May 22, 2013

In the News

Posted by David Corfield

Applications of category theory are described by Julie Rehmeyer in ScienceNews under the banner

One of the most abstract fields in math finds application in the ‘real’ world.

Now, how about applications in the real world?

Posted at May 22, 2013 9:17 AM UTC

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Re: In The News

Going to the extreme, does anyone know of examples for which some of the simplest category theory concepts (perhaps in the form of diagrams) could be used as a teaching tool to elucidate some fundamental mathematical principles (for example, the distributive property) that are typically taught in secondary school? Then perhaps one could use those examples as a wedge to more easily introduce concepts of a less elementary nature. Students would then have a chance to become aware of category theory and even start to use it as a tool of their own. Now that could lead to big changes.

Posted by: Charlie Clingen on May 26, 2013 4:39 PM | Permalink | Reply to this

Re: In The News

My First-off thought was that distributivity is a tricky thing, because I’ve become rather used to a rather fancy sort of distributivity, where pullbacks of a colimit diagram $D$ along a map to $D$ is again a colimit diagram; the dual thing tends not to be true, and things that don’t dualize have usually taken me a lot of acclimatization.

But, then, recalled (what I think you’re aluding to) that the statement of distributivity for rings (and the ring action part of a module) is that multiplication is a $+$-*homomorphism*; and further that there is a good reason for this in the case of $\mathbb{Z}$ having to do with $\mathbb{Z}$ representing the underlying set functor in Abelian Groups… I’ve no hypothesis either way whether teaching this earlier will be generally helpful, except that different students learn differently, so having as many different ways to become familiar with something can’t really hurt; expecting nonspecialists to teach them all correctly, on the other hand… Generally, the trouble I find with distributivity is that students learning calculus want it to apply *all the time*; but the set of solutions to the equation $sin(x+y) = sin(x) + sin(y)$ is remarkably thinner than they’d like. It’s not that they don’t know of distributivity as a thing, it’s that they’ve forgotten it’s rather special.

I expect the simplest actually category-theoretic thing that grade-school students deal with is the notion of greatest common divisor; the thing I certainly didn’t learn about gcd until university Algebra is that the gcd is also the least linear combination… But to clarify, the relation “evenly divides” makes $\mathbb{Z}$ a poset in an interesting way, such that gcd is exactly $lim$ (or maybe $colim$, depending on orientation):

$gcd(a,b)$ is a thing that both divides both $a,b$ and is divisible by everything that divides both $a,b$.

The other examples I tend to hear about are actually examples of teaching a decategorified thing in grade school (natural numbers come from finite sets!), and later on motivating a categorical something by pointing out that it categorifies something you learned in grade-school.

Posted by: Jesse C. McKeown on May 27, 2013 4:24 AM | Permalink | Reply to this

Applications

Does anyone know what she’s talking about in this paragraph?

Category theory’s spread has continued. Many results in quantum information theory turn out to follow directly from category theory. Category theory’s hierarchical structure has made it useful for modeling complex biological systems. Category theoretic models of language have outperformed conventional ones in distinguishing, for example, the meaning of “saw” in sentences like “I saw a man with a saw.” It’s even proving valuable in developing rigorous models of music theory.

Posted by: Sebastien on June 5, 2013 7:34 PM | Permalink | Reply to this

Re: Applications

See for instance page 8 of this and this scan of a previous NS article feature this work.

Posted by: David Roberts on June 6, 2013 1:06 AM | Permalink | Reply to this

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