‘Category Theory for the Sciences’ by David I. Spivak *(via Source)*

‘Category Theory for Scientists’ by David I. Spivak *(via Source, see comments from John Baez)*

‘Category theory in context’ by Emily Riehl *(via Source < via Source)*

Nice comments, Todd!

]]>I also find it hard to describe the power of categorical thinking, but one thing that a committedly categorical approach helps to do is to clear away conceptual clutter and distill mathematical situations down to their essences. One aphorism for this is due to Peter Freyd: “Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial.” This sounds like he’s making a little joke, but he has a serious point: that one thing category theory does is help make the softer bits seem utterly natural and obvious, so as to quickly get to the heart of the matter, allowing one to isolate the hard nuggets, which one may then attack with abandon. (I’m paraphrasing the nLab, but that’s okay since I wrote the relevant passage.)

One of the great exemplars and high artists of the categorical spirit was Alexandre Grothendieck, who had a way of dissolving problems, not by attacking them head-on — what he called the hammer and chisel approach — but by soaking them within a vast sea of theory. This is even more mysterious than Freyd’s dictum, and very hard to describe, but the style consists much more in finding the right kinds of *definitions*, lifting problems to a higher plane where former obstacles become spurious, are swept away. Few mathematicians can really pull this off, but in the end a thoroughgoingly Grothendieckian approach will also give an impression of utter naturality — what John Baez fondly calls the ‘Tao’ of mathematics. Pierre Deligne said this: “a typical Grothendieck proof consists of a long series of trivial steps where “nothing seems to happen, and yet at the end a highly non-trivial theorem is there”.

It’s a bit tricky to explain how category theory actually helps. It’s sort of like having an extra pair of eyes that can see patterns that would otherwise be invisible. It’s sort of like knowing an extra language that lets you think thoughts you just couldn’t think otherwise. And it’s sort of like having an overall map of mathematics.

Sometimes category theory makes long and clumsy proofs short and efficient. Sometimes it lets you guess what you should try to prove, but you still have to struggle through some grunge to prove those things. But most of all it lets you state theorems that you just couldn’t state without category theory—at least not without lots of clumsy circumlocutions.

]]>Unfortunately I dislike programming, and I’ve never programmed in Haskell, so I’m not the right person to write a book that connects category theory and computer science. The closest I came is this paper, which you may already know:

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, in *New Structures for Physics*, ed. Bob Coecke, Lecture

Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95–172.

Mike supplied the practical expertise in programming, and I supplied the expertise in symmetric monoidal closed categories, and together we put together a pretty nice section on how they fit together. But this was really just scratching the surface of categories and computation! To really get serious I think one would need to bring in 2-categories, and nobody has fully worked out the details yet. I tried here:

• John Baez, Categories and computation.

See especially the references to Seeley near the end.

I wish someone would work out all the details and write a great book about it, full of pictures!

]]>Given the pervasiveness of programming (languages) in all disciplines, I think writing a book with some bias towards a computer science audience would channel the interest of quite a vast range of backgrounds (math, physics, CS, chemistry, bioinformatics…). Substitute “computer science” by “Haskell” and then it might be clearer what I mean perhaps. I would buy a book on the subject written by you with Dan Piponi and maybe Mike Stay as well. Riehl gives has some examples from Haskell but there could be way many more. I’m thinking an intro level book for people with some solid background in undergrad algebra.

]]>You don’t actually need to mention Hermite functions or use a formula for the Hermite functions to prove this result; one can do everything a bit abstractly and just *define* the th Hermite function to be the th eigenvector of the harmonic oscillator (possibly times normalization factor, if you want to make life more complicated than it strictly needs to be). No grungy formulas or grungy manipulations are required, since this is part of the math gods’ favored territory.

John

What you have just done above is an exercise in Eli Stein’s and Rami Shakarchi’s Fourier Series book which drags in Hermite functions as part of the overall proof. No mention of category theory of course. All bare hands stuff. I can see how category theory can illuminate structural things across seemingly disparate areas. I’ll bat on with David’s book to get a bit more proficient with the detail..

Cheers

Peter

]]>I was a pure mathematician a long time ago. At least this means I am not frightened by the purely abstract nature of Emily Riehls’ book. David Spivak has applications, but they mostly involve databases. I saw one mention of a Markov chain. My impression is that his applications are almost all about structured information, not processes.

I am not really looking for concrete examples drawn from my own field of evolutionary biology. I can recognise Markov processes and chemical reaction networks as relevant to my work, and electrical circuits and even Feynman diagrams as relevant to others in the field. There are plenty of databases used in evolutionary biology too, but they don’t interest me!

Should it come to pass that you write a vaguely similar book to Spivak’s, perhaps it should have chapters with titles like ‘Structured information’, ‘Transforming information’, ‘Continuous time processes’, or some such.

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