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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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  • $\begingroup$ I think this question is too vague, so I voted to close. $\endgroup$
    – Andy Putman
    Commented Apr 24, 2012 at 16:03
  • $\begingroup$ You could produce a long list of instances where algebraic and topological structure are combined: topological/normed vector spaces, operator semigroups, C*-algebras etc. $\endgroup$
    – Michael Renardy
    Commented Apr 24, 2012 at 16:12
  • $\begingroup$ Every locally compact group contains an open closed subgroup, which is a projective limit of Lie groups. $\endgroup$
    – Marc Palm
    Commented Apr 24, 2012 at 16:29
  • $\begingroup$ 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. $\endgroup$
    – David Roberts ♦
    Commented Apr 24, 2012 at 21:02
  • $\begingroup$ 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. $\endgroup$
    – Gil Kalai
    Commented Apr 25, 2012 at 14:22

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You could look up the interaction of groupoids and smooth structures, for example in

Pradines, J. In Ehresmann's footsteps: from group geometries to groupoid geometries. (English summary) Geometry and topology of manifolds, 87 - 157, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007. (arXiv:0711.1608)

There is a lot of literature on Lie groupoids. Noncommutative geometry uses measured groupoids, which arose in work of Mackey on what came to be called ergodic groupoids.

In fact there is a lot of literature on structured groupoids, usually thought of as groupoids internal to a category. These are often more interesting than group objects internal to a category- thus group objects in the category of groups are just abelian groups, but groupoid objects in the category of groups are equivalent to crossed modules, which are thought of as 2-dimensional groups.

One reason for this interest is that groupoids generalise equivalence relations, which are related to quotients-- and quotienting is part of the "bigger picture" in mathematics.

See also this web page on Groupoids in Mathematics.

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