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?



Examples in the vein of smooth manifold + group = Lie group
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](http://www.groupoids.org.uk/gpdsweb.html). 


