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.