# Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually:

Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion.

Let me call it Dedekind's lemma. Next to Cayley's theorem:

Every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$.

it is a prominent special case of Yoneda's lemma.

All proofs are constructive, and so is the proof of Erdős' lemma on intersection graphs:

Every graph $G$ is isomorphic to a family of subsets $S_i$ of a set S such that $v_i$ and $v_j$ are joined by an edge iff $S_i \cap S_j \neq \emptyset$

What I wonder about is:

Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

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Well, posets and groups are both special cases of categories, but graphs aren't, so I'm not sure what category you'd be applying Yoneda to. –  Qiaochu Yuan Feb 7 '12 at 18:23
Note that if Erdös Lemma isn't a special case of Yoneda's Lemma (which I suspect it isn't, for reasons that Qiaochu mentions), this will be very hard to prove - it will require some Gödel-like effort on a meta-level. –  dominiczypen Sep 2 at 10:22