Cohomology and Extensions
Given a group and a subgroup , then their coset is the quotient .
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
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
Similarly there is a coset on the right .
For Lie groups and Klein geometry
If is an inclusion of Lie groups then the quotient is also called a Klein geometry.
More generally, given an (∞,1)-topos and a homomorphism of ∞-group ojects , hence equivalently a morphism of their deloopings , then the homotopy quotient is given by the homotopy fiber of this map
See at ∞-action for more on this definition. See at [[higher Klein geometry] and higher Cartan geometry for the corresponding concepts of higher geometry.
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