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December 28, 2012

A Higher Stacky Perspective On Chern-Simons Theory

Posted by Urs Schreiber

We are finalizing a contribution for a book on mathematical aspects of quantum field theory:

Domenico Fiorenza, Hisham Sati, Urs Schreiber,

A higher stacky perspective on Chern-Simons theory

Abstract. This text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized (“extended”, “mutli-tiered”) to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. At the end we provide a brief review and outlook of the higher prequantum field theory of which this is a first example. This includes a higher geometric description of supersymmetric Chern-Simons theory, generalized geometry, higher Spin-structures, anomaly cancellation and various other aspects of quantum field theory.

Comments are welcome! (See the pdf at the above link.)

Posted at 3:19 PM UTC | Permalink | Followups (1)

December 24, 2012

Category Theory 2013

Posted by Tom Leinster

Every year or two, there’s a major category theory conference. The last one was in 2011 in Vancouver, and the next one is in Sydney from 7 to 13 July 2013.

The invited speakers are:

  • Eugenia Cheng (Sheffield)
  • Pieter Hofstra (Ottawa)
  • Zurab Janelidze (Stellenbosch)
  • Emily Riehl (Harvard)
  • our very own Mike Shulman (IAS)
  • Ross Street (Macquarie)

More information is at the conference website. There, you can also find out about the workshop the week beforehand, which features mini-courses by Marcelo Aguiar, Richard Garner, Scott Morrison, and again Mike Shulman.

Early in the new year, the website will be updated with info on registration and accommodation, and there’ll be a call for contributed talks.

Posted at 11:35 PM UTC | Permalink | Followups (6)

December 20, 2012

Names For Equivalences

Posted by Mike Shulman

Homotopy type theory has a problem: we need names for a bunch of slightly different kinds of “equivalences”. Until now, we’ve been muddling along with some fairly ad hoc choices. Can you help think of better names?

Hardly any knowledge of type theory is necessary in order to help. Just read on…

Posted at 12:23 AM UTC | Permalink | Followups (9)

December 18, 2012

Category-Theoretic Foundations in Irvine

Posted by John Baez

There will be a two-day workshop:

Posted at 6:45 PM UTC | Permalink | Followups (22)

Rethinking Set Theory

Posted by Tom Leinster

Over the last few years, I’ve been very slowly working up a short expository paper — requiring no knowledge of categories — on set theory done categorically. It’s now progressed to the stage where I’d like to get some feedback. Here’s the latest draft. (Edit: Now revised in the light of your helpful comments below — thanks! — and posted as arXiv:1212.6543.) Typos, clumsy wording, mathematical matters: I want to hear it all.

I have one request, though. If you do leave a comment, please take more time than you usually would to make sure it’s (1) carefully worded, (2) respectful to other people, and (3) scrupulously polite. Sorry to ask this: normally I wouldn’t feel the need, but there’s an unfortunate history of discussions of categorical set theory turning bad-tempered, and I really want to avoid that happening here.

So, if you’re composing a comment and you feel yourself getting hot under the collar, please save a draft, sleep on it, and come back later when your temperature has returned to normal. There’s no hurry: this blog isn’t going anywhere.

I’d also like to hear about anything I’ve written in my draft that you think is overstated. (I’m particularly keen to hear about this from people who fundamentally share my views.) This paper is intended to be thought-provoking, and I know there are parts with which some people will disagree. However, it’s definitely not supposed to be inflammatory. I want every statement I’ve made to be careful and measured, so I’ll be grateful if you can help me find places where I might have slipped up.

Posted at 2:47 AM UTC | Permalink | Followups (237)

Enriched Indexed Categories

Posted by Mike Shulman

Here’s the second of the three papers about generalized category theory that I promised you last time:

This one is about a notion of “category” which simultaneously generalizes enriched categories, internal categories, and indexed categories.

Posted at 1:05 AM UTC | Permalink | Followups (4)

December 14, 2012

The Additivity of Traces in Stable Monoidal Derivators

Posted by Mike Shulman

By a strange coincidence, three new papers that I’ve been working on (two of them coauthored) are going to appear this month (I hope). They are all very different papers, but each of them is about some kind of generalized category theory. The first one is out today:

This paper is about derivators. I blogged a bit about derivators a few years ago, but at the time I didn’t have anything concrete to point to in terms of their usefulness, only some thoughts about how they might be convenient for various things. This paper makes some progress on that front, by proving the additivity of traces in the context of derivators.

Posted at 3:34 AM UTC | Permalink | Followups (16)

December 9, 2012

Universe Polymorphism and Typical Ambiguity

Posted by Mike Shulman

Sorry I’ve been kind of quiet recently. There’s been a lot of good stuff happening at IAS this year, but (aside from my being busy helping to make it happen), not so much of it has been easily bloggable. But here’s something that I’ve just learned: do you know the difference between universe polymorphism and typical ambiguity (or even what either of them means)?

Posted at 8:11 PM UTC | Permalink | Followups (10)

December 8, 2012

Bob on TAC

Posted by Tom Leinster

Every month, the Notices of the American Mathematical Society carries a piece on mathematics publishing: journals, parasitic publishers, and the like. This month, it’s by Bob Rosebrugh, founder and managing editor of Theory and Applications of Categories (TAC).

Bob’s piece relates how TAC was set up, and why, despite TAC being one of the first electronic mathematics journals, he wishes he’d followed his gut feeling and done it even earlier. He briefly mentions TAC’s sister publication, Reprints in Theory and Applications of Categories, which seems to me to represent an idea that should have caught on much more widely.

But most of all, he urges action —

So, if your subject area of mathematics doesn’t yet have a free electronic journal, it’s time to start!

— and tells us that it’s actually much less work than you think:

Colleagues in my field often suppose that managing a subject area electronic journal is a heroic endeavour. The truth is very different.

Posted at 12:11 AM UTC | Permalink | Followups (6)

December 7, 2012

Research Fellowships in Edinburgh

Posted by Tom Leinster

Looking for a postdoc in a city built round a volcano? Look no further. We have five two-year positions available, across mathematics, statistics and operational research:

5 Whittaker Research Fellowships

(By the way, the volcano is extinct. Or so they say.)

Posted at 12:19 PM UTC | Permalink | Post a Comment

December 6, 2012

What Can Category Theory Do For Philosophy?

Posted by David Corfield

I’m thinking of organising a meeting next Summer, perhaps in July, here in Canterbury to address the question in the title.

What feels like an age ago, I wrote that I wasn’t looking for (higher) category theory to be the tool to spark off an analytic philosophy Mark II. I meant by this that I wasn’t looking for a rerun of the transformation in philosophy brought about by Frege’s logic as taken up by Russell, Carnap, etc. Rather, I looked on the reconceptualisation of the basic notions of mathematics brought about by category theory largely as a sign that we should never forget that we are historically situated beings, aiming for something of enduring value, but in the full expectation that our successors will see what we achieve as limited in many respects.

I’m still sympathetic to that point of view, but clearly it shouldn’t stand in the way of work that actively uses the formalisms provided by (higher) category theory to allow progress to be made on what are counted as philosophical issues. It has seemed to me for a while somewhat arbitrary which formalisations of which subject matters fall under the remit of philosophy. Search for a probabilistic logic and you’re in, rethink basic geometric concepts and you’re not.

Anyhow, here are some topics which might be treated at such a meeting:

  1. Categorical logic & type theory (especially of the intensional dependent variety)
  2. Modal logic
  3. Physics & structuralism
  4. Other metaphysics, e.g., identity, space, quantity
  5. Probability theory, stochastic mechanics & machine learning
  6. Other: biology, natural language, Arrow’s theorem,…

I’d be interested to hear any thoughts on such a meeting and about anything else you might like to see. I’m maintaining a page at my personal wiki to gather some ideas. If you’d rather not write to a public blog, you can contact me via here.

Posted at 9:46 AM UTC | Permalink | Followups (34)