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July 30, 2011

Coinductive Definitions

Posted by Mike Shulman

I’ve come to believe, over the past couple of years, that anyone trying to study ω\omega-categories (a.k.a. (,)(\infty,\infty)-categories) without knowing about coinductive definitions is going to be struggling against nature due to not having the proper tools. But although coinductive definitions are a basic notion in mathematics, for some reason they don’t seem to be taught, even to graduate students. Write something like

A 1-morphism f:xyf\colon x\to y in an (n+1)(n+1)-category is an equivalence if there exists a 1-morphism g:yxg\colon y\to x and equivalences 1 xgf1_x \to g f and fg1 yf g\to 1_y in the relevant hom-nn-categories; and every 1-morphism in a 0-category is an equivalence

and any mathematician (who has some inkling of what nn-categories are) will be happy. If you ask why this definition isn’t circular, since it defines the notion of “equivalence” in terms of “equivalence”, the mathematician will say “it’s an inductive definition” and expect you to stop complaining. But if you write something like

A 1-morphism f:xyf\colon x\to y in an ω\omega-category is an equivalence if there exists a 1-morphism g:yxg\colon y\to x and equivalences 1 xgf1_x \to g f and fg1 yf g\to 1_y in the relevant hom-ω\omega-categories

the same mathematician will object loudly, saying that this definition is circular. (In fact, not very long ago, that mathematician was me.) But actually, this latter is a perfectly valid coinductive definition.

Posted at 8:11 PM UTC | Permalink | Followups (14)

July 27, 2011

Local and Global Supersymmetry

Posted by Urs Schreiber

The field of fundamental high energy physics – that part of physics that deals with fundamental particles probed in particle accelerators – is witnessing interesting developments these days: after decades of only a minimum of new experimental observations of interest, finally plenty of data has been collected and now analyzed. And finally a multitude of theoretical models that have been developed over the years can be tested against experiment.

Apart from lots of new information about which mass the hypothetical Higgs particle – if it indeed exists – does not have, one of the striking experimental results is that they increasingly – and by now strongly – disfavour what are called supersymmetric extensions of the standard model of particle physics . Well-informed discussion of these developments can for instance be found on this blog.

In the course of these developments, I see and hear a lot of discussion around me of whether the concept of “supersymmetry” as such is thus experimentally ruled out. There is an enormous amount of literature revolving around the concept of supersymmetry quite independently of the “supersymmetric standard model of particle physics”. Is all that now proven to be ill-conceived? Is “supersymmetry” being shown to play no role in nature?

We almost had a discussion of this kind also here on the blog recently. Since this is a widespread misunderstanding, I thought I’d try to say something about it here.

A little appreciated but important fact is this: there is a crucial distinction between what is called local supersymmetry and what is called global supersymmetry and between target space supersymmetry and worldvolume supersymmetry. I have tried to say a bit about this in the nnLab entry

supersymmetry.

Even less widely appreciated seems to be the following noteworthy fact: local worldline supersymmetry is experimentally verified since 1922 – when the Stern-Gerlach experiment showed that there are fundamental particles with a property called spin : these spinning fermion particles – the electrons and quarks that you, me, and everything around us is made of – happen to have worldline supersymmetry .

I have tried to give an indication of this in the entry

spinning particle

which also collects a bunch of original references and textbook chapters where this fact is discussed in detail.

So the assumption that there is local worldvolume supersymmetry in nature is not speculation, but experimental fact as soon as there is any spinor in the world. Of course this is not the global target space supersymmetry that is currently being experimentally ruled out at the LHC. So it is good to distinguish these concepts. And indeed, despite of what many people are on record as having said: nothing at all in sigma-model theory implies that a supersymmetric sigma-model (such as the spinning particle, or the spinning string, for that matter) has target space backgrounds that generically are globally supersymmetric. On the contrary: the generic background will not be!

This simple fact seems not to be widely appreciated, either. It is the direct analog of the following self-evident bosonic statement: while ordinary gravity is a locally Poincaré-invariant theory (a Poincaré-gauge theory) its generic solution – a given pseudo-Riemannian manifold – does not have a nontrivial action of the Poincaré group or of any of its nontrivial subgroups. It will only have such actions if it has flows of isometries given by Killing vectors. Analogously, the generic solution to a theory of supergravity – which is a locally super-Poincaré-invariant theory– does not have any covariantly constant spinor, hence the perturbative quantum field theory on this background does not have a global supersymmetry.

This has always been clear. Some more sophisticated discussion of this point is for instance in

Dienes, Lennek, Sénéchal, Wasnik, Is SUSY natural? (arXiv:0804.4718)

which is effectively a detailed expansion of the statement about generic absence of global symmetries in backgrounds.

There’d be much more to say (and there’d be need to expand the above nnLab entries much more), but I must stop here and take care of other tasks. The upshot is:

  1. there is still all the reason in the world to believe that the concept of local supersymmetry (aka: supergravity) is fundamental for our world – not the least because 1-dimensional worldline supergravity is an experimentally observed fact;

  2. the models of global supersymmetry that are currently being ruled out by experiment are not rooted in theory, but in phenomenological model building. The general theory of supersymmetry is as unaffected by these models being ruled out as the theory of gravity is unaffected by a given cosmological model being ruled out.

Posted at 2:10 AM UTC | Permalink | Followups (45)

July 25, 2011

Bohr Toposes

Posted by Urs Schreiber

To every quantum mechanical system is associated its Bohr topos : a ringed topos which plays the role of the quantum phase space . The idea of this construction is that it naturally captures the geometric and logical aspects of quantum physics in terms of higher geometry/topos theory.

Below the fold I try to give an exposition of the facts that motivate the construction, the construction itself, and an indication of the resulting notion of presheaves of Bohr toposes associated with every quantum field theory. For more details and further links see the nnLab entry Bohr topos .

See also the previous entry A Topos for Algebraic Quantum Theory .

Posted at 12:48 AM UTC | Permalink | Post a Comment

July 11, 2011

Doctrinal and Tannakian Reconstruction

Posted by David Corfield

Interest in doctrines at the Café goes right back to one of its earliest posts nearly five years ago, and even to one a few days earlier. We’ve begun to record some material at the doctrine page in nLab, but I’m sure there’s more wisdom from the blog to be extracted. Looking over the material again raised a question or two in my mind, which I’d like to pose now.

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the recovery of a theory from the category of models of that theory.

Posted at 11:01 AM UTC | Permalink | Followups (42)

July 7, 2011

Operads and the Tree of Life

Posted by John Baez

Tomorrow I’m giving a talk about an operad that shows up in biology. I wrote my lecture notes in the form of a blog entry:

A remark by Tom Leinster was important in helping me figure out this stuff.

Posted at 5:59 PM UTC | Permalink

July 2, 2011

Definitions of Ultrafilter

Posted by Tom Leinster

One of these days I want to explain a precise sense in which the notion of ultrafilter is inescapable. But first I want to do a bit of historical digging.

If you’re subscribed to Bob Rosebrugh’s categories mailing list, you might have seen one of my historical questions. Here’s another: have you ever seen the following definition of ultrafilter?

Definition 1  An ultrafilter on a set XX is a set UU of subsets with the following property: for all partitions X=X 1⨿⨿X n X = X_1 \amalg \cdots \amalg X_n of XX into a finite number n0n \geq 0 of subsets, there is precisely one ii such that X iUX_i \in U.

This is equivalent to any of the usual definitions. It’s got to be in the literature somewhere, but I haven’t been able to find it. Can anyone help?

Posted at 11:35 PM UTC | Permalink | Followups (25)

July 1, 2011

Nikolaus on Higher Categorical Structures in Geometry

Posted by Urs Schreiber

Yesterday Thomas Nikolaus – former colleague of mine in Hamburg – has defended his PhD.

His nicely written thesis

Higher categorical structures in geometry – General theory and applications to QFT

discusses plenty of subjects of interest here; the main sections are titled:

  1. Bundle gerbes and surface holonomy

  2. Equivariance in higher geometry

  3. Four equivalent versions of non-abelian gerbes

  4. A smooth model for the string-group

  5. Equivariant modular categories via Dijkgraaf-Witten theory .

Have a look at his slides for a gentle overview.

Myself, I have to dash off now. Maybe I’ll say a bit more about what Thomas did in these chapters a little later. Or maybe he’ll do so himself…

Posted at 5:27 PM UTC | Permalink | Followups (9)