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?

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    $\begingroup$ I'd have to check, but the Yoneda embedding is probably it. It definitely covers many of those at least. $\endgroup$ – Derek Elkins left SE Mar 2 '16 at 0:50
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    $\begingroup$ The (enriched) Yoneda lemma. $\endgroup$ – Qiaochu Yuan Mar 2 '16 at 1:34
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    $\begingroup$ 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? $\endgroup$ – Vik78 Mar 2 '16 at 4:51
  • $\begingroup$ Are you asking for a unified statement of these embedding results or are you also asking for a unified proof? $\endgroup$ – J.-E. Pin Mar 2 '16 at 11:31
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    $\begingroup$ I asked a very similar question on mathoverflow a while back: mathoverflow.net/questions/136832/… $\endgroup$ – Alex Kruckman Mar 2 '16 at 19:54

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