# Most general form of Cayley's theorem?

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?

• I'd have to check, but the Yoneda embedding is probably it. It definitely covers many of those at least. Commented Mar 2, 2016 at 0:50
• The (enriched) Yoneda lemma. Commented Mar 2, 2016 at 1:34
• 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? Commented Mar 2, 2016 at 4:51
• I asked a very similar question on mathoverflow a while back: mathoverflow.net/questions/136832/… Commented Mar 2, 2016 at 19:54
• @QiaochuYuan: I wouldn't say the Yoneda lemma generalizes Cayley's theorem, but rather the Yoneda embedding. Commented Mar 4, 2016 at 6:19

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.