nLab
coset

Contents

Idea

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

Definition

Internal to a general category

In a category CC, for GG a group object and HGH \hookrightarrow G a subgroup object, the left/right object of cosets is the object of orbits of GG under left/right multiplication by HH.

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

H×Gμproj GG H \times G \underoverset{\mu}{proj_G}\rightrightarrows G

where μ\mu is (the inclusion H×GG×GH\times G \hookrightarrow G\times G composed with) the group multiplication.

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

G×Hproj GμG G \times H \underoverset{proj_G}{\mu}\rightrightarrows G

Internal to SetSet

Specializing the above definition to the case where CC is the well-pointed topos SetSet, given an element gg of GG, its orbit gHgH is an element of G/HG/H and is called a left coset.

Using comprehension, we can write

G/H={gH|gG} G/H = \{g H | g \in G\}

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

For Lie groups and Klein geometry

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

For \infty-groups

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

G/H BH BG. \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 HH is a normal subgroup.

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

The natural projection GG/HG\to G/H, mapping the element gg to the element gHg H, realizes GG as an HH-principal bundle over G/HG/H. We therefore have a homotopy pullback

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

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

G/H BH * BG \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)