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English:

I am doing a paper on Yoneda's lemma and its corollaries, for example the corollary to see when two objects in a category are isomorphic. I want to know if there are other applications of this Lemma in categories or theorems.

Thank you very much.

Spanish:

Estoy haciendo un trabajo sobre el lemma de Yoneda y sus corolarios, por ejemplo el corolario para ver cuando dos objetos en una categoría son isomorfos. Quiero saber si hay otras aplicaciones de este Lemma en categorías o en teoremas.

Muchas gracias

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  • $\begingroup$ The calculation of the units and counits of adjunctions from the hom-isomorphism presentation of an adjunction is essentially an application of the Yoneda lemma. It's used to show that the "presheaf of sieves" is a subobject classifier in a presheaf category. Just as two quick examples. $\endgroup$ – Malice Vidrine Sep 18 '18 at 20:41
  • $\begingroup$ This might be a tough post to get a single answer on, though, because the Yoneda lemma comes up so often, but not always in particularly interesting sounding ways. $\endgroup$ – Malice Vidrine Sep 18 '18 at 20:45
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    $\begingroup$ To expand on Malice Vidrine's comment, it is a very common and effective tactic to prove an isomorphism of objects by embedding in the category of presheaves, performing some manipulations to get an isomorphism of representables, and then using Yoneda. From this, your question becomes a bit like asking for applications of the commutativity of addition... $\endgroup$ – Derek Elkins Sep 18 '18 at 21:07
  • $\begingroup$ These might be helpful : math.stackexchange.com/questions/95712/…, mathoverflow.net/questions/3184/… $\endgroup$ – Arnaud D. Sep 19 '18 at 9:04

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