> "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 motivatio

> "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.