group theory

# Contents

## Definition

### Internal to a general category

In a category $C$, for $G$ a group object and $H↪G$ 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

$H×G\underset{\mu }{\overset{{\mathrm{proj}}_{G}}{⇉}}G$H \times G \underoverset{\mu}{proj_G}\rightrightarrows G

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

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

$G×H\underset{{\mathrm{proj}}_{G}}{\overset{\mu }{⇉}}G$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 $\mathrm{Set}$, given an element $g$ of $G$, its orbit $\mathrm{gH}$ is an element of $G/H$ and is called a left coset.

Using comprehension, we can write

$G/H=\left\{gH\mid g\in G\right\}$G/H = \{g H | g \in G\}

Similar situation on the right.

## 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 $gH$, realizes $G$ as an $H$-principal bundle over $G/H$. We therefore have a homotopy pullback

$\begin{array}{ccc}G& \to & *\\ ↓& & ↓\\ G/H& \to & BH\end{array}$\array{ G & \to&* \\ \downarrow && \downarrow \\ G/H &\to& \mathbf{B}H }

where $BH$ is the delooping groupoid of $H$. By the pasting law for homotopy pullbacks then we get the homotopy pullback

$\begin{array}{ccc}G/H& \to & BH\\ ↓& & ↓\\ *& \to & BG\end{array}$\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)