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For many classes of algebraic structures, there exists a family of structures such that any member of the class can be embedded in some member of the family (groups and symmetric groups, unital rings and rings of endomorphisms of abelian groups, distributive lattices and sets under union and intersection). Is there some construction from category theory or universal algebra that counts all of these examples as special cases?

Vik78
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    I'd have to check, but the Yoneda embedding is probably it. It definitely covers many of those at least. – Derek Elkins left SE Mar 02 '16 at 00:50
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    The (enriched) Yoneda lemma. – Qiaochu Yuan Mar 02 '16 at 01:34
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    Thanks for the responses. Is there any way a person who only knows the basic definitions of category theory (category, morphism, functor) can understand the Yoneda embedding in any meaningful way? I can just barely parse the wikipedia article stating the Yoneda lemma. I know it's a fundamental result in category theory-- what are some of its applications, beyond generalizing Cayley's theorem? – Vik78 Mar 02 '16 at 04:51
  • Are you asking for a unified statement of these embedding results or are you also asking for a unified proof? – J.-E. Pin Mar 02 '16 at 11:31
  • I won't understand a proof, so I would just appreciate an informal explanation of what the Yoneda lemma means, if it's at all possible for someone who knows some algebra but hasn't studied category theory to understand. – Vik78 Mar 02 '16 at 14:09
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    I asked a very similar question on mathoverflow a while back: http://mathoverflow.net/questions/136832/is-there-a-general-theory-of-representation-theorems – Alex Kruckman Mar 02 '16 at 19:54
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    @QiaochuYuan: I wouldn't say the Yoneda lemma generalizes Cayley's theorem, but rather the Yoneda embedding. – Omar Antolín-Camarena Mar 04 '16 at 06:19
  • Does this answer your question? [Yoneda-Lemma as generalization of Cayley`s theorem?](https://math.stackexchange.com/questions/1701/yoneda-lemma-as-generalization-of-cayleys-theorem) – user557 Mar 27 '21 at 02:34

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Just as Cayley's theorem tells us that any group can be embedded in the symmetric group $\operatorname{Sym}G$, the Yoneda lemma of category theory says that any category $D$ can be embedded into a category of functors.

It turns out Cayley's theorem can be viewed as a special case of the Yoneda lemma. There is a way to view a group as a category, and in this context the Yoneda embedding can be seen to be the Cayley embedding.