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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?

user557
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MJD
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1 Answers1

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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.

Qiaochu Yuan
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    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 06:30
  • Would you mind elaborating (here or in you blog) a bit more how went from set-theoretic presheaves to 2-valued presheaves? Did you embed 2 into $\operatorname{Set}$? I understand that once you realise that a presheaf can be identified with a downward closed set, and a representable presheaf corresponds to $D_y$, that you get the result - but I don't understand how the presheaves look like originally, meaning before we consider them as functors from $P^{\operatorname{op}}$ to $2$. – Sha Vuklia Nov 16 '20 at 12:53
  • @Sha: yes, $2$ embeds into $\text{Set}$ as the full subcategory on ${ 0, 1 }$ (where $0$ is the initial object / the empty set and $1$ is the terminal object / the one-element set). The Yoneda embedding of a preorder canonically takes values in this full subcategory. – Qiaochu Yuan Nov 16 '20 at 18:02
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    Right, thanks! I got a bit confused when you worked with $2$, so I wrote it out in the 'general notation', and now I see it; the point is that each hom-set is either empty or not (consisting of one element). – Sha Vuklia Nov 16 '20 at 20:49