Note the word "brief" here. This request is easily fulfilled for fields ranging from linguistics to algebraic geometry to general relativity to molecular biology. Curious if there is such a result for category theory.
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Most scientific & mathematical disciplines I know of have results that educated outsiders can appreciate and go "wow" after a brief acquaintance, even without understanding the details. Does anyone know of such a result for category theory?
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Eg general relativity: gravitational lensing, the existence of black holes, the fact the planets are really going in "straight lines" (geodesics) around the sun, in the right geometry... each has a great 2-minute explanation. (And dozens of other phenomena, too.)
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The one thing I know of that sort of fits - the definition of the tensor product of modules in terms of a universal mapping principle is clearly inspired by category-theoretic ideas (and is very nice). But not a big "wow", & needs quite a bit of background to be "brief"
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An example, from general relativity: "If you're on a spaceship, and carry a really good clock down near to the event horizon of a black hole, and come back, you'll find that time has slowed down relative to people who didn't go on the trip". Dwell on that, & it's mindblowing.
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Or an example from complex analysis: suppose you have a function that's analytic everywhere in the complex plane. Than integrate it round a closed curve in the complex plane, and you'll find that you get 0. Again, it's mindblowing!
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For the last, you need to know what analytic means: (f(z+epsilon)-f(z))/epsilon approaches a constant limit as epsilon approaches 0, everywhere. Intuitively, the key thing going beyond real differentiability is that epsilon can approach 0 from any direction in the complex plane
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I've been searching for such a thing for a long time, and it's nice to see the responses here. But I'd ask for something similar for Group Theory. What is your "Brief, nice thing that makes people go Wow!" for Group Theory?
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I'm not a group theorist, nor a mathematician, so I'm the wrong person to ask. I do find the Solovay-Kitaev thm wonderful: informally, if you take products of elements in (many) compact Lie groups, they fill in the group exponentially quickly, & surprisingly near to uniformly
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A picture is better to explain what this means. And probably best explained concretely using, e.g., rotations and a few similar examples.
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I don't quite understand it well enough to make the attempt, but I'll bet there's a mindblowing 5-minute explanation of what Gromov's theorem on groups of polynomial growth says.
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Hmm. I'll have to think about that. I certainly don't know enough to be able to see immediately how or why that's interesting, so I'd have to dig into it. One result I did use to impress some 16 year olds was that in a group of even order there is an element a with a^2 = 1.
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That's in a completely different league, but it served to show them something that without group theory wasn't obvious, but with group theory was completely obvious. That hinted at the power of the topic, but took about 2 minutes from nothing.
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The things you're talking about would not be accessible to the youngsters I deal with, so I'm looking for something they wouldn't be able to show, but which becomes obvious with just basic group theory.
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It's not quite that level, but Fermat's little theorem of course follows from Lagrange's theorem. I'll bet you could do an amazing elementary video that was Euclid's algorithm, Fermat's little theorem, Lagrange's theorem, and Pratt's theorem (!), all in about 15 minutes!
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The Yoneda lemma is the obvious answer I think
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What's the brief explanation of why I should care?
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It’s a precise way to say “anything is perfectly determined by its relationship to everything else”
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Can you say why that's interesting?
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You know a mathematical object by the company it keeps. The yoneda lemma essentially means an object is what it does: See here for better explanation:https://www.math3ma.com/blog/the-yoneda-perspective …
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That and the followup post (with more details) look very interesting!
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