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 lik