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In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof:

If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma.

I understand that a poset can be considered to be a category with at most one arrow between any two objects, and I understand the statement of the Yoneda lemma, although I have little experience in using it. But I do not understand this proof. How does the Yoneda Lemma help?

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The [silly] tag is... silly. – Asaf Karagila May 24 '12 at 20:59
For some related examples in posts here, see the floor function and gcds and lcms. – Bill Dubuque May 24 '12 at 21:23

This is explained in my blog post on the Yoneda lemma.

By the way, I do not consider this argument "nuking mosquitos." The Yoneda lemma is hardly a nuke; I would reserve that term for a highly technical result which requires a long proof. The proof of the Yoneda lemma is extraordinarily short and elegant. Besides, even this seemingly trivial special case can be surprisingly useful.

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+1 I hadn't known about the blog before this. – Mark Bennet May 24 '12 at 21:15
The nuke isn’t specifically the Yoneda lemma: it’s employing the wholly unnecessary language and machinery of category theory to prove a trivial result that follows immediately from the definition (each is a lower bound of the other, and the order is antisymmetric). Possibly it’s useful to see this fact as an instance of something more general $-$ I’m agnostic on that point $-$ and perhaps the machinery is something less than a nuke, but it’s certainly using a bulldozer to move a chickpea. – Brian M. Scott May 25 '12 at 6:30

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