Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Saturday, December 30, 2006

M Theory Lesson 8

By now hopefully we suspect that the categorical concept of monad is important for probing possible definitions of observable. A monad $T$ naturally defines $T$-algebras. Let's look at an example from Mac Lane's classic Categories for the Working Mathematician (p 138, 1st edition).

Define a functor $P$ on Set as follows. On sets, $P$ sends $X$ to the set of all subsets of $X$. A function $f$ gets sent to $Pf$, which sends $S$ to the direct image of $S$ under $f$, as a subset of $X$. There is a natural transformation whose components are arrows from $X$ to $PX$ which take elements of $X$ to one point sets, and yet another natural transformation with arrows from $PPX$ to $PX$ which takes sets of sets to a union of sets. This data makes $P$ a monad, called the power set monad.

Recall that a complete semi-lattice $C$ satisfies that every subset $S$ has a least upper bound in $C$. A $P$-algebra is a complete semi-lattice with $x \leq y$ given by $h \{ x,y \} = y$ where $h$ is part of the data for a $P$-algebra, and it also gives the least upper bound for $S$. So the category of $P$-algebras is the category of all complete semi-lattices along with the appropriate arrows.

This has been mentioned a number of times before, so I hope I'm not boring you to death. Alas, I must run again.

Tuesday, December 26, 2006


A quick and jolly hello to my dear blogosphere friends and other friends who are far away. Happy New Year! Special wishes to those still struggling with the effects of the tsunami, which occurred exactly two years ago today, the 26th. I'm sitting in an air-conditioned internet cafe near the beach - can't complain. And there was snow in the mountains down south yesterday.

Friday, December 22, 2006

Awaiting Fairies

We all await the new constraints on the SM Higgs mass, which will follow from the new measurement of W mass. Tommaso has already given away that the first four digits are 8040 which can only mean $m_W$ = 80.40 GeV. Meanwhile, the no Higgs vote has stabilised at just above 50%. Will all CDF worker family members please refrain from feeding the CDF worker too much food over the next couple of weeks. They have a lot of work to do.

Update: It's now official ...
$m_W = 80.413 \pm 0.048$ GeV


Posts on this blog will probably be less frequent over the next few weeks. This is not due to the contagious lethargy of the season, but rather to the lack of reliable computing facilities in the neighbourhood.

For a good laugh, take a look at Scott Aaronson's post on becoming a mercenary in the String Wars after a cushy visit to Stanford. Personally, I'd settle for a bug infested hovel and some chips to eat, if anyone feels like flying me somewhere. I'd love to talk about quantum topos theory, M Theory and operads and the calculation of LHC amplitudes, but no one seems interested in such things.

Oh, my! I've been tagged by Mahndisa.
Six Weird Things About Me:
1. I own more ice tools than handbags.
2. I am extremely anxious about being even one minute late.
3. It's unusual amongst my peers that I don't think it's unusual that I don't have any substantial assets, such as a car.
4. I tend to be succinct.
5. When I was barely two years old I went missing and was found in a pantry, having just polished off a large jar of black olives.
6. I've been missing a few times. Last time it involved quite a number of search teams and an airforce iroquois, but I'll tell that story some other time.

Hmmm. I guess I'll tag Louise, Carl B, Matti, Nigel, Tommaso and Jonathon (but I don't mind if you ignore it).

Tuesday, December 19, 2006

Machian Madness

In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations.

These are the words of Ernst Mach (1838-1916). Amongst physicists, Mach is known as the last anti-atomist, persisting in his view well after the 1905 papers of Einstein. But Mach's views were not philosophically trivial, based on a line of reasoning going back to the Monadology of Leibniz. In his point of view Relationalism was being neglected in favour of the classical reductionism long in vogue.

From The Analysis of Sensations, published in 1897: The popular notion of an antithesis between appearance and reality has exercised a very powerful influence on scientific and philosophical thought. We see this, for example, in Plato's pregnant and poetical fiction of the Cave, in which, with our backs turned towards the fire, we observe merely the shadows of what passes (Republic, vii 1). But this conception was not thought out to its final consequences, with the result that it has had an unfortunate influence on our ideas about the universe. The universe, of which nevertheless we are a part, became completely separated from us, and was removed an infinite distance away.

Mach greatly influenced Einstein's thinking about Relativity. Ironically, the unfinished task of understanding Mach's principle for inertia must bring together both Atomism (in a more monadic guise) and Relativity.

Monday, December 18, 2006

Freezing Over

It might not happen as in the movie, but a possible consequence of changing ocean currents is a fast onset ice age. It is well known that rapid glaciation has occurred in the past. Who will be ready for that?

One respected colleague appears to think that girls will be on the beach in their bikinis, while the hardier men will be stuck in a crevasse on a large glacier. Well, the former would be preferable, I can assure you. It does not take long to make oneself some clothing. Sea levels will fall, but glaciers will quickly thicken and bury anyone who happens to find themselves near a current surface. Having myself fallen 20 metres to the bottom of a large crevasse on the Grosser Aletsch glacier, I can testify to the fact that infalling water and snow is unhelpful in fighting off the effects of hypothermia.

For those who are wondering, the new background photo is the view of Mt Cook from the terminal lake of the Hooker glacier. This glacier has been retreating rapidly for some time.

Sunday, December 17, 2006

M Theory Lesson 7

Craig Pastro says it should be called $M^2$ Theory (after the two conspirators) but the conventional name will suffice. Recall that in Lesson 6 we discussed points and how one really should worry about one's concept of point in thinking about quantum gravity. The star student charged ahead to think about generalised idempotents in relation to parity.

So let's go back to the relation $T^2 = T$. The reason for the capital $T$ (besides our swanky new latex capabilities) is that rather than sources or targets for arrows in a category, we would now like to weaken the relation and talk about monads.

A monad is a functor $T: C \rightarrow C$ with natural transformations $\mu: TT \rightarrow T$ and $\eta: 1 \rightarrow T$. Think of these as multiplication and unit. They satisfy an associativity and unit law. The square that represents associativity may have its vertices labelled by signs --, -+, +- and ++ where the source -- is the composition TTT before bracketing. Such parity cubes appear naturally in higher categorical contexts.

Saturday, December 16, 2006

Testing Testing

Thanks to the helpful gebar and Asymptotia I can now attempt ...

$\int_0^1 f(u_{ij}) \omega = \zeta(1,2) + \zeta(3)$

$H^{2}(\mathbf{T}^{+} \times \mathbf{T}^{+} , S_{m,n}(- \mu - 2 , - \eta - 2))$

Yipee!!! That was fun. Unfortunately, I haven't got categorical diagrams going yet, and only moduli integrals are interesting, but it's a start. Let's toast a great day: the day Blogger nerds start free publishing in mathematics! In other news, the no Higgs vote is growing, now at 57%.

Friday, December 15, 2006

Closer to the Top

Fermilab have announced evidence for the single top quark. In the blogosphere, Tommaso Dorigo and Tony Smith have recently discussed single top production at D0. The parameter determined was the Vtb, which lies between 0.68 and 1.0. This is consistent with the Standard Model, which predicts the blue line in the diagram.

In other news today, a story on sea level rise, which may be more imminent than many appreciate. There were also some nice pictures in New Scientist last week from Antarctica.

Update 16.12: Tommaso has some further cautionary comments to make about the D0 diagram above.

Tuesday, December 12, 2006

Weak Bimonoids

Without going into the diagrams, which unfortunately is most of the story, I will say a little about the recent AusCat talk of Ross Street on weak bimonoids in braided monoidal categories, which he also spoke about at ANU recently. The study of these new diagrams was partly motivated by the work of the mathematical physicist Robert Coquereaux.

A monoid is given by an object A and arrows m and eta for multiplication and unit. The braiding enters in considering AoB to be a monoid. Dually, a comonoid has a comultiplication and counit. A bimonoid is a monoid A which also has a comonoid structure. The trick to defining a weak bimonoid is to carefully choose a self dual set of diagrams such that bimonoids are always weak bimonoids.

The aim is to relate this definition to a concept of quantum category. In this setting the term quantum involves a linearisation process (so this is not really a quantum gravity kind of quantum). A category is usually specified by source and target maps from C1 to C0, the arrow and object sets. On linearisation one moves into a category of vector spaces rather than sets, and the objects C1 and C0 are comonoids in this category.

How about working with more general monoidal categories? A quantum category is by definition such a category V, with maps s and t into opp(C0) and C0 respectively. Note that the condition of oppositeness is null in the case of vector spaces. This is a natural definition because it looks like the dual of a diagram defining a bialgebroid (I looked on Wiki but they haven't quite got this far). In this dual diagram we consider objects A and R, where A is a weak bimonoid and R is the analogue of C0.

The nice diagrams allow one to show that R can be recovered from the structure of A. The proof crucially uses the idempotents t and s, and the splitting of idempotents in the sense of a Karoubi envelope.

Monday, December 11, 2006

It's Now Official

Update Tuesday: My sincerest apologies to Sean and friends at Cosmic Variance, who appear to have reinstated my posting rights (for now at least).

Update Sunday: Sigh. Didn't last long.


On Cosmic Variance today Sean asks the reader how they might better moderate the comment section on their blog. Of course, banning crackpots would do the trick. And on CV they seem to know exactly who the crackpots are, those CV guys being so much smarter than everybody else. Sean has failed to acknowledge in the post that they have already banned a number of crackpots. And lest there be any doubt that yours truly qualifies, I did the Brutally Honest Personality Test (courtesy of Chad Orzel), so I'm officially a Crackpot (INTJ) personality. On the age-gender balanced score for extroversion I actually managed 0%. You might as well shoot me now.

Sunday, December 10, 2006

M Theory Lesson 6

In the Higgs poll over at PF a whopping 50% of people are voting for no Higgs found in the next three years. Well, back to M Theory again.

Points take many guises. Eventually we will have the option of throwing them away altogether, using topos theory. After all, the question that really needs answering is, what is an observable? But things get pretty interesting before then. In Brannen's Clifford algebra one hunts out idempotents, which satisfy the simple projector relation tt=t. Similarly, one might use points in projective (twistor) space, described by the Jordan algebra of Michael Rios.

Are there other ways in which points are naturally associated to operators satisfying tt=t? Why yes, using categories in a fairly simple way, as follows. One can think of the objects of a category as the identity arrows on those objects. For example, a set being a discrete category which only has objects, only has identity arrows. Now let t be the target map, sending an arrow f to its target identity arrow tf. Then clearly tt=t. Similarly, one can discuss a source map s.

For those who are really keen on playing, remember that a topology t in the form of an arrow from Omega to Omega in an elementary topos also obeys the relation tt=t.

Saturday, December 09, 2006

Review Reviewed

I'm not the first, but it looks like fun: the String theorist Polchinski has a review in American Scientist about the two books The Trouble with Physics and Not Even Wrong. I agree with much of his critique, so I will focus on reviewing the String aspects of the review.

The Standard Model is a quantum field theory...

OK, we have a problem right here at the start. The Standard Model is just that: a model. But we'll let this go, since it's just semantics and such language is common usage. which particles behave as mathematical points...

OK, this one is a bigger problem. What does he mean by point? How are Strings background independent if particles have no meaning independent of classical points? Somehow I don't think Polchinski is thinking of functors between toposes when he uses the word point.

Smolin presents the rise and fall of string theory as a morality play... But this story, however grippingly told, is more a work of drama than of history.

Yes, this is quite true, but it isn't really surprising for a String theorist to think so. And why do they all keep going on about this vacuum energy idea?

The review then moves on to background independence. No wonder these guys can't stop arguing about this. Both sides think their ideas are more background independent than the other. So long as spin foams ignores things like AdS/CFT and T-duality they cannot possibly be talking about gravity, and on the other hand, as mentioned above, particles may be localised but a point is not just some piece of classical moduli. If one is willing to dive head first into Derived Categories it is difficult to understand why this is not understood. Sigh.

New physical theories are often discovered using a mathematical language that is not the most suitable for them.

Hang on a minute! What theory? I don't see any theory. Some nice maths, sure. But when you talk about T-duality there is never any discussion of Machian principles or experimental evidence for them. If I've missed such papers, please direct me to them! Ahh, wait a minute. They call it the Holographic Principle. Nice idea, but we don't actually get any physics from the way it is formulated. Again, if I'm wrong about this, please tell me!

Extending this principle to spaces with the edges free will require a major new insight.

Er, like a new Machian principle maybe? Would it be a problem if we threw out the Stringy particle zoo?

It is possible that the solution to this problem already exists among the alternative approaches that Smolin favors.

Er, well, no. That's not to say that it does not exist. Later on, the review sinks back into a discussion on Dark Energy.

Friday, December 08, 2006

Back to Business

Hello again! Connection problems here are finally sorted out, so we can get back to business. It was too nice a day yesterday to stay inside anyway. I went with some kea friends to the Basin in Ku-ring-gai where we went swimming and visited a local wallaby, who had a little baby joey in her pouch.

It is very easy to grow impatient in a city. Some things just cannot happen unless conditions are exactly right. Take Fyfe Pass, for example. In Fyfe's day the ice was much thicker than it is now. These days it isn't exactly what one would call a pass, but one can still cross, given the right conditions: a fine day in late spring after the cirque has avalanched out and left a high pile of debris almost to the top of the bluff at the base of the gut. It's simple climbing from there really, with only one flaky abseil across the waterfall. A lot of things are like that. Not so hard when you look at them the right way.

Sunday, December 03, 2006

Bloggy Bloom

Having installed a CQ Counter a week or two ago, I have been able to watch what sort of people visit this site. Very entertaining. I have also been getting emails from complete strangers about comments on my blog. Isn't it nice to know that someone is paying attention? In mountaineering there is a term for getting someone to follow you casually, secure in their knowledge that they know more than you do: it's called sandbagging.

Other physics blogs have been rather quiet lately, with the exception of Babe in the Universe and Clifford, who likes to chat. Clifford says he's going to remember one of my nutty lines for his next Hollywood party. He's so sweet. People must be busy working productively, as they put it. Well, I guess I should get back to the abstract nonsense then.

Saturday, December 02, 2006

Back to the Future

Well, I feel that one's Morgan-Phoa blogging duties must not be neglected. It was a wonderful week at ANU, thanks to the hospitality of Amnon Neeman, James Borger and Boris Chorny. It was very hot and dry, except for one cool evening, which we spent feasting at Tosolini's. Most of us stayed at Toad Hall, which is a whole 150m from the Maths building!

The format was informal, the idea being to speak about some big problem in one's own area of interest that should have wider appeal. I have never been to a workshop where this worked so well. Tuesday kicked off with Steve Lack giving an introductory talk on Topos Theory. For this week the term introductory assumes that one is either really into Category Theory or really, really good at Algebraic Geometry or Homotopy Theory.

James Borger followed with some revision on Algebraic Spaces and a question about related, weaker topos like structure. To begin with, an old problem with Rng, the category of rings, is that it doesn't have nice limits for doing Algebraic Geometry. The traditional solution is to use schemes. But then one ends up relying on the Axiom of Choice even though it isn't really needed, and the whole theory is very complicated. So instead of a category of schemes one defines a category AlgSp of Algebraic Spaces. This is the opposite category to Rng (= affine schemes) equipped with an etale topology. So things start to look more topos theoretic. James wants to think of a subfunctor from AlgSp into etale sheaves as a kind of half topos.

Boris Chorny modestly launched into some pretty technical ideas on small presheaves. A motivation for his questions was the Motivic Homotopy of Voevodsky, something I would dearly like to understand if I lived another 100 years. This means looking at functors from (wait for it) the opposite category of finite smooth schemes over S with the Nisnevich topology into the category of simplicial sets. Boris says that the new understanding of Motivic Spaces means that we need to redo the classical theory. For instance, is there a way of doing Motivic Homotopy in bigger categories?

Mark Weber spoke about 2-toposes. He has been doing some very interesting work on this. A nice example should be a category of categories CAT. So what is the analogue of the subobject classifier and truth arrow? One thing that works is the forgetful functor from pointed sets into Set. So the suboject classifier becomes the whole category Set! Mark's notion of 2-topos has the advantage of being able to axiomatise the notion of size. Other examples are based on internal categories in globular sets.

On Tuesday afternoon we had a departmental seminar by Ross Street on his recent work (with Craig Pastro) on quantum categories and weak Hopf algebras in braided monoidal categories. I will talk about this at some later date.

Wednesday morning was Operad Time. Yummy! After a short talk by a certain disreputable physicist, Michael Batanin asked the participants about homotopy types and the search for an ideal theorem about their characterisation by higher groupoids. Fortunately this involved some introductory words on higher operads. Things get particularly interesting when one gets to dimension three. 3-homotopy types require Gray groupoids. These arise as algebras of the Gray 3-operad G. Let's briefly describe this. For the 0-tree and for any 1-tree, G is the singleton. For a 2-tree which is labelled by ordinals (m1,m2,...,mk), G is the shuffles on (m1,m2,...,mk). And finally, for a 3-tree G is given by a Cartesian product of G for the boundary 2-trees. This structure is due to the weakened interchange law of Gray categories. The question is: what are the higher dimensional analogues of this operad? Batanin also gave the Colloquium talk on Thurdsday afternoon, about the relation of his work to Deligne's conjecture.

Simona Paoli was wondering about a model structure for internal categories. She discussed a 2005 paper by Everaert et al and open questions related to the Tamsamani approach to higher categories. This involves some intriguing looking cubical structures, which I don't understand at all. Alexei Davydov then spoke about autoequivalences, and Amnon Neeman about equivalences for derived categories.

On Thursday morning we were visited by Peter Bouwknegt, who posed the question of a good definition for C* algebra objects in monoidal categories, with motivation from his work on T-duality. At this point there was some laughter on the invasion of physics into such a pure mathematics workshop. Later on we actually found some time for more introductory talks. Boris Chorny told us about the Calculus of Functors. He focused on a dictionary between the traditional calculus of functions and the Homotopy Calculus. To begin with, instead of a function from a manifold M into R one has a functor from pointed topological spaces into either a category of topological spaces or a category of spectra. The notion of |x - y| small becomes f: X --> Y is highly connected. The replacement of derivatives defined using h-->0 is quite abstract: the derivative of a functor F with respect to a pointed space X is the homotopy colimit (as n --> oo) of the n-fold loop space of the homotopy fibrations of F(X v S^n) --> F(X). Er, yeah, OK. Anyway, the point is that spheres S^n for large n are like highly contractible spaces. So we had a real h-->0 and now we have a discrete n --> oo.

My favourite talk for the week was the last: James Borger speaking about Lambda Rings and related goodies. But enough blathering from me on all this!