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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 (15)

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 (8)

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 (and updated after the original appearance of this post):

Posted at 11:31 PM UTC | Permalink | Followups (34)

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)