The main site will be temporarily down for 48 hours starting at 6pm Central Standard Time on Friday 6 January. In Greenwich Mean Time, that’s midnight at the end of 6 January and the beginning of 7 January. They’re shutting off the power to the building at the University of Texas where the server lives.

We’ve worked hard to get a good mixture of mathematicians on the one hand, and theoretical computer scientists on the other. Two of the speakers, Antony Maciocia and Peter Kropholler, are mathematicians — Maciocia is in algebraic geometry, and Kropholler at the intersection of group theory and topology. The other two, Dirk Pattinson and Thorsten Altenkirch, are in theoretical computer science — though their talks should definitely be interesting to mathematicians.

All the speakers have been firmly requested to make their talks broadly accessible to what we hope will be a very mixed audience. So I’m looking forward to an excellent afternoon.

Here are the talks. To see the abstracts, click on the titles.

• Antony Maciocia (Edinburgh): Triangulated categories in algebraic geometry
• Thorsten Altenkirch (Nottingham): Monads need not be endofunctors
• Peter Kropholler (Glasgow): My favourite adjunctions
• Dirk Pattinson (Imperial): Category-theoretic proof theory of modal logics

Incidentally, I think the Glasgow speaker has been inspired by some other talks in Glasgow…

If you think you’d like to come, send a mail to me (T.Leinster#maths,gla,ac,uk) or scotcats#cis,strath,ac,uk. It’s not essential that you do so — you can just turn up — but it would help, and you definitely should if you’re intending to come for dinner.

Category Theory and Philosophy of Mathematics Today
(a satellite event associated with the workshop Foundations of
mathematics: What and Why?, Paris, May 18 – June 25 ).

When: Three days: May 17, 2010 from 9 A.M. to 4 P.M, May 31, 2010 from 9.30 A.M. to 4 P.M, and June 14, 2010 from 9.30 A.M. to 6.30 P.M.

Where: Paris, Ecole Normale Supérieure (45, rue d’Ulm, 75005).  Salle “W” (staircase B, 3d floor): May 17, 31 and June 14 in the morning.   Salle Beckett (ground floor): June 14 afternoon.

Languages: French and English

Here’s the program:

le 17 mai / May 17 (ENS, salle “W”):

9h – 9h30 / 9 – 9.30 A.M.:
Andrei Rodin:
Welcoming address

9h30 – 10h50 / 9.30 – 10.50 A.M.
Yuri Manin
Languages of Mathematics

11h – 12h20 / 11A.M. – 12.20 P.M.
Christian Houzel (*)
TBA

12h20 – 14h  / 12.20 P.M. – 2 P.M.
LUNCH

14h-15h / 2 – 3 P.M.
René Guitart (Paris-Diderot)
Topos and Algebraic Universe

15h – 16h / 3 – 4 P.M.
Noson Yanofsky
On the Utility of Category Theory

le 31 mai / May 31 (ENS, salle “W”):

9h30 – 10h30  /  9.30 – 10.30 A.M.
Alberto Peruzzi (*)
Categorical philosophy, rather than philosophy of category theory

10h40 – 12h  /  10.40 A.M. – 12 P.M.
Jean Sallantin & Dominique Luzeaux
Ideosphères et mathèmes : utilisation d’outils catégoriques

12h – 14h  / 12 P.M. – 2 P.M.
LUNCH

14h – 15h  /  2 – 3 P.M.
Alain Prouté
On the link between topoi and the vernacular of mathematics

15h – 16h  / 3 – 4 P.M.
Andrei Rodin
Towards categorical foundations of geometry: a historical approach

le 14 juin / June 14: (ENS, salle “W”, salle Beckett)

9h30 – 10h50  /  9.30 – 10.50 A.M. (salle “W”)
Jean Bénabou
“Transcendent” methods in Category  Theory

11h – 12h  / 11 A.M. – 12 P.M. (salle “W”)
Marc Lachièze-Rey (CNRS & Paris-Diderot),
Catégories et physique: un example de la gravitation quantique

12h20 – 14h  /  12.20 P.M. – 2 P.M.
LUNCH

14h – 15h  /  2 – 3 P.M.  (salle Beckett)
Giuseppe Longo
L’esprit des categories et la vie

15h – 16h  /  3 – 4 P.M. (salle Beckett)
Pino Rosolini (Genoa)
Categories and sets

16h15 – 17h15  / 4.15 – 5.15 P.M. (salle Beckett)
Jean Petitot
TBA

17h15 – 18h30  / 5.15 – 6.30 P.M. (salle Beckett)
Table Ronde / Round Table

(*) sous réserve d’une confirmation / to be confirmed

Until then, you can leave comments at the loose ends thread here.

Thanks again for your patience.

Update (6 May) The cafe is now visible again, but not fully operational. With any luck it will be working properly in a couple of days.

The software problems seem to be mostly fixed. One or two people still find themselves unable to leave comments. If this happens to you, please tell me or one of the other hosts. We’ll post the comment for you and see if we can get the problem sorted out. (And if you just can’t wait to get that comment into cyberspace, you can put it at the loose ends entry at this site.)

Thanks for your patience.

In my paper A diagrammatic approach to Hopf monads I had lots and lots of surface diagrams — these are the one-dimension higher version of string diagrams which can be useful for notating morphisms in monoidal 2-categories or 3-categories. So I had to draw a hundred or so pictures like this one.

A surface diagram involving a Hopf monad

[I hope to write something more about Hopf monads at the café at some point very soon.]

I drew all of these pictures in xfig — a standard 2-dimensional drawing package in which you have to do all of the 3-d stuff by eye and can’t do any fancy shading or anything. It would be really nice if I could actually create these as 3-d objects and then just export some projection of them, possibly suitably ray traced. I had an attempt at creating some categorical surfaces using blender, which is a splendid piece of open source software which you can use to make movies like Shrek or Monsters Inc. Here are two of my efforts.

The first represents the associator natural transformation in a monoidal category (see the Hopf monads paper linked above for the details).

The second is the so-called swallowtail relation (see Figure 25 on page 40 of HDA4).

As well making movies with such models you can also embed the 3d model in to pdf files and, provided you’re using acrobat your readers can rotate the surface for themselves. Here’s an example from meshlab.

Whilst this is all well and good, there are at least two things I would like to be able to do that I currently can’t.

• Draw on the surfaces. I want to be able to draw what are essentially morphisms, like in the Hopf monad picture above, and also to be able to label them.
• Integrate with mathematics software such as sage or maple, or possibly even with some scripting language such a python. Then I wouldn’t have to input the surfaces with a mouse incredibly slowly in blender, but would be able to somehow code up the surfaces.

One thing to note is that these “surfaces” can be singular, as in the associator example above, so are not so easy to handle in standard packages with tend to expect non-singular things.

So, has anyone else tried using 3d software for modeling surface diagrams?

And not-entirely-coincidentally, at long last I’ve put online a draft of my (first) paper about the stack semantics and comparing material and structural set theories. You can get it from my nlab page:

• The stack semantics

There are also slides from today’s talk and one from last November.

In brief, the idea of the stack semantics is to extend the internal logic of a topos to a language which can talk about unbounded quantifiers (quantifiers of the form “for all sets” rather than “for all elements of A” for some fixed set A). In this extended language, we can then state topos-theoretic axiom schemas which are as strong as the full separation and replacement axioms of ZF. (Ordinary topos theory is only equiconsistent with bounded Zermelo set theory, which is much weaker than ZF.) This generalization is extremely easy—even easier than some presentations of the ordinary internal logic—and is in fact implicit throughout topos theory, but has seemingly never been written down precisely before.

If that intrigues you, then you may want to look first at the talk from November; it’s aimed at category theorists without much experience in categorical logic. Then you can go on to look at the paper itself, most of which should (I hope) also be fairly accessible. Comments are welcome!

Note: This entry and its comments have been copied back to the main n-Category Cafe site. Please submit further comments there (unless the comment system there starts to have issues again).

• John Baez and John Huerta, An Invitation to Higher Gauge Theory.

Tim van Beek wanted a heads-up when it was done, so he could make another round of corrections. Indeed, I’d love comments from all of you!

There are lots of juicy example applications in this paper! The idea is to get lots of theoretical and mathematical physicists to see those examples without requiring them to already be experts on category theory. They will then fall in love with -categories, turn into an army of Urs Schreiber clones, and take over the universe. My duty to mathematical physics thus fulfilled, I can focus my energies on saving the planet.

That’s the plan, anyway. Sounds like a comic book. Here’s how the paper goes:

In Section 1 we start by explaining categories, functors, and how parallel transport for particles can be seen as a functor taking any path in a manifold to the operation of parallel transport along that path. In < a href http://math.ucr.edu/home/baez/invitation_2.pdf#page=11″>Section 2 we ‘add one’ and explain how parallel transport for particles and strings can be seen as ’2-functor’ between ’2-categories’. This requires that we generalize Lie groups to ‘Lie 2-groups’. In Section 3 we describe many examples of Lie 2-groups, and sketch some of their applications:

• shifted abelian groups, gerbes, and their role in string theory and multisymplectic geometry
• the Poincaré 2-group and the spin foam model for 4d Minkowski spacetime
• tangent 2-groups, 4d theory and topological gravity
• inner automorphism 2-groups and 4d theory with cosmological constant term
• automorphism 2-groups, nonabelian gerbes, and the gravity 3-group
• string 2-groups, string structures, the passage from Lie -algebras to Lie -groups, and the Lie 3-superalgebra governing 11-dimensional supergravity

Finally, in Section 4 we discuss gauge transformations, curvature and nontrivial 2-bundles.