Skip to main content
MathOverflow

Return to Answer

added a link
Source Link
Ronnie Brown
Ronnie Brown
  • 12.3k
  • 1
  • 63
  • 81

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.

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.

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.

Source Link
Ronnie Brown
Ronnie Brown
  • 12.3k
  • 1
  • 63
  • 81

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.