The n-Category Café

Nils Baas has been talking to me about the idea of “hyperstructures” a lot lately – and I have tried to think hard about it.

To me it seems like the main basic idea is this:

We might want to have something like an $n$-graph and equip it with a notion of composition (“fusion”) which does not distinguish between source and targets.

Tom Leinster once told me that this is pretty close to saying “cyclic operad”, as far as I rememeber. But it seems to me that there might still be a good point in looking for more:

whatever the right notion of “$n$-graphs with fusion” is, the “morphisms” between them should not simply be morphisms, but should be “bonds” ($n$-graph elements), too.

With Konrad Waldorf I was talking about this a bit. We came up with the following idea which might be a good guiding example:

Let $C$ be any category with all pullback. Then we know that spans in $C$ form a bicategory.

But now, what if I considered multispans in $C$?

Here a multispan is, clearly, an object on $C$ equipped with an arbitrary number of morphisms out of it.

Given two multispans, I can check if they have any “leg” in common, pull them back along this common leg and obtain a new multispan.

Clearly, the structure of multispans together with this fusion operation does not form a category – unless one artificially introduces labels that mark certain legs as incoming and other legs as outgoing.

Moreover, it is pretty clear that we can consider multispans of multispans in the obvious way, ad infionitum.

So I am guessing that multispans in a category $C$ might be a good guiding example for a definition of hyperstructure.

Even more so, since the other main motivating example that Nils Baas is emphasizing is hyperstructures of cobordisms with corners and singularities. Given that a cobordism can be regarded as a cospan, and naturally actually as a multi-cospan if we stop distinguishing between in- and out-parts of its boundary, this might actually be a special case of the more general multi(co)-span situation I just described.

Does anyone have any ideas on this??

I remeber John Baez pointing out how mutlispans of groupoids naturally appear as higher-rank tensors in the groupoidification program. Possibly there is some room for convergence here.

Where do multispans live?

Hello,

Could you give “us” some advice pertaining to where undergraduates would start if they were interested in someday working in this area of mathematics (namely Lie algebra and categorical theory). Courses available to the uninformed include linear algebra, algebraic structures, field theory… etc. What books would one read as a starting point?

Are there resources available online? I’ve never heard these topics mentioned in class and wonder when I would come across them. When do people who pursue such topics in graduate school learn about these topics?

Grant Malcolm and the late Ray Paton at the University of Liverpool explored the application of category theory to biology.

See, for example:

Michael J. Fisher, Grant Malcolm and Ray C. Paton: Spatio-logical processes in intracellular signalling. Biosystems, 55(1-3): 93–105, 2000.

available from:

http://www.csc.liv.ac.uk/~grant/bib.html

This book might be worth looking at:

• Corrado Priami and Gordon D. Plotkin, eds. Transactions on Computational Systems Biology VI, Lecture Notes in Computer Science 4220, Springer, 2006.

I know Plotkin’s work on category theory and computer science, but lately he’s been applying some of the same ideas to biology.

Lately Mike Stay has been trying to understand and explain the pi calculus, which is some mutant version of the lambda calculus. He might enjoy this article in the above volume:

• Céline Kuttler, Simulating bacterial transcription and translation in a stochastic pi calculus, pp. 113-149.

I think this is a very good idea.
I remain curious about math [BA] with an MD.

I have been reading both physics and engineering literature.
Biomathematics appears to be more oriented toward engineering than physics.
For example, pharmacology has some mechanisms of action that are poorly understood, yet somehow work.
This is more consistent with a Richard Bellman insight than with rigor or precision.

Something that intrigues me about the adoption of a mathematical theory, such as category theory, into a discipline like biology is that it illuminates understandings, and tensions between these understandings, of what constitutes success in that discipline.

As disciplines unfold, new understandings of success develop, old tensions may be resolved, and new tensions appear. I think the key task for the philosophy of science is to bring out what it is about this process, if anything, which is rightly described as rational.

Of course, this characterisation of the key task may put me at odds with other philosophers, so we have our own tensions to resolve. And we have to do our own thinking about the rationality of philosophy.

Rashevsky and Rosen influenced my choice of PhD Thesis topic, indirectly, through reading their work. Particularly Dynamical Systems Theory in Biology [1970] by Rosen.

Since I was doing what today would be called metabolic network systems biology, and taking a couple of Graph Theory courses, and a Category Theory course in grad school (1973-1977) I was motivated to establish that Birth and Death were adjoint (a hunch that never panned out).

I went after explaining dynamical properties of metabolisms with the a priori belief that they were complex systems, in the sense that Rosen described.

I used classical means in my proof – Krohn Rhodes decomposition of the semigroup of differential operators of the nonlinear system of Michaelis-Menten equations of the metabolism, and finding eigenfunctions corresponding to waves in phase space (waves later rediscovered by Prigogine).

My first proof of my main result was 30 pages of matrix exponentials and the like, but that collapsed into a one-liner with the proper representation and semigroup decomposition, and restriction to “Physical Systems” (i.e. causality).

But, back then, only a handful of biologists had enough Math to follow this (some corresponded with me from Edinburgh and Russia). Only a handful of mathematicians knew enough biology (John Holland, in particular, on the Genetic Algorithm side).

But I think that Biology is ready for Category Theory today, due to the number of people in Complex Systems who know enough Biology and enough math, and care enough about system dynamics.

I urge that John Baez or someone else associated with this blog to make such a presentation at the 8th International Conference on Complex Systems, May 2009 (18 months from now), sponsored by the New England Complex Systems Institute.

The head of NCSI (Yaneer Bar-Yam) is a hard scientist, with several publications in Science. His father (also on the Board) has a named chiar of Physics at U.Massachusetts. Yaneer’s brother-in-law is also published in Science. The quality of the people and papers at ICCS is high.

Flimsy as my Physics credentials are, I chaired both Physics sessions at the 7th International Conference on Complex Systems (28 Oct-2 Nov 2007) because, in part, I have gotten to know the Executive Committee and staff rather well, and believe in extensive preparation for session chairs (i.e. reading several papers by each presenter, so as to introduce them well and in context of conference and session themes).

This interdisciplinary conference is the perfect venue for such a paper and dialogue. The default attendee is a recovering Physicist. Computer and Math (Graph Theory, Chaos) background is common. Appreciation of aesthetically strong theory is present, as is an appreciation for how complex Biological systems really are. Top biologists, from Harvard Medical School and many countries, are present.

Stuart Kauffman’s ideas are familar (he’s been a plenary speaker there).

I’m not in a position this time to make a formal invitation, as I have other responsibilities assigned for ICCS-2009, but do encourage this community to google “NECSI ICCS” and explore the notion.

I have more a philosophical comment regarding David’s statement “I would imagine that a higher proportion of biologists are bewildered by this kind of work than their physicist colleagues by similar work in their field”.

I remember about 10 years ago, people started talking about something “biological physics” or “biophysics”. Someone wrote a letter to Physics Today claiming that biophysics was a subfield of physics, in the same way as condensed matter physics or particle physics, and that it should be a required class for all physics majors. I love biology, but clearly “biophysics” is a subfield of biology, not physics. If your goal is to increase our understanding of life on this planet, you’re doing biology. If your goal is to increase our understanding of the Universe, you’re doing physics.

So why did people invent the term “biophysics”? Throughout history, from Ancient Greece to the late 20th Century, physics required knowledge of advanced mathematics. In fact, the driving force behind the advancement of mathematics was physicists. However, biology required very little mathematical knowledge. Charles Darwin, even though he was a genius in his field, could not possibly have begun to understand the mathematics of Felix Klein or the physics of James Clerk Maxwell, even though Felix Klein and James Clerk Maxwell could easily have understood Darwin’s work. One of the reasons why the work of Gregory Mendel languished in obscurity is because it was a work in biology that required mathematical knowledge and understanding.

In fact all the way up until the late 20th Century, biology required very little mathematics. Physicists would be astonished how little mathematics biology required. However, in the late 20th Century, that changed. Finally, a measurable amount of biology required advanced mathematics. However, the biologists did not possess the mathematical knowledge to do it. Most biologists can’t even do highschool calculus. Therefore, they had to get physicists to do it for them. The physicists would not do it if you told them they were doing “biology” so you have to tell them they are doing “physics”. That’s why they invented the term “biophysics”.

I think the solution is not to get physicists to do biology. If they want to do it, that’s fine, but be honast that it’s a field of biology, not physics. The solution is to require biology students to take more math and physics classes. Right now you can get a BS in biology with no math classes at all. I think biology majors, actually any science major, including chemistry, geology, etc. should be required to take all the same math classes that physics majors are required to take.

Jeffery Winkler

http://www.geocities.com/jefferywinkler

@Tom Leinster & John Baez:

Thank you for your helpful suggestions! My background (and current area of study) is computer science and pure mathematics. I’m interested in exploring the possibility of applying new concepts to the field of neuroscience (not to mention broadening my horizons in algebra for other research).

I will (try to) read Conceptual Mathematics and come back booming with questions.

Thanks Again!

I’ve been thinking lately that category theory might be a viable way to make more rigorous the talk of “emergent properties” one hears so often in biophysics and other areas of the amorphous domain called “complex systems”. See John Armstrong’s random thoughts on Hofstadter.

Even if the biologists find themselves bewildered by category theory, they’re wanting to steal the word functor.