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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 (15)

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 (13)

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 (4)

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 (6)

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 (25)

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)

June 7, 2014

Categorical Homotopy Theory

Posted by Emily Riehl

In my first year at Harvard, I had an opportunity to teach a graduate-level topics course entitled “Categorical Homotopy Theory.” Its aim was to highlight areas in which category theoretic abstractions provide a particularly valuable insight into classical homotopy theoretic constructions. Over the course of the semester I gave lectures that focused on homotopy limits and colimits, enriched category theory, model categories, and quasi-categories.

In hopes that attendees would be able to drop in and out without feeling totally lost, I decided to write lecture notes. And now they have just been published by Cambridge University Press as an actual physical book and also as an ebook (or so I’m told).


One of the wonderful things about working with CUP is that they have given me permission to host a free PDF copy of the book on my website. At the moment, this is the pre-copyedited version. There is an extra section missing from chapter 14 and various minor changes made throughout. In a few years time, I’ll be able to post the actual published version.

Posted at 8:22 PM UTC | Permalink | Followups (7)

June 5, 2014

Should Mathematicians Cooperate with GCHQ? Part 3

Posted by Tom Leinster

Update (6 July 2014)  A much shorter version of this post appears in the July edition of the LMS newsletter, along with a further opinion from Trevor Jarvis (Hull).

In April, the newsletter of the London Mathematical Society published my piece “Should mathematicians cooperate with GCHQ?”, which mostly consisted of factual statements based on the Snowden leaks, followed by the mild opinion that as individuals and institutions, we can choose whether to give GCHQ our cooperation. Two mathematicians associated with GCHQ, Richard Pinch and Malcolm MacCallum, have now replied. I will address their points, then make some suggestions for mathematics departments in the post-Snowden era.

Logo of US spy satellite NROL-39
Real, not-made-up logo of US spy satellite

Posted at 12:59 PM UTC | Permalink | Followups (12)

June 2, 2014

Codescent Objects and Coherence

Posted by Emily Riehl

Guest post by Alex Corner

This is the 11th post in the Kan Extension Seminar series, in which we will be looking at Steve Lack’s paper

A previous post in this series introduced us to two-dimensional monad theory, where we were told about 22-monads, their strict algebras, and the interplay of the various morphisms that can be considered between them. The paper of Lack has a slightly different focus in that not only are we interested in morphisms of varying levels of strictness but also in the weaker notions of algebra for a 22-monad, namely the pseudoalgebras and lax algebras.

An example that we will consider is that of the free monoid 22-monad on the 22-category Cat\mathbf{Cat} of small categories, functors, and natural transformations. The strict algebras for this 22-monad are strict monoidal categories, whilst the lax algebras are (unbiased) lax monoidal categories. Similarly, the pseudoalgebras are (unbiased) monoidal categories. The classic coherence theorem of Mac Lane is then almost an instance of saying that the pseudoalgebras for the free monoid 22-monad are equivalent to the strict algebras. We will see conditions for when this can be true for an arbitrary 22-monad.

Thanks go to Emily, my supervisor Nick Gurski, the other participants of the Kan extension seminar, as well as all of the participants of the Sheffield category theory seminar.

Posted at 12:46 AM UTC | Permalink | Followups (21)

May 25, 2014

Spans and the Categorified Heisenberg Algebra (Part 2)

Posted by John Baez

Last summer I gave a little course on something I really like: Jeffrey Morton and Jamie Vicary’s work on the ‘categorified Heisenberg algebra’ discovered by Mikhail Khovanov. It ties together combinatorics and the math of quantum theory in a fascinating way… related to nice old ideas, but revealing a new layer of structure. I blogged about that course here, with links to slides and references.

The last two weeks I was in Paris attending a workshop on operads. I learned a lot, and it was great to talk to Mathieu Anel, Steve Awodey, Benoit Fresse, Nicola Gambino, Ezra Getzler, Martin Hyland, André Joyal, Joachim Kock, Paul-André Melliès, Emily Riehl, Vladimir Voevodsky… and many other people to whom I apologize for not including in this prestigious list! (The great thing about senility is never having to say you’re sorry, but I haven’t quite reached that stage.)

There is a lot I could say… but that will have to wait for another time. For now I just want to point out this annotated video:

Spans and the categorified Heisenberg algebra.

of a talk at the Catégories, Logiques, Etc… seminar at Paris 7, run by Anatole Khelif. This should be a fairly painless introduction to the subject, since I sensed that lots of people in the audience wanted me to start by explaining prerequisites: categorification, TQFTs, 2-Hilbert spaces and the Heisenberg algebra.

That means I didn’t manage to discuss other interesting things, like the definition of symmetric monoidal bicategory, or the role of combinatorics, especially Young diagrams. For those, go here and check out the links!

Posted at 12:36 PM UTC | Permalink | Followups (5)

May 22, 2014

Enrichment and the Legendre-Fenchel Transform II

Posted by Simon Willerton

Last time, I described the Legendre-Fenchel transform. For a real vector space VV, the transform gives a bijection between the convex functions on VV and the convex functions on its dual V #V^{#}. This time I’ll show how this is naturally an example of the profunctor nucleus construction in enriched category theory.

The key to this relationship is that the natural pairing V×V #V\times V^{#}\to \mathbb{R} between a vector space and its dual can be thought of as profunctor between discrete ¯\overline{\mathbb{R}}-categories, where ¯=[,]\overline{\mathbb{R}}=[-\infty ,\infty ] is the extended real numbers thought of as a closed, symmetric monoidal category.

As I’ll explain, we can then apply the machinary of the nucleus construction to get a pair of maps between the ¯\overline{\mathbb{R}}-presheaves on VV, which just means the ¯\overline{\mathbb{R}}-valued functions on VV, and the ¯\overline{\mathbb{R}}-copresheaves, ie the ¯\overline{\mathbb{R}}-valued functions, on V #V^{#}. Restricting to the part on which the pair of maps is bijective, we get that the transform gives a bijection between the respective sets of convex functions.


This category theoretic treatment gives us a little more than the standard treatment. Usually people put an ordering on the functions and observe that Legendre-Fenchel transform is order reversing. The category theory tells us that we can refine this and put a generalized, generalized metric space structure on the space of functions so that the transform is distance non-increasing, and is in fact distance preserving in the case of convex functions. This is illustrated by the above picture, as we will discover below!

Posted at 9:58 PM UTC | Permalink | Followups (13)