group theory

# Contents

## Idea

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

## Definition

### Internal to a general category

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

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

### Internal to $Set$

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 $H \backslash G$.

### For Lie groups and Klein geometry

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

### For $\infty$-groups

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.

## Properties

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 June 24, 2015 15:49:17 by Urs Schreiber (195.113.30.252)