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Does Shelah's categoricity conjecture for abstract elementary classes have applications in other branches of mathematics?

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I’m not sure if it counts as “applications in other branches of mathematics”.

The other branch could be category theory. The conjecture was translated in the category-theoretic framework:

Beke, Tibor, and Jirí Rosický. "Abstract elementary classes and accessible categories." Annals of Pure and Applied Logic (2012) (arxiv link)

The above work can be seen as a part of research in categorical model theory which was also mentioned on the MO before (see here and here).

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