I am currently writing a thesis and got to thinking about the bigger picture of mathematics in the following sense. Both manifolds and groups have highly developed theories in their own rights. When combined, in the appropriate way, we arrive at the theory of Lie groups. This theory is more than just the sum of its parts and there are many interesting interpretations of the algebra in terms of geometry, and vice-versa.
I wanted more examples of this sort of phenomenon in mathematics. Not all groups and not all manifolds are Lie groups, and I would like to find examples of this specific situation.
For example: in algebraic geometry we associate a geometric object to a commutative ring. We obtain insights into the geometry from the algebraic theory and also vice versa. This is not the same situation, as the construction works for any commutative ring.
Is category theory the way of thinking about this? A Lie group is a group object in the category of smooth manifolds. If this is the go to source for examples like this where can I find a list of examples of A-objects in the category B?