- #151

- 859

- 0

**event in July**

Oxford have something cool coming up: an event under the name

**CATS, KETS and CLOISTERS**. See

http://se10.comlab.ox.ac.uk:8080/FOCS/CKCinOXFORD_en.html [Broken]

Last edited by a moderator:

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter setAI
- Start date

- #151

- 859

- 0

Oxford have something cool coming up: an event under the name

http://se10.comlab.ox.ac.uk:8080/FOCS/CKCinOXFORD_en.html [Broken]

Last edited by a moderator:

- #152

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,852

- 10

- #153

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,852

- 10

A good discussion on categrification here: http://www.dcorfield.pwp.blueyonder.co.uk/2006/05/scope-of-categorification.html.

"Every object in mathematics is the Euler characteristic of some complex". Warms the cockles of my old topologist's heart.

"Every object in mathematics is the Euler characteristic of some complex". Warms the cockles of my old topologist's heart.

Last edited by a moderator:

- #154

- 859

- 0

selfAdjoint said:Warms the cockles of my old topologist's heart.

With good reason, selfAdjoint. The statement is highly non-trivial. Deeper than the deep blue sea. What is cohomology about, really? One certainly needs some sort of [itex]d[/itex] operator with a property like

[itex]d^2 = 0[/itex]. Well, let's use the symbol [itex]1[/itex] instead of [itex]0[/itex] because, after all, we don't just want Abelian cohomology.

An example of a nice monad is the

There is a progression of ideas here:

1. everything is an object (in a set)

2. everything is an arrow (in a category, or points becoming Strings)

3. everything is a functor (by lifting an arrow into

4. everything is a ....

Last edited:

- #155

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,852

- 10

Kea said:With good reason, selfAdjoint. The statement is highly non-trivial. Deeper than the deep blue sea. What is cohomology about, really? One certainly needs some sort of operator with a property like

. Well, let's use the symbol instead of because, after all, we don't just want Abelian cohomology. Some things in mathematics are non-commutative. Then clearly we are talking about monads. Yes, the category theoretic kind.

That's great Kea! Does the monad concept also embody the cobordism aspect of cohomology?

- #156

- 859

- 0

selfAdjoint said:Does the monad concept also embody the cobordism aspect of cohomology?

Oh, yes, although as a measly physicist I don't claim to understand much. But the following might be of some interest:

Imagine we could show that the category of vector spaces was a quantum topos. The truth arrow would be a map from [itex]\mathbb{C}[/itex] into [itex]\mathbb{C} \oplus \mathbb{C}[/itex], the qubit. It's OK to projectivise and get the 2-sphere going to [itex]\mathbb{C}\mathbb{P}^{3}[/itex], which might be familiar to people who know a little bit about classical causality.

Anyway, if one has a pair of

Exercise: recover the Sundance preons from these sort of diagrams. Of course, mass hasn't been taken into account yet so there will only be one generation.

Note that the D-brane people have also vigorously studied the recovery of the SM from [itex]\mathbb{Z}_{6}[/itex] orbifolds and products of three 2-tori. See Honecker-Ott and Bailin.

- #157

- 859

- 0

You know, octopi are very strange creatures.

- #158

- 859

- 0

A. D. Lauda

http://www.tac.mta.ca/tac/volumes/16/4/16-04.pdf

Anyway, he looks at these ambijunctions in 2-categories first, which are secretly

I wonder what a categorified CFT would be physically?

- #159

- 859

- 0

- #160

- 4

- 0

- #161

- 4

- 0

- #162

- 4

- 0

oops sorry about that

- #163

arivero

Gold Member

- 3,363

- 102

selfAdjoint said:

It is OT, I know, but you have intriged me: are you sure about date and formula?

- #164

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,852

- 10

arivero said:It is OT, I know, but you have intriged me: are you sure about date and formula?

As you say, OT. Maybe we should move these two posts to the History forum?

They didn't have the notation, and their expression was in terms of "forms" which were functions over the Euclidean (or Eudoxian) proportions. A modern view of the proportions sees them as positive real numbers, but that was way in the future for these medieval thinkers. Because the proportions are developed rigorously in Euclid's book, the Merton scholars were able to do careful thinking about their forms. They had names instead of notation: A "uniform form" was a constant function from the proportions to the proportions; a "diffform form" was a linear function, and a difformly difform" form was a constantly accelerated function, i.e. a quadratic, although I don't think they ever spotted the link between difformlly difform and the square function.

The domain variable of the form (corresponding to t in the formula) was called Latitude, and the range (corresponding to v) was called Longitude. And the Merton College mean speed rule was "The total Longitude made good by a difform form (i.e. linear function) over a range of Latitude is the same as made good by a uniform form (i.e. constant function) of value the mean of the two Longitudes at the end of the range". Work it out.

Nicole Oresme, a wonderful French mathematician contemporary with these Merton scholars, proved this theorem by graphing the Latitude versus the Longitude; the graph came out a right triangle (on top of a rectangle) and he applied the rule for the area of a triangle (area = one half base X height) to demostrate the mean speed rule.

{Added} Another Frenchman, the philosopher Jean Buridan, asserted that objects in motion have a quality he called impetus which was propotional to the speed of the object and to its weight. This is not to be confused with the stupid thing also called impetus taught by the scholastics of Galileo's time. And Buridan also claimed that the impetus of a falling body was a difform form of time. All any of these people would have had to do would be to apply the Merton rule to Buridan's impetus to derive the law of falling bodies. But right about this time two things happened. The Black Death raged across Europe, and it is conjectured that Buridan died of it. And the long standing schism in the Catholic Church, which saw two separate papacies, was healed and the reunified church took the opportunity to crack down on dangerous thought. Oresme was offered a bishop's hat, at a remote country diocese in Normandy - he was a Norman by birth. And Merton College, whose statutes were much more liberal than the typical Oxford college, was "normalized", ending its research program.

Last edited:

- #165

marcus

Science Advisor

Gold Member

Dearly Missed

- 24,757

- 788

Kea, if you have a free moment you might want to glance atKea said:Care of David Corfield, there is a new Category blog: http://bosker.wordpress.com/

http://arxiv.org/abs/quant-ph/0606114

q-Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation

Louis H. Kauffman, Samuel J. Lomonaco Jr

87 pages, 58 figures

========

this isn't a recommendation but it has 58 figures

and the title contains several hot keywords.

besides being by Louis Kauffman how can you miss, it's got to be worth a look

Last edited:

- #166

- 859

- 0

Aaron D. Lauda, Hendryk Pfeiffer

http://www.arxiv.org/abs/math.QA/0606331

This has to do with loop+arc diagrams for state sums - really cool stuff - and the mathematicians should be impressed too!

- #167

- 859

- 0

Aaron D. Lauda, Hendryk Pfeiffer

http://www.arxiv.org/abs/math.QA/0606331

This has to do with loop+arc diagrams for state sums - really cool stuff - and the mathematicians should be impressed too! Roughly speaking, it's about piecing together two Frobenius type structures - one commutative and one symmetric - and characterising their interaction. The author's call this latter bit

- #168

- 859

- 0

Bee's Top Ten https://www.physicsforums.com/showthread.php?t=125537

BeeF for Dinner https://www.physicsforums.com/showthread.php?t=125494

Stroop Theory https://www.physicsforums.com/showthread.php?t=122662

f-h/Distler chat https://www.physicsforums.com/showthread.php?t=125336

Thread for Careful https://www.physicsforums.com/showthread.php?t=125556

Against Symmetry https://www.physicsforums.com/showthread.php?t=122422

Sundance and PI https://www.physicsforums.com/showthread.php?t=113185

The same Elephant https://www.physicsforums.com/showthread.php?t=117286

Last edited:

- #169

selfAdjoint

Staff Emeritus

Gold Member

Dearly Missed

- 6,852

- 10

Interesting new paper on the arxiv:

gr-qc/0607038

From: Ioannis Raptis

Date: Mon, 10 Jul 2006 14:38:35 GMT (28kb)

A Dodecalogue of Basic Didactics from Applications of Abstract Differential Geometry to Quantum Gravity

Authors: Ioannis Raptis

Comments: 16 pages, preliminary version

We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide-cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of `poetic language'

From "lessonet 2:

The sole dynamical variable in ADG-gravity is an algebraic A-connection field D5 acting (on the local sections of) a vector sheaf E defined on an in principle arbitrary topological space X. The physical kinematical configuration space in the theory is the moduli space AA(E)/AutE of the affine space of connections A modulo the (local) gauge transformations in the principal sheaf AutE. The ADG-formalism on gravity is called

sort. At the same time, (the products of) measurements (:numbers) are our own actions (and numbers our own artifacts/inventions), hence no physical reality, and no interpretation as the gravitational field living ‘out there’, should be given to the spacetime metric, like in the original formulation of GR. This is consistent with our viewing gravity as a pure gauge theory—ie, that the gravitational field is simply the connection D.

(The bolding is mine).

- #170

- 859

- 0

Marcus has found a new gem

http://arxiv.org/abs/quant-ph/0608035

Bob Coecke, Dusko Pavlovic

36 pages, 46 pictures

which definitely belongs here. It extends the categorical diagram techniques of Kindergarten Quantum Mechanics to discuss measurement in a novel way in terms of special internal objects. This really is a good way to think of quantum mechanical information processes.

- #171

- 859

- 0

There is a new paper by Fuchs et al (the categorical CFT people)

http://www.arxiv.org/PS_cache/hep-th/pdf/0607/0607247.pdf [Broken]

Last edited by a moderator:

- #172

- 859

- 0

- #173

marcus

Science Advisor

Gold Member

Dearly Missed

- 24,757

- 788

I've been reading your posts there with considerable interest. You've recorded your thoughts there for some time without making a general announcement of it. Thanks for sharing with us!Kea said:

- #174

- 859

- 0

- #175

- 470

- 0

whoa! nice blog Kea!

Share: