coset

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

Given a group $G$ and a subgroup $H$, then their *coset* is the quotient $G/H$.

In a category $C$, for $G$ a group object and $H \hookrightarrow G$ a subgroup object, the left/right *object of cosets* is the object of orbits of $G$ under left/right multiplication by $H$.

Explicitly, the left coset space $G/H$ coequalizes the parallel morphisms

$H \times G \underoverset{\mu}{proj_G}\rightrightarrows G$

where $\mu$ is (the inclusion $H\times G \hookrightarrow G\times G$ composed with) the group multiplication.

Simiarly, the right coset space $H\backslash G$ coequalizes the parallel morphisms

$G \times H \underoverset{proj_G}{\mu}\rightrightarrows G$

Specializing the above definition to the case where $C$ is the well-pointed topos $Set$, given an element $g$ of $G$, its orbit $gH$ is an element of $G/H$ and is called a *left coset*.

Using comprehension, we can write

$G/H = \{g H | g \in G\}$

Similarly there is a coset on the right $H \backslash G$.

If $H \hookrightarrow G$ is an inclusion of Lie groups then the quotient $G/H$ is also called a *Klein geometry*.

More generally, given an (∞,1)-topos $\mathbf{H}$ and a homomorphism of ∞-group ojects $H \to G$, hence equivalently a morphism of their deloopings $\mathbf{B}H \to \mathbf{B}G$, then the homotopy quotient $G/H$ is given by the homotopy fiber of this map

$\array{
G/H &\longrightarrow& \mathbf{B}H
\\
&& \downarrow
\\
&& \mathbf{B}G
}
\,.$

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 $H$ is a normal subgroup.

Unless $G$ is abelian, considering both left and right coset spaces provide different information.

The natural projection $G\to G/H$, mapping the element $g$ to the element $g H$, realizes $G$ as an $H$-principal bundle over $G/H$. We therefore have a homotopy pullback

$\array{
G & \to&*
\\
\downarrow && \downarrow
\\
G/H &\to& \mathbf{B}H
}$

where $\mathbf{B}H$ is the delooping groupoid of $H$. By the pasting law for homotopy pullbacks then we get the homotopy pullback

$\array{
G/H & \to&\mathbf{B}H
\\
\downarrow && \downarrow
\\
* &\to& \mathbf{B}G
}$

Revised on December 17, 2014 21:30:04
by Urs Schreiber
(127.0.0.1)