April 30, 2015

Breakfast at the n-Category Café

Posted by Emily Riehl

Michael Harris recently joined us for breakfast at the n-category café. Perhaps some readers here would be interested in following the postprandial discussion that is underway on his blog: Mathematics Without Apologies.

Posted at 6:30 PM UTC | Permalink | Followups (2)

April 28, 2015

Categories in Control

Posted by John Baez

To understand ecosystems, ultimately will be to understand networks. - B. C. Patten and M. Witkamp

A while back I decided one way to apply my math skills to help save the planet was to start pushing toward green mathematics: a kind of mathematics that can interact with biology and ecology just as fruitfully as traditional mathematics interacts with physics. As usual with math, the payoffs will come slowly, but they may be large. It’s not a substitute for doing other, more urgent things—but if mathematicians don’t do this, who will?

As a first step in this direction, I decided to study networks.

This May, a small group of mathematicians is meeting in Turin for a workshop on the categorical foundations of network theory, organized by Jacob Biamonte. I’m trying to get us mentally prepared for this. We all have different ideas, yet they should fit together somehow.

Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now I want to say a bit about what I’ve been doing with Jason Erbele.

Posted at 10:42 PM UTC | Permalink | Followups (39)

April 24, 2015

A Synthetic Approach to Higher Equalities

Posted by Mike Shulman

At last, I have a complete draft of my chapter for Elaine Landry’s book Categories for the working philosopher. It’s currently titled

• Homotopy Type Theory: A synthetic approach to higher equalities. pdf

As you can see (if you read it), not much is left of the one fragment of draft that I posted earlier; I decided to spend the available space on HoTT itself rather than detour into synthetic mathematics more generally. Although the conversations arising from that draft were still helpful, and my other recent ramblings did make it in.

Comments, questions, and suggestions would be very much appreciated! It’s due this Sunday (I got an extension from the original deadline), so there’s a very short window of time to make changes before I have to submit it. I expect I’ll be able to revise it again later in the process, though.

Posted at 4:06 AM UTC | Permalink | Followups (84)

April 12, 2015

The Structure of A

Posted by David Corfield

I attended a workshop last week down in Bristol organised by James Ladyman and Stuart Presnell, as part of their Homotopy Type Theory project.

Urs was there, showing everyone his magical conjuring trick where the world emerges out of the opposition between $\emptyset$ and $\ast\;$ in Modern Physics formalized in Modal Homotopy Type Theory.

Jamie Vicary spoke on the Categorified Heisenberg Algebra. (See also John’s page.) After the talk, interesting connections were discussed with dependent linear type theory and tangent (infinity, 1)-toposes. It seems that André Joyal and colleagues are working on the latter. This should link up with Urs’s Quantization via Linear homotopy types at some stage.

As for me, I was speaking on the subject of my chapter for the book that Mike’s Introduction to Synthetic Mathematics and John’s Concepts of Sameness will appear in. It’s on reviving the philosophy of geometry through the (synthetic) approach of cohesion.

In the talk I mentioned the outcome of some further thinking about how to treat the phrase ‘the structure of $A$’ for a mathematical entity. It occurred to me to combine what I wrote in that discussion we once had on The covariance of coloured balls with the analysis of ‘the’ from The King of France thread. After the event I thought I’d write out a note explaining this point of view, and it can be found here. Thanks to Mike and Urs for suggestions and comments.

The long and the short of it is that there’s no great need for the word ‘structure’ when using homotopy type theory. If anyone has any thoughts, I’d like to hear them.

Posted at 2:01 PM UTC | Permalink | Followups (144)

April 7, 2015

Information and Entropy in Biological Systems

Posted by John Baez

I’m helping run a workshop on Information and Entropy in Biological Systems at NIMBioS, the National Institute of Mathematical and Biological Synthesis, which is in Knoxville Tennessee.

I think you’ll be able to watch live streaming video of this workshop while it’s taking place from Wednesday April 8th to Friday April 10th. Later, videos will be made available in a permanent location.

To watch the workshop live, go here. Go down to where it says

Investigative Workshop: Information and Entropy in Biological Systems

Then click where it says live link. There’s nothing there now, but I’m hoping there will be when the show starts!