Given two structures (in the model theory sense), I believe the existence of an isomorphism between them implies that they are elementarily equivalent. The implication of elementary equivalence is what I understand to be the most important property of isomorphisms, so when I'm attempting to understand natural isomorphisms from category theory, I'm looking for a similar implication.

Does a natural isomorphism between functors imply that the functors satisfy the same set of first order sentences?

Does my understanding of the importance of isomorphisms between structures make sense?

If natural isomorphisms have nothing to do with anything like elementary equivalence, then what makes them important?

  • $\begingroup$ First order sentences in what language? $\endgroup$ – Eric Wofsey Aug 8 '18 at 21:54
  • $\begingroup$ I was assuming that there was a standard language where facts about functors etc were expressed, so I guess I have to say I don't know. $\endgroup$ – etha7 Aug 8 '18 at 22:06
  • $\begingroup$ Well, that language is some form of set theory, but obviously the point of isomorphism is that it gives a notion of sameness a bit more refined than set-theoretic sameness. But functors are not (generally) models of first order theories, so the same reasoning doesn't really go through. $\endgroup$ – Malice Vidrine Aug 8 '18 at 22:11
  • $\begingroup$ If not elementary equivalence or something similar, then what notion of sameness do natural transformations provide? $\endgroup$ – etha7 Aug 8 '18 at 22:45
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    $\begingroup$ Elementary equivalence is a fairly involved concept that requires a lot of structure to even be defined, namely a notion of semantics for classical first-order logic. There are many concepts (e.g. basically all of algebra, or, relevantly, the theory of functors) that can be defined in contexts that don't support a (meaningful) semantics for first-order logic, classical or otherwise. This all to say that it is bizarre and limiting to demand that every notion of equivalence corresponds to elementary equivalence for some theory. $\endgroup$ – Derek Elkins Aug 8 '18 at 22:54

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