coset

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

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

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

Similar situation on the right.

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 October 30, 2013 23:24:47
by Colin Tan
(137.132.250.13)