# Examples in the vein of smooth manifold + group = Lie group [closed]

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?

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## closed as off topic by Andy Putman, Benoît Kloeckner, Marc Palm, Igor Rivin, Gjergji ZaimiApr 25 '12 at 14:13

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I think this question is too vague, so I voted to close. – Andy Putman Apr 24 '12 at 16:03
You could produce a long list of instances where algebraic and topological structure are combined: topological/normed vector spaces, operator semigroups, C*-algebras etc. – Michael Renardy Apr 24 '12 at 16:12
Every locally compact group contains an open closed subgroup, which is a projective limit of Lie groups. – Marc Palm Apr 24 '12 at 16:29
Look up the phrase Lawvere theory. It won't tell you interesting things a bout how the algebraic structure interacts with the ambient structure, but it is a very general version of what you are talking about. That said, I also think this question is too vague. – David Roberts Apr 24 '12 at 21:02
Ordered fields is an example of this nature. To quote wikipedea "In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. In 1926, this grew eventually into the Artin–Schreier theory of ordered fields and formally real fields." I am not sure there are many definite examples of this kind so this can be a useful question in spite of being vague. – Gil Kalai Apr 25 '12 at 14:22