology

I wrote:

Yeah, it’s still somewhat mysterious. But much less than it was yesterday. I am even feeling inclined to now get back to my orginal motivation of looking at dendroidal presheaves on the orbit category…

Mike, here is an observation.

Possibly the point to notice in view of naïve and genuine GG-spectra is that in the context of lined (,1)(\infty,1)(pre)sheaf (,1)(\infty,1)-toposes H\mathbf{H} there are two different notions of paths and spheres:

  • there is the categorical nn-sphere Δ n/Δ n\Delta^n/\partial \Delta^n;

  • there is the geometric nn-sphere I n/I nI^n/\partial I^n, where II is a singled out interval object.

For XX an object in the (,1)(\infty,1)-topos, the first is what is seen when forming its loop space object of XX in terms of the homotopy pullback,

Ω catX X \array{ \Omega_{cat} X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& X }

while the second is seen, in contrast, by forming the categorical loop space object of the path \infty-groupoid Π(X)\Pi(X) that is built from the geometric line Π(X)=lim [Δ I ,X]\Pi(X) = \lim_\to [\Delta^\bullet_I, X].

Ω geomX Π I(X). \array{ \Omega_{geom} X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \Pi_I(X) } \,.

Coincidentally (or not so coincidentally, since this is what’s on my mind these days) we were just the other day talking about this also at smooth loop space. As amplified there, too, it is easy to miss the pattern here by looking just at H=Top\mathbf{H} = Top, due to the fact that TopTop is really two different things here, that look alike.

We can make more explicit what is going on by using the fact that TopTop is the localization of Sh (,1)(Top)Sh_{(\infty,1)}(Top) at maps X×IXX \times I \to X. Under this equivalence

Top=Sh (,1)(*)Sh (,1)(Top) I(sPSh(Top) proj loc,I) Top = Sh_{(\infty,1)}(*) \stackrel{\simeq}{\leftarrow} Sh_{(\infty,1)}(Top)^I \simeq (sPSh(Top)_{proj}^{loc,I})^\circ

which is induced by the evident restriction of presheaves to the point, a topological space is identified, going the other way round, with the constant simplicial presheaf with value the simplicial set corresponding to that space. But that’s of course

Π I(X)=Sing(X). \Pi_I(X) = Sing(X) \,.

So under this equivalence categorical spheres in the above sense are turned into geometric ones. Since it’s an equivalence, this is hard to notice, though! :-)

But if we move away from plain TopTop now, for instance to GTopG Top, the difference may start to show.

So one thing I am wondering about is this: in analogy to the above equivalence, might we have a localization of Sh (,1)(GTop)Sh_{(\infty,1)}(G Top) that makes the evident restriction functor

PSh (,1)(O G)Sh (,1)(GTop) PSh_{(\infty,1)}(O_G) \leftarrow Sh_{(\infty,1)}(G Top)

an equivalence? I guess you see what I am getting at: if we have such a localization, then it would mean that when forming spheres and spectrum objects in PSh (,1)(O G)PSh_{(\infty,1)}(O_G) we should think about if we possibly mean really geometric spheres in the equivalent Sh (,1)(GTop) locSh_{(\infty,1)}(G Top)^{loc}. These geometric spheres are reasonably taken to be the objects I n/I nI^n / \partial I^n equipped with a GG-action structure – but that’s what we get from the one-point compactifications S VS^V of linear representations of VV, in particular.

We would then form in H=Sh (,1)(GTop) loc\mathbf{H} =Sh_{(\infty,1)}(G Top)^{loc} not the stabilization under taking categorical loops and spheres (the ordinary Stab(H)Stab(\mathbf{H})) but the stabilization with respect to taking the given geometric spheres. And this should be the genuine GG-spectra.

And we naturally expect that on geometric kk-fold loop objects not a E kE_k-Top-operad, but an E kE_k-H\mathbf{H}-operad acts. In fact I think, that’s what I kept mentioning, that what this should be is a dendroidal presheaf.

The kind of reasoning behind this is along the lines of the construction that I recently added to interval object, here, where from a geometric path object/interval object a dendroidal internal object is formed. I’d think that taking fiberwise the BV tensor product on dendroidal sets of this construction kk-times should yield the relevant E kE_k-operad-dendroidal presheaf that acts on the geometric loop space objects.

Well, I need to work this out in more detail. But currently my idea is that this, or something like this, is what is underlying Peter May’s construction of the GTopG Top-operadic recognition principle, that we are talking about at the nnForum.

Do you see what I am trying to get at?

Posted by: Urs Schreiber on January 24, 2010 7:28 PM | Permalink | Reply to this

Re: The Sacred and the Profane

In his excellent book about the modern rebirth of abstract mathematics entitled “Equations from God: Pure Mathematics and Victorian Faith”, Daniel J. Cohen writes twice (on pages 15 and 19) about the view of mathematics as a courier between the sacred (abstract or divine) and the profane (practical or material) and says that this view became central to Procline Neoplatonism.

My friend is a philosopher of physics, and so I told him that quantum theory is the most accurate scientific theory we have and that the combination of these 11 fundamental and still unsolved mysteries of quantum theory are why I remain an advocate for neoplatonism:

1) Is there a correct theory of quantum gravity?

2) Does supersymmetry (susy) actually exist in Nature, and if so then which form of susy?

3) What is the origin of mass?

4) What are the causes and the ubiquity of quantum decoherence?

5) What is the origin of quantum-mechanical wave / particle duality for fermions?

6) Does dark energy actually exist, and if so then what is it comprised of?

7) In 2009, the Princeton mathematicians J.H. Conway and S. Kochen proved that not everything within physical reality can be physically predetermined nor random [1], thus establishing that there cannot be a final and complete physical theory of Nature. What is the third option that they have discovered and which they poorly name “free will”?

8) In the journal Science is a perspective piece [2] by N. Gisin which correctly cites evidence that quantum nonlocal entanglement must originate outside of Einsteinian spacetime. I would also add that the very common entanglement sudden death (ESD) must originate outside of spacetime too because ESD is not a stochastic nor random phenomenon [3]. Thus, from what do entanglement and ESD originate?

9) Is the wavefunction of quantum mechanics fundamentally non-sequential as argued in [4] or is the wavefunction still somehow sequential in its behavior?

10) In 1989, J. Tate and B. Cabrera found that the mass of the Cooper pairs in a rotating superconductor is greater than predicted by BCS or Ginzburg-Landau theory [5]. The work of M. Tajmar [6] has confirmed the existence of this anomaly, and so what is the origin of this anomaly?

11) What is the correct mathematical-physical explanation for the origin of geometric phase? (Please see [7] for one new attempt at an explanation).

References


[1] J.H. Conway and S. Kochen, “The Strong Free Will Theorem”, Notices of the AMS
v56(2), pp.226-232 (2009).

[2] N. Gisin, “Quantum Nonlocality: How Does Nature Do It?”, Science v326(5958),
pp.1357-1358 (2009).

[3] T. Yu and J.H. Eberly, “Sudden Death of Entanglement”, Science v323(5914),
pp.598-601 (2009).

[4] S. Dolev and A.C. Elitzur, “Non-sequential Behavior of the Wave Function”,
http://arxiv.org/abs/quant-ph/0102109

[5] J. Tate et al., “Precise Determination of the Cooper-Pair Mass”, Phys. Rev. Lett. 62,
pp.845-848 (1989).

J. Tate et al., “Determination of the Cooper-pair Mass in Niobium”, Phys. Rev. B 42,
pp.7885-7893 (1990).

[6] M. Tajmar and F. Plesescu, “Fiber-Optic-Gyroscope Measurements Close to Rotating
Liquid Helium”, http://arxiv.org/abs/0911.1033

[7] K. Fujikawa, “Geometric Phases and Hidden Gauge Symmetry”,
http://arxiv.org/abs/0910.0396


Posted by: Charlie Stromeyer on January 19, 2010 4:05 PM | Permalink | Reply to this

Re: The Sacred and the Profane

This comment stems from mentioning the change in focus from quantum gravity to n-category theory made by John Baez. JB presented a very
plausible argument as to why Set should be replaced by nCob and Hilb because of their “deep analogy”. (Quantum Quandaries:
A Category-Theoretic Perspective April 2004)

In 2009 I read a claim here at the Cafe that Category theory was quite likely the solution to resolve the problem of quantum gravity. (The comment started off with “dare I say it” I think.) This remark impressed me at the time as having the nature of a sacred pronouncement, not sufficiently grounded in tangible evidence.

From the 2004 paper regarding time reversal:
“The big difference is that in topological quantum field theory we cannot measure time in seconds, because there is no background metric available to let us count the passage of time. We can only keep track of topology change. …
This fact reinforces a point already well- known from quantum field theory on curved spacetime, namely that unitary time evolution is not a built-in feature of quantum theory but rather the consequence of specific assumptions about the nature of spacetime[13].”

“To conclude, it is interesting to contrast nCob and Hilb with the more familiar category Set, whose objects are sets and whose morphisms are functions. There is no way to make Set into a ∗-category, since there is no way to ‘reverse’ the map from the empty set to the one-element set.
So, our intuitions about sets and functions help us very little in understanding ∗-categories. The problem is that the concept of function is based on an intuitive notion of process that is asymmetrical with respect to past and future: a function f: S → S′ is a relation such that each element of S is
related to exactly one element of S′, but not necessarily vice versa. For better or worse, this built-in ‘arrow of time’ has no place in the basic concepts of quantum theory.”

In 1952, David Bohm, before he started pushing his own theory, wrote a textbook name “Quantum Theory”. He said quantum theory was not considered a universal theory because the notion of entropy, time, and causality as a series of related events is fundamental to the human perception of the universe. The ‘arrow of time’ is considered fundamental.

Thus I think for Category theory to earn the description of the correct tool to use to resolve the problem of quantum gravity, a 2010 update of the view expressed in 2004 is called for. What additional pragmatic evidence has been garnered since the 2004 paper?

At one time the wave function collapse was a favored theory, but now it mostly considered not part of reality, but an aspect of quantum decoherence. What elevates a category theory solution to quantum gravity out of the level of plausibility arguments into tangibility?

Posted by: Stephen Harris on January 21, 2010 10:12 AM | Permalink | Reply to this

Re: The Sacred and the Profane

As James points out, the interaction between the sacred and the profane is obviously complex. In regard to the status of category theory, it might be useful to refer to ‘pure=sacred’ and ‘applied=profane.’ The closest analogy for this particular distinction might be the pure and applied model theories, that I briefly mentioned in an earlier post. At level zero, every mathematician uses logic all the time. Applied model theory means something more substantial, such as the Ax-Kochen theorem. Hrushovki’s theorem, on the other hand, goes much further. The current day infusion of categories is probably for most people like the basic language of logic. Applications that I know of are perhaps at the level of Ax-Kochen. Since I’m a fan of category-theorists, I like to think it’s a matter of time before someone does something as unambiguously deep and broad in scope as Hrushovski using category theory. As was the case there, it’s rather plausible that some people attaining greater depth in the sacred tradition will eventually be useful for this development.

I know even less the meaning of pure category theory than I do that of pure model theory. But it does seem to touch more often on questions like ‘what is the *right* definition of an n-category’ than pragmatists might be comfortable with. This is maybe the kind of question that’s closest to a sacred tradition in the sense formulated by Van Dalen. My feeling is most people at this cafe rather enjoy this line of questioning. On the other side, I hope Bertrand Toen doesn’t mind if I mention that he, for example, declared himself in no uncertain terms to be a ‘profane’ category-theorist when I spoke to him last November. (One mathematical consequence is that his definition of an n-stack does nothing at all for algebraic topology. He’s interested in applications to algebraic geometry.)

I realize I’m not contributing much the discussion with the paragraphs above, and I’m presently a bit too occupied to think more deeply on this issue. But perhaps I can proffer some humble advice to pure category-theorists. Many of us consider important some sense of the sacred in all spheres of life. However, it’s clear that most people are struck far more often by the *operation* of the sacred than by its philosophy. So it may be with the doctrines of category theory. When words are necessary, brief allusions are usually preferable to sermons.

Posted by: Minhyong Kim on January 21, 2010 2:12 PM | Permalink | Reply to this

Re: The Sacred and the Profane

As is my habit, let me reformulate at least one sentence:

‘There’s no point in complaining if people are struck far more often by the *operation* of the sacred than by its philosophy.’

Posted by: Minhyong Kim on January 21, 2010 2:20 PM | Permalink | Reply to this

Re: The Sacred and the Profane

By “the *operation* of the sacred” do you mean what I was describing in my post, where the use of something for some purpose is taken to reflect the glory of that thing?

Something necessary for this is that the thing be used in the ‘right’ way. Using a Bible to keep the door open is evidently a profane use, just as ripping off the 3-dimensional vector portion of the quaternions must have felt for their supporters like Tait.

There must be subtle gradations of profanity of use. Didn’t Grothendieck consider parts of Deligne’s use of his work profane?

I’m looking forward to some sacred use of 2-groups in number theory.

Posted by: David Corfield on January 21, 2010 2:54 PM | Permalink | Reply to this

Re: The Sacred and the Profane

MK wrote: ‘There’s no point in complaining if people are struck far more often by the *operation* of the sacred than by its philosophy.’

I’m not so sure how well the sacred and the operation of the sacred can be distinguished or even comprehended separately. This concept doesn’t seem analogous to how a theory uses the scientific method to make testable experimental predictions in order to establish its felicity and which do seem hospitable to reduction.

I didn’t get what he meant so I’ll mention a few things that crossed my mind. Take some sacred philosophy such as karma which uses the vehicle of reincarnation to accumulate merit. This requires the soul to exist in some realm outside of physical reality pending incarnations. This operation of the philosophy behind reincarnation surely appears outside the scope of the scientific method.

The philosophy of Platonism includes eternal mathematical verities which supposedly exist in a realm outside physical reality. Again there are no experiments to test the operation or existence of this claim.

JB wrote: “It has been known for quite some time in category theory that each category has its own ‘internal logic’, and that while we can reason externally about a category using classical logic, we can also reason within it using its internal logic — which gives a very different perspective. For example, our best understanding of intuitionistic logic has long come from the study of categories called ‘topoi’, for which the internal logic differs from classical logic mainly in its renunciation of the principle of excluded middle [9, 11, 30].”

This paragraph caused me to wonder how the internal logic can abdicate LEM (which looks like an operation to me) but at the same time makes use of the Platonic mathematical device of the Axiom of Choice to prove at least some Category theory results by using LEM. Maybe there is a simple explanation for this but now it looks to me like a tension of inconsistency due to attempting to unravel the sacred from the operation of the sacred not found when using testable experimental predictions to confirm a theory. Validation by experiments seems to be an increasingly rare requirement for quantum theories/interpretations as if they are no longer a bastion of faith.

Posted by: Stephen Harris on January 21, 2010 6:31 PM | Permalink | Reply to this

Re: The Sacred and the Profane

This paragraph caused me to wonder how the internal logic can abdicate LEM (which looks like an operation to me) but at the same time makes use of the Platonic mathematical device of the Axiom of Choice to prove at least some Category theory results by using LEM.

AC is not actually necessary for most results in category theory; you just have to use anafunctors instead of functors. Don’t let the historical accident that anafunctors have a funny name throw you off; really what we ordinarily call a “functor” should have been called a “strict functor,” with anafunctors getting the unadorned name of “functor”.

Posted by: Mike Shulman on January 22, 2010 6:08 AM | Permalink | Reply to this

Re: The Sacred and the Profane

I noticed from your link that the paper I had read must be quite old because Makkai was still promoting the adoption of the anafunctor concept. My reading of his paper is that he felt the “principle of isomorphism” was quintessential to Category theory and that the Axiom of Choice diluted or made impure the principle of isomorphism. His word was “violates”. So I thought this situation made a good example of the sacred idea profaned. You probably know the content of the following (historical) notes but probably someone doesn’t.

www.math.uni-hamburg.de/home/schreiber/anafun2.pdf
“Avoiding the axiom of choice in general category theory”
by M. Makkai (McGill University)

“In Category Theory, there is an underlying principle according to which the right notion of “equality” for objects in a category is isomorphism. Let me refer to the principle as the principle of isomorphism. Therefore, when singling out an object with a certain property, we should be content with determining the object up to isomorphism only.

General category theory in its usual form does not quite live up to the principle of isomorphism; the ubiquitous use of the Axiom of Choice in general category theory is a related fact. Whether or not an explicit choice of products is available, something of the canonicity of the resulting entity (functor) is lost when we make a particular choice of products.

The general form of the above type of use of the Axiom of Choice is in taking “the” adjoint of a functor on the basis of the representability of a family of Set-valued functors derived from the given functor. Every time we use the Adjoint Functor Theorem to get an adjoint, we use the Axiom of Choice in the described manner. There are similar violations of canonicity and attendant uses of the Axiom of Choice in the definitions of various concrete monoidal categories, and higher dimensional categorical objects.

In this paper, I propose a revision of the notion of functor, that of anafunctor, and consequent revisions of certain higher dimensional concepts, that makes possible a theory based more thoroughly on canonical constructions than ordinary category theory, and specifically, that rectifies the violations described above of the principle of isomorphism. The resulting theory avoids Choice to a large extent (although not completely; see below), and still has the same general form as classical general category theory.

The use of the prefix “ana” has been suggested by Dusko Pavlovic. He noted the use of “pro-” in category theory (profunctor, proobject), and noted that in biology, the terms “anaphase” and “prophase” are used in the same context.”


Posted by: Stephen Harris on January 22, 2010 4:37 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Complete non-sequitur, but karma is entirely profane as far as philosophical ideas go, which is why Buddhists and Hindu’s have theories and practices that try to transcend the wheel of karma. While the Christian category of sacred doesn’t map on to karma or nirvana, karma is very much embedded in the material world; the main claim being that actions have necessary consequences. Reincarnation is a theoretical postulate for explaining a conservation law: if all acts have consequences, and an agent dies before the consequences of a particular act have fruitioned, then the agent must be reborn in order to be a bearer of those consequences. In none of the various Indian schools of philosophy is there a halfway house where the soul rests between incarnations. Reincarnation as you imagine it is far closer to the Platonic idea of recapitulation. For the same reasons, classical Indian philosophers aren’t platonists about mathematics since the idea of a world outside the senses is a Greek/Christian idea.

Posted by: RajeshKasturirangan on January 22, 2010 1:00 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I enjoyed reading your comment. It was useful.

Posted by: Eugene Lerman on January 22, 2010 3:15 PM | Permalink | Reply to this

Re: The Sacred and the Profane

RajeshKasturirangan wrote:
“In none of the various Indian schools of philosophy is there a halfway house where the soul rests between incarnations. Reincarnation as you imagine it is far closer to the Platonic idea of recapitulation. For the same reasons, classical Indian philosophers aren’t platonists about mathematics since the idea of a world outside the senses is a Greek/Christian idea.”

SH: Isn’t the sacred mother tongue of Hinduism, Sanskrit? “The Tibetan word Bardo (Sanskrit: Antara-bhava) means literally “intermediate state” - also translated as “transitional state” or “in-between state” or “liminal state”. In Sanskrit the concept has the name antarabhava.”

I used the expression “a realm outside physical reality’. I’m aware that non-dualism states that there is only one reality. Does that make the following description untrue?

http://www.hinduwebsite.com/heavenhell.asp
“The universe consists of multiple worlds, layers and planes of existence, some known and some unknown, some within the field of awareness and sensory knowledge and some much beyond.”

Platonism, transmigration, and “innate knowledge”
… “This soul, according to Platonic thought, once separated from the body, spends an indeterminate amount of time in “formland” (see The Allegory of the Cave in The Republic) and then assumes another body. Therefore, according to Plato, we need only recall our buried memories to manifest innate knowledge.”

Posted by: Stephen Harris on January 22, 2010 5:16 PM | Permalink | Reply to this

Re: The Sacred and the Profane


http://www.nonduality.com/whatis9.htm
“Brahman is outside time, space, and causality, which are simply forms of empirical experience. No distinction in Brahman or from Brahman is possible.”

Posted by: Stephen Harris on January 22, 2010 6:04 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Minhyong said:

it might be useful to refer to ‘pure=sacred’ and ‘applied=profane.’

I’m not sure I agree with this. The pure/applied seems to be more about objectively what you do with the thing, while sacred/profane seems more about what subjective attitude you have toward it. I agree there does some to be some connection, but I suspect that it’s more along the lines of familiarity breeding contempt: if you use something all the time, it’s more difficult to regard it with sacred awe than if you only take it out occasionally to gaze upon in uncomprehending admiration. However, some parts of maths seem to naturally attract more of a sense that they are in some way profoundly at the bottom of everything, more pure and elevated than other parts, the mathematics of mathematics.

But since this is a matter of attitude, I suppose many people won’t feel this way.

David said:

Using a Bible to keep the door open is evidently a profane use

Using a Bible for divination or as a source of phrases to use in magic spells might also be regarded as a profane use, but one more closely related to its sacred use. In fact, this feels to me more like the sort of “sacrilege” that is involved in treating logic or category theory as just another mathematical subject.

Posted by: Tim Silverman on January 21, 2010 7:47 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I agree with you. I was writing quite hastily, but I think what I had in mind was in fact the association between attitude and usage you refer to. ‘Contempt’ however seems to be too strong a word in this connection. Van Dalen also seemed to be linking the profane with using the constructions of logic quite casually, and considered it a good thing up to a point.

Posted by: Minhyong Kim on January 21, 2010 10:46 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Minhyong wrote:

I agree with you.

Oh! What a pleasant surprise! You often say things I find interesting, but then we seem to end up talking past each other, rather frustratingly. I’m most chuffed that this didn’t happen this time.

‘Contempt’ however seems to be too strong a word in this connection.

Oh, yes, I quite agree. But “Familiarity breeds contempt” is a well-known proverb, while “familiarity breeds casualness” isn’t :-)

Posted by: Tim Silverman on January 22, 2010 2:03 PM | Permalink | Reply to this

Re: The Sacred and the Profane

Sorry to interrupt, but I still don’t get it, what’s the essential difference of the following dual pairs:

  • profane / sacred
  • means / ends
  • use as intended / misuse
  • applied / pure

?

When I tried to formulate short explanations of each pair I found the easisiest way was to stick with the example of the bible, my apologies for the cultural egocentrism:

misuse: You use a book to keep a door open. use : You read a book.

(Some people don’t see any use in books, at least since telephone books became obsolete, so we have to agree on some cultural background to agree on the meaning of this example).

profane: You study the bible to learn about the history of the Mediterranean region. sacred: You study the bible to learn about god’s message to his creation.

(This is from the point of view of e.g. catholic scholars, note that most do both at the same time).

means: You tell an adventurous story from the bible to get kids interested in religion. ends: You tell a story from the bible to convey a message from god to your audience.

applied: You learn bible verses in order to be able to disposses others. pure: You learn bible verses in order to be able to recite them when you crave to be closer to god.

The last examples will only make sense if you look at them from a certain point of view, e.g. that of a priest that is a member of the vatican and adheres to catholic dogma.

Posted by: Tim vB on January 22, 2010 4:41 PM | Permalink | Reply to this

Re: The Sacred and the Profane

I think my meaning is a bit different. I meant that inspiration often comes from brief intimations of the sacred as it manifests itself in daily affairs, even more than during a conscious ritual of some sort. The practical point was that if we refer to the awesome nature of some categorical construction too often, the idea might lose its appeal. This is one of the reasons I’m also a bit against referring too often to the beauty of mathematics as a whole.

Perhaps I was merely repeating the sentiment that many people have already epxpressed, that the thoughtful practice of the profane can be an effective vehicle for communicating the sacred.

Posted by: Minhyong Kim on January 21, 2010 10:56 PM | Permalink | Reply to this
Read the post Equivariant Stable Homotopy Theory
Weblog: The n-Category Café
Excerpt: Trying to understand equivariant stable homotopy theory from a higher-categorical perspective.
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