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I'm not an expert. I'm just as confused as you are about that.

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It depends what you mean by "successful". Will it ever be as powerful as our set theories? Answer: yes, it already is. Will it ever be accepted socially as a superior alternative to set theory? Lawls, no way.

Really, category theory *does* have sets. It just call them (and everything else) by a different name. The analog for a set is essentially the object. Morphisms are analogous to functions. While functions map members of sets to member of other sets, morphisms map members of objects to other objects.

The big difference is that CT almost *never* talks about the "members of an object". They don't value the members. Instead, they focus their energy on finding relations between morphisms, especially when chained together through function composition.

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Hurkyl

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My view of the future isn't quite so pessimistic. :tongue:Will it ever be accepted socially as a superior alternative to set theory? Lawls, no way.

A set is a member of a set-theoretic universe. Your statement is only true if we're studying some set-theoretic universe and the category happens to be a subcategory of it.Really, category theory *does* have sets. ... morphisms map members of objects to other objects.

Yes, it's true that category theory has

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For the unillustrated :) , could you show an example? In which realms category theory do provide a better start point than set theory or other tradition?But the set-theoretic approach of trying to describe everything as a 'set with structure' is often misleading, awkward, or simply doesn't work.

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Hurkyl

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Wikipedia also gives two examples of categories whose objects can't be viewed as sets with structure.

I should state a caveat, though -- if you add a 'large cardinal axiom' to your set theory, then while hTop cannot be viewed as 'small sets' with structure, you can represent them as 'large sets' with structure. However, the general recipe for such a construction is wholly unenlightening -- it's simply a restatement of the category structure. (It's directly analogous to the fact that any group can act on itself)

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