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August 20, 2014

Holy Crap, Do You Know What A Compact Ring Is?

Posted by Tom Leinster

You know how sometimes someone tells you a theorem, and it’s obviously false, and you reach for one of the many easy counterexamples only to realize that it’s not a counterexample after all, then you reach for another one and another one and find that they fail too, and you begin to concede the possibility that the theorem might not actually be false after all, and you feel your world start to shift on its axis, and you think to yourself: “Why did no one tell me this before?”

That’s what happened to me today, when my PhD student Barry Devlin — who’s currently writing what promises to be a rather nice thesis on codensity monads and topological algebras — showed me this theorem:

Every compact Hausdorff ring is totally disconnected.

Posted at 11:54 PM UTC | Permalink | Followups (11)

August 12, 2014

The Ten-Fold Way (Part 4)

Posted by John Baez

Back in 2005, Todd Trimble came out with a short paper on the super Brauer group and super division algebras, which I’d like to TeXify and reprint here.

In it, he gives extremely efficient proofs of several facts I alluded to last time. Namely:

• There are exactly 10 real division superalgebras.

• 8 of them have center \mathbb{R}, and these are Morita equivalent to the real Clifford algebras Cliff 0,,Cliff 7Cliff_0, \dots, Cliff_7.

• 2 of them have center \mathbb{C}, and these are Morita equivalent to the complex Clifford algebras liff 0\mathbb{C}liff_0 and liff 1\mathbb{C}liff_1.

• The real Clifford algebras obey

Cliff i Cliff jCliff i+jmod8 Cliff_i \otimes_{\mathbb{R}} Cliff_j \simeq Cliff_{i + j mod 8}

where \simeq means they’re Morita equivalent as superalgebras.

It easily follows from his calculations that also:

• The complex Clifford algebras obey

liff i liff jliff i+jmod2 \mathbb{C}liff_i \otimes_{\mathbb{C}} \mathbb{C}liff_j \simeq \mathbb{C}liff_{i + j mod 2}

These facts lie at the heart of the ten-fold way. So, let’s see why they’re true!

Posted at 1:46 AM UTC | Permalink | Followups (3)

August 7, 2014

The Ten-Fold Way (Part 3)

Posted by John Baez

My last article on the ten-fold way was a piece of research in progress — it only reached a nice final form in the comments. Since that made it rather hard to follow, let me try to present a more detailed and self-contained treatment here!

But if you’re in a hurry, you can click on this:

and get my poster for next week’s scientific advisory board meeting at the Centre for Quantum Technologies, in Singapore. That’s where I work in the summer, and this poster is supposed to be a terse introduction to the ten-fold way.

Posted at 4:00 AM UTC | Permalink | Followups (34)

August 2, 2014

Wrestling with Tight Spans

Posted by Tom Leinster

I’ve been spending some time with Simon Willerton’s paper Tight spans, Isbell completions and semi-tropical modules. In particular, I’ve been trying to understand tight spans.

The tight span of a metric space AA is another metric space T(A)T(A), in which AA naturally embeds. For instance, the tight span of a two-point space is a line segment containing the original two points as its endpoints. Similarly, the tight span of a three-point space is a space shaped like the letter Y, with the original three points at its tips. Because of examples like this, some people like to think of the tight span as a kind of abstract convex hull.

Simon’s paper puts the tight span construction into the context of a categorical construction, Isbell conjugacy. I now understand these things better than I did, but there’s still a lot I don’t get. Here goes.

Posted at 3:31 AM UTC | Permalink | Followups (21)

July 27, 2014

Basic Category Theory

Posted by Tom Leinster

My new book is out!

Front cover of Basic Category Theory

Click the image for more information.

It’s an introductory category theory text, and I can prove it exists: there’s a copy right in front of me. (You too can purchase a proof.) Is it unique? Maybe. Here are three of its properties:

  • It doesn’t assume much.
  • It sticks to the basics.
  • It’s short.
Posted at 8:42 PM UTC | Permalink | Followups (12)

July 22, 2014

The Ten-Fold Way (Part 2)

Posted by John Baez

How can we discuss all the kinds of matter described by the ten-fold way in a single setup?

It’s bit tough, because 8 of them are fundamentally ‘real’ while the other 2 are fundamentally ‘complex’. Yet they should fit into a single framework, because there are 10 super division algebras over the real numbers, and each kind of matter is described using a super vector space — or really a super Hilbert space — with one of these super division algebras as its ‘ground field’.

Combining physical systems is done by tensoring their Hilbert spaces… and there does seem to be a way to do this even with super Hilbert spaces over different super division algebras. But what sort of mathematical structure can formalize this?

Here’s my current attempt to solve this problem. I’ll start with a warmup case, the threefold way. In fact I’ll spend most of my time on that! Then I’ll sketch how the ideas should extend to the tenfold way.

Fans of lax monoidal functors, Deligne’s tensor product of abelian categories, and the collage of a profunctor will be rewarded for their patience if they read the whole article. But the basic idea is supposed to be simple: it’s about a multiplication table.

Posted at 11:02 AM UTC | Permalink | Followups (41)

July 21, 2014

Pullbacks That Preserve Weak Equivalences

Posted by Mike Shulman

The following concept seems to have been reinvented a bunch of times by a bunch of people, and every time they give it a different name.

Definition: Let CC be a category with pullbacks and a class of weak equivalences. A morphism f:ABf:A\to B is a [insert name here] if the pullback functor f *:C/BC/Af^\ast:C/B \to C/A preserves weak equivalences.

In a right proper model category, every fibration is one of these. But even in that case, there are usually more of these than just the fibrations. There is of course also a dual notion in which pullbacks are replaced by pushouts, and every cofibration in a left proper model category is one of those.

What should we call them?

Posted at 11:03 PM UTC | Permalink | Followups (17)

July 20, 2014

The Place of Diversity in Pure Mathematics

Posted by Tom Leinster

Nope, this isn’t about gender or social balance in math departments, important as those are. On Friday, Glasgow’s interdisciplinary Boyd Orr Centre for Population and Ecosystem Health — named after the whirlwind of Nobel-Peace-Prize-winning scientific energy that was John Boyd Orr — held a day of conference on diversity in multiple biological senses, from the large scale of rainforest ecosystems right down to the microscopic scale of pathogens in your blood.

Cartoon of John Boyd Orr

I used my talk (slides here) to argue that the concept of diversity is fundamentally a mathematical one, and that, moreover, it is closely related to core mathematical quantities that have been studied continuously since the time of Euclid.

Posted at 2:10 PM UTC | Permalink | Followups (7)

July 19, 2014

The Ten-Fold Way (Part 1)

Posted by John Baez

There are 10 of each of these things:

  • Associative real super-division algebras.

  • Classical families of compact symmetric spaces.

  • Ways that Hamiltonians can get along with time reversal (TT) and charge conjugation (CC) symmetry.

  • Dimensions of spacetime in string theory.

It’s too bad nobody took up writing This Week’s Finds in Mathematical Physics when I quit. Someone should have explained this stuff in a nice simple way, so I could read their summary instead of fighting my way through the original papers. I don’t have much time for this sort of stuff anymore!

Posted at 11:33 AM UTC | Permalink | Followups (10)

July 15, 2014

Math and Mass Surveillance: A Roundup

Posted by Tom Leinster

The Notices of the AMS has just published the second in its series “Mathematicians discuss the Snowden revelations”. (The first was here.) The introduction to the second article cites this blog for “a discussion of these issues”, but I realized that the relevant posts might be hard for visitors to find, scattered as they are over the last eight months.

So here, especially for Notices readers, is a roundup of all the posts and discussions we’ve had on the subject. In reverse chronological order:

Posted at 11:31 PM UTC | Permalink | Followups (28)

July 10, 2014

Describing PROPs Using Generators and Relations

Posted by John Baez

Here’s another post asking for a reference to stuff that should be standard. (The last ones succeeded wonderfully, so thanks!)

I should be able to say

CC is the symmetric monoidal category with the following presentation: it’s generated by objects xx and yy and morphisms L:xyyL: x \otimes y \to y and R:yxyR: y \otimes x \to y, with the relation

(L1)(1R)α x,y,x=(1R)(L1)(L \otimes 1)(1 \otimes R)\alpha_{x,y,x} = (1 \otimes R)(L \otimes 1)

Here α\alpha is the associator. Don’t worry about the specific example: I’m just talking about a presentation of a symmetric monoidal category using generators and relations.

Right now Jason Erbele and I have proved that a certain symmetric monoidal category has a certain presentation. I defined what this meant myself. But this has got to be standard, right?

So whom do we cite?

Posted at 11:34 AM UTC | Permalink | Followups (9)

July 8, 2014

The Categorical Origins of Lebesgue Integration

Posted by Tom Leinster

I’ve just come back from the big annual-ish category theory meeting, Category Theory 2014 in Cambridge, also attended by Café hosts Emily and Simon. The talk I gave there was called The categorical origins of Lebesgue integration — click for slides — and I’ll briefly describe it now.

There are two theorems.

Theorem A The Banach space L 1[0,1]L^1[0, 1] has a simple universal property. This leads to a unique characterization of integration on [0,1][0, 1].

Theorem B The functor L 1:L^1: (finite measure spaces) \to (Banach spaces) has a simple universal property. This leads to a unique characterization of integration on finite measure spaces.

Posted at 12:54 AM UTC | Permalink | Followups (79)

July 1, 2014

The Linearity of Traces

Posted by Mike Shulman

At long last, the following two papers are up:

I’m super excited about these, and not just because I like the results. Firstly, these papers are sort of a culmination of a project that began around 2006 and formed a large part of my thesis. Secondly, this project is an excellent “success story” for a methodology of “applied category theory”: taking seriously the structure that we see in another branch of mathematics, but studying it using honest category-theoretic tools and principles.

For these reasons, I want to tell you about these papers by way of their history. (I’ve mentioned some of their ingredients before when I blogged about previous papers in this series, but I won’t assume here you know any of it.)

Posted at 4:35 AM UTC | Permalink | Followups (12)

June 28, 2014

Kan Extension Seminar Talks at CT2014

Posted by Emily Riehl

The International Category Theory Conference will take place this coming week, Sunday June 29 - Saturday July 4th, in (old) Cambridge. To those readers who will be in attendance, I hope you’ll stop by to visit the Kan Extension Seminar, which will present a series of eight 15-minute expository talks this coming Sunday (June 29) at Winstanley Lecture Theatre in Trinity College.

We will have tea starting at 2pm with the first talks to commence at 2:30. There will be a short break around 3:50pm with the second series of talks to begin at 4:10. The talks should finish around 5:30, at which point we will walk together to the welcome reception for the CT.

Please join us! We have a fantastic line-up of talks that promise to be interesting and yet understandable with very little assumed background. I’ve listed the speakers and titles below the break. Abstracts and more information can be found here.

Posted at 12:56 AM UTC | Permalink | Followups (2)

June 27, 2014

Enriched Indexed Categories, Again

Posted by Emily Riehl

Guest post by Joe Hannon.

As the final installment of the Kan extension seminar, I’d like to take a moment to thank our organizer Emily, for giving all of us this wonderful opportunity. I’d like to thank the other participants, who have humbled me with their knowledge and enthusiasm for category theory and mathematics. And I’d like to thank the nCafé community for hosting us.

For the final paper of the seminar, we’ll be discussing Mike Shulman’s Enriched Indexed categories.

The promise of the paper is a formalism which generalizes ordinary categories and can specialize to enriched categories, internal categories, indexed categories, and even some combinations of these which have found use recently. In fact the paper defines three different notions of such categories, so-called small 𝒱\mathcal{V}-categories, indexed 𝒱\mathcal{V}-categories, and large 𝒱\mathcal{V}-categories, where 𝒱\mathcal{V} is an indexed monoidal category. For the sake of brevity, we’ll be selective in this blog post. I’ll quickly survey the background material, the three definitions, and their comparisons, and then I want to look at limits in enriched indexed categories. Note also that Mike himself made a post on this paper here on the nCafe in 2012, hence the title.

Posted at 6:24 PM UTC | Permalink | Followups (18)