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For some context, Fermat's last theorem has a beautiful statement, which can be explained to a high schooler, it appears naturally when teaching the pythagorean theorem and just a bit of generalization gets you all the way to its statement. Its proof uses connections between various parts of modern mathematics.

What about a theorem that goes even further, its statement is arithmetic, can be naturally motivated at a high school level and its proof goes into the realm of $\infty$-categories. Are there such statements? Maybe the answer is much simpler and the statement is Fermat's last theorem as I am not an expert.

For context, my experience with category theory comes from formal verification of software and modularity issues therein. While one might argue this can be motivated to a high schooler, the issues come in at a different scale. I would argue that these issues would show up in a well-designed computer science class for high schoolers, but that the objects in any concrete theorems directly depending on them will be much harder to motivate at this level without a course in rigorous formal logic or some amount of software engineering practice. Compared to these, in Fermat's last theorem all the objects and operations in the statement come without going beyond the standard curriculum.

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    $\begingroup$ I removed my upvote after the most recent edit; I think it is a big but reasonable ask for a theorem explainable to high schoolers that uses category theory in an essential way, but asking for $\infty$-categories to be used is absurd imo. $\endgroup$
    – Alec Rhea
    Commented Apr 27, 2022 at 1:10
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    $\begingroup$ @AlecRhea I don't think this is absurd. Just asking for category theory is a trivial question - Fermat is already an example of this unless you want to be absurdly stringent about "essential". For infinity-categories, I don't think there exists an example now but probably there will be in 50 years. $\endgroup$
    – Will Sawin
    Commented Apr 27, 2022 at 1:13
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    $\begingroup$ I'd say that $(\infty, n)$-category theory is still pretty young, and you shouldn't expect it as of today to solve problems explainable to high-school students that are beyond the reach of old established fields like algebraic number theory and algebraic geometry. Let's be clear not to mythologize higher category theory as blowing other powerful fields out of the water, at least not yet. $\endgroup$
    – Todd Trimble
    Commented Apr 27, 2022 at 1:25
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    $\begingroup$ Infinity categories show up in Gaitsgory and Lurie's proof that the Tamagawa number of a simply connected simple algebraic group over a function field over a finite field is equal to 1. This is still quite far from a statement that you could explain to a high schooler, but for all I know it could imply something nontrivial about some elementary diophantine equation. It's sort of an accident that Fermat's last theorem is connected to elliptic curves and modular forms, and if there's an answer to your question then it's probably also an accident of that sort. $\endgroup$
    – Paul Siegel
    Commented Apr 27, 2022 at 1:29
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    $\begingroup$ @AlecRhea that's correct, theorem that's easily stated as a teaser for motivated students. I found that motivations for various fields, i.e. insolubility of quintics -> galois theory, which was used to play around with groups as an abstraction, helped me with motivation to go further into maths. So in this case it is look we can take processes of generalization very far, it leads to interesting results. Showing the process of generalization by looking at adding arrows between arrows and saying we can keep doing this and it will lead to something we can already talk about. But now I am unsure. $\endgroup$
    – Ilk
    Commented Apr 27, 2022 at 2:59

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