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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.
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.)
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"
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.
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!
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
Relevant Chomsky quote: https://www.stephenhicks.org/2013/11/27/chomsky-on-postmodernism/ …
Yes, I think often of the start of that Chomsky note (it's attributed, AFAIK) when someone waxes poetic about category theory. But see my more positive remarks elsewhere in this thread.
Yeah, sorry, I don't think there is one. Category theory is some serious math major stuff, and even most math major-track people never see much of it.
You know in topology tea cups and donuts are equivalent? You can morph from one the other smoothly without tearing or creasing, like shaping clay or a rubber sheet. The rules about creasing come from topological smoothness ideas...
More generally, instead of shapes or surfaces you can handle arbitrary mathematical objects and their morphisms. Whether objects can morph to each other is determined by the rules of those objects, like how the smoothness ideas lead to rules about creasing in topology.
The Haskell functional programming language is a triumph of category theory.
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