TITLE: Shelah's categoricity conjecture
QUESTION [7 upvotes]: Does Shelah's categoricity conjecture for abstract elementary classes have applications in other branches of mathematics?

REPLY [3 votes]: 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).