I think it is much clearer to say that a functor is representable if it is naturally isomorphic to Hom(A,-) for some A; which is why I rewrote it that way. I think the way it is written now obscures this fact, and makes the notion seem more complicated that it really is. The fact that every functor can be represented by a pair (A,φ) is important, but secondary. -- Fropuff 14:59, 2004 Jul 28 (UTC)
- My feeling was only that the natural transformation should be specified as given, because the functors have to be naturally isomorphic in that way. Isn't that right? If not, then your edits are how the page should be. If so, maybe something like '... naturally isomorphic to Hom(A,-) for some A. This natural isomorphism should be...'
- If it was like that already, I apologize. I can't remember. -- mat_x 21:17, 28 Jul 2004 (UTC)
Every natural isomorphism will necessarily be of the stated form. To see this let Φ : Hom(A,-) → F be any natural isomorphism. Define φ ∈ F(A) by φ = ΦA(idA). Then for any morphism u : A → X one can show that ΦX(u) = (Fu)(φ) by following idA around the commutative square induced by Hom(A,u) and ΦA. -- Fropuff 23:04, 2004 Jul 28 (UTC)
- OK. Reminds me of Yoneda's lemma. mat_x 08:00, 29 Jul 2004 (UTC)
Ha. You are right of course. The argument I gave above is just one version of Yoneda's lemma. I think I finally understand that lemma. Perhaps something like that should go in the article too. Fropuff 00:21, 2004 Jul 30 (UTC)
- Yes, I guess they are pretty intertwined. I think we both have the same idea about how this article should go. I'm happy to let you make the changes above. mat_x 10:50, 30 Jul 2004 (UTC)