Cohomology and Extensions
Internal to a general category
In a category , for a group object and a subgroup object, the left/right object of cosets is the object of orbits of under left/right multiplication by .
Explicitly, the left coset space coequalizes the parallel morphisms
where is (the inclusion composed with) the group multiplication.
Simiarly, the right coset space coequalizes the parallel morphisms
Internal to Set
Specializing the above definition to the case where is the well-pointed topos , given an element of , its orbit is an element of and is called a left coset.
Using comprehension, we can write
Similar situation on the right.
The coset inherits the structure of a group if is a normal subgroup.
Unless is abelian, considering both left and right coset spaces provide different information.
The natural projection , mapping the element to the element , realizes as an -principal bundle over . We therefore have a homotopy pullback
where is the delooping groupoid of . By the pasting law for homotopy pullbacks then we get the homotopy pullback
Revised on October 30, 2013 23:24:47
by Colin Tan