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I think this is a good point. I find the language of category theory an invaluable tool in my own understanding of abstract algebra, and universal properties ar
by joppy 8y ago
I think this is a good point. I find the language of category theory an invaluable tool in my own understanding of abstract algebra, and universal properties are a large part of that. I think that a lot of undergraduate mathematicians could benefit from seeing the universal properties of a product/coproduct, and reflecting on what this means in categories that they're comfortable working in. Universal properties are usually quite useful.
My specific gripe is with expecting that a statement like "a Lie group is a group object in the category of manifolds" is going to lead to any new theorems in the theory of Lie groups. It's just not, and you are not going to become any better at Lie groups by understanding that statement. It's good to understand at some point, because it unifies the definition of group / Lie group / algebraic group, etc, but it will really not give you anything new to work with.
- danharaj 8y ago> It's good to understand at some point, because it unifies the definition of group / Lie group / algebraic group, etc, but it will really not give you anything new to work with. I think that's true at an elementary level, but on the other hand when one doesn't know what concepts they're looking for in a new area, category theory can be a machete cutting a swathe through the jungle. I think Grothendieck was right :)
- zozbot123 8y ago> "a Lie group is a group object in the category of manifolds" The point is of course to motivate the definition of a Lie group. Whether such strong motivation counts as "a better understanding of Lie groups" is largely a matter of semantics that there's some room to disagree over. Same goes for the categorical definition of a ring that's been mentioned at the start of this thread, or indeed the definition that OP talks about.