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Random early-morning thought: are there any "classification theorems" in category theory? Theorems to the effect that any object of type XYZ has either form A, B, or C? Eg the classification of finite simple groups
Maybe these don't exist because category theory can only prove neat results 🤔
I guess any equivalence of categories is like this, but I'm wondering whether we have more sort of "case analysis"-type results. Not "An X is the same as a Y" but "An X is either a Y or a Z".